Quantum Gibbs Sampling Citation Chen, Chi-Fang (2025) Quantum Gibbs Sampling. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/dy0f-3216. https://resolver.caltech.edu/CaltechTHESIS:08152024-132609600 Abstract Markov Chain Monte Carlo algorithms are indispensable in classical thermodynamic simulation, perhaps due to their mathematical simplicity, algorithmic efficiency, and physical origin. In particular, Glauber dynamics is a detailed-balanced continuous-time Markov chain that fixes the Gibbs distribution and also serves as a mathematically succinct model of classical thermalization. In this thesis, we proposed a quantum computation analog of Glauber dynamics that is exactly detailed balanced yet algorithmic efficient, inherits the locality of the target Hamiltonian, and resembles Davies'-like generators physically derived from a weak system-bath coupling. We hope our proposal will serve as a quantum algorithm for quantum thermodynamic simulation and a model of open system thermalization where a suitable construction has been lacking for noncommuting Hamiltonians. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: thermalization, quantum algorithms, quantum simulation, quantum detailed balance, Markov Chain Monte Carlo Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Brandao, Fernando Thesis Committee: Preskill, John P. (chair) Motrunich, Olexei I. Tropp, Joel A. Brandao, Fernando Defense Date: 9 July 2024 Funders: Funding Agency Grant Number Eddleman Graduate Fellowship UNSPECIFIED Record Number: CaltechTHESIS:08152024-132609600 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08152024-132609600 DOI: 10.7907/dy0f-3216 Related URLs: URL URL Type Description https://arxiv.org/abs/2303.18224 arXiv Article adapted for ch.2 https://arxiv.org/abs/2311.09207 arXiv Article adapted for ch.3 https://doi.org/10.1145/3618260.3649675 DOI Article adapted for ch.4 ORCID: Author ORCID Chen, Chi-Fang 0000-0001-5589-7896 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16640 Collection: CaltechTHESIS Deposited By: Chi Fang Chen Deposited On: 21 Aug 2024 22:04 Last Modified: 21 May 2025 22:25 Thesis Files PDF - Final Version See Usage Policy. 4MB Repository Staff Only: item control page

Quantum Gibbs Sampling Thesis by Chi-Fang Chen In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2025 Defended July 9th, 2024 ii © 2025 Chi-Fang Chen ORCID: 0000-0001-5589-7896 All rights reserved iii ACKNOWLEDGEMENTS Graduate school has been full of turns. I am heavily in debt to those who have lent their hands in times of uncertainty and who have fought together in pursuit of truth. I want to especially thank my advisor, Fernando Brandão, for his incredibly fast and brutally honest advice about research and just anything. He always provides critical input at the right moment. He has also put no constraints on my travel and interests, which allows me to be delocalized among many great places and talents. I feel lucky to have been exposed to the ideas about thermalization and Gibbs sampling in this quantum computing wave. Without Fernando, I also wouldn’t have met my fellow Gibbs samplers: András Gilyén, from whom I learned so much about quantum algorithms and mathematical writing, and Michael Kastoryano, from whom I learned about open systems and mixing times. I also want to thank Joel Tropp, who introduced me to the beauty of randomness and matrices and whom I regarded as my underground co-advisor. Without his generosity and openness to crazy ideas, it would not have been possible to bridge ideas from random matrices to quantum information and back. I also feel privileged to work with Joel’s postdoc, Jorge Garza-Vargas, and Ramon van Handel at Princeton, who guided me into the mathematical land of random matrix theory. I want to thank John Preskill for collaborations and many fun lightning discussions. The wisdom in his words always returns to me. I also wanted to thank him for making his group meeting and IQIM such a thrilling, open, and engaging hub filled with great people. I also wanted to express my gratitude to my committee members at Caltech, Olexei Motrunich and Manuel Endres, for many helpful discussions along the way. I realized I hadn’t had a chance to officially thank those who really set me on this path. During my undergraduate years, I feel privileged to have worked with Andrew Lucas, my long-time collaborator, on earlier projects on Lieb-Robinson bounds since junior year. He was so tolerant, truthful, and open to my random questions about career and science, and he was hospitable during my visits to his group at CUBoulder. I also want to thank my undergrad mentors and advisors, Geoffrey Pennington, Grant Salton, Sean Hartnoll, and Patrick Hayden, who perhaps planted a little seed of quantumness. In retrospect, this all began with the Science Class program at Chien Kuo High School in Taiwan with a bunch of crazy smart peers, where we first tasted iv the fun of math and science as we solved the Hydrogen atom wave function from scratch and implemented Fast Fourier transform in C. It was really the vision of our teachers and Prof. Ming-Jui Lin, who created such a stimulating scientific culture. My travels would not have been so much fun and smooth without enthusiastic hosts: Umesh Vasirani and Lin Lin at Simons’ Institute and Berkeley, Aram Harrow and Soonwon Choi at MIT, Anurag Anshu at Harvard, Alex Dalzell, and Sam Mcardle and Mario Berta at AWS, Patrick Hayden and Adam Bouland at Stanford, Minh Tran and Kunal Sharma at IBM, and Nobuyuki Yoshioka at TokyoU, Ying-Jer Gao and Hsi-Sheng Goan at NTU, Kai-Min Chung at Sinica, Han-Shuan Lin and Chung-Yu Mou at NTHU, and Ching-Yi Lai at NCYU. I should also thank Bonnie and Marica for handling much of the logistics and my friends who shared their sofas (for extended periods of time): Pei-Ming Chen, Isaac Yu, Ching-Ting Tsai, Kai-Yuan Cheng, and Vinh Nguyen around the Bay Area; Cyuan-Han Chang, Hau-wei Lin, and Han-Hsin Lin around Caltech; Kai-Feng Chen, and Hsin-Yuan Huang and Chi-Yun Cheng around Boston. Thank you to all my friends. My journey is always filled with random fun stuff; let me just list the activities we did together: board games, video games, bubble tea, hiking, thanksgiving, hotpot, etc. Lastly, I owe much to my family and especially my parents, Tzu-Chih Chen and Yu-Chu Chiu, for their unbounded kindness and forgiveness, which make my life fault-tolerant to this day. v ABSTRACT Markov Chain Monte Carlo algorithms are indispensable in classical thermodynamic simulation, perhaps due to their mathematical simplicity, algorithmic efficiency, and physical origin. In particular, Glauber dynamics is a detailed-balanced continuoustime Markov chain that fixes the Gibbs distribution and also serves as a mathematically succinct model of classical thermalization. In this thesis, we proposed a quantum computation analog of Glauber dynamics that is exactly detailed balanced yet algorithmic efficient, inherits the locality of the target Hamiltonian, and resembles Davies’-like generators physically derived from a weak system-bath coupling. We hope our proposal will serve as a quantum algorithm for quantum thermodynamic simulation and a model of open system thermalization where a suitable construction has been lacking for noncommuting Hamiltonians. vi PUBLISHED CONTENT AND CONTRIBUTIONS The following articles are included (with minor edits) as chapter 2, chapter 3, and chapter 4, in order. [1] Chi-Fang Chen, Michael J Kastoryano, Fernando GSL Brandão, and András Gilyén. Quantum thermal state preparation. arXiv preprint arXiv:2303.18224, 2023. URL https://arxiv.org/abs/2303.18224. CFC participated in the conception of the project, the development of the proofs, and the writing of the manuscript. [In preparation for submission]. [2] Chi-Fang Chen, Michael J Kastoryano, and András Gilyén. An efficient and exact noncommutative quantum gibbs sampler. Short plenary talk, QIP2024. arXiv preprint arXiv:2311.09207., 2023. URL https://arxiv.org/abs/ 2311.09207. CFC conceived of the project and participated in the development of the proofs and the writing of the manuscript. [In preparation for submission]. [3] Chi-Fang Chen, Hsin-Yuan Huang, John Preskill, and Leo Zhou. Local minima in quantum systems. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1323–1330, 2024. URL https://doi.org/10. 1145/3618260.3649675. CFC participated in the conception of the project, the proofs, and the writing of the manuscript, particularly in the analysis of energy gradients. vii TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . vi Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Nature’s cooling process . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Markov Chain Monte Carlo algorithms . . . . . . . . . . . . . . . . 3 1.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter II: Algorithmic primitives for quantum Thermal state preparation . . 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Approximate stationarity of the Gibbs state . . . . . . . . . . . . . . 30 2.3 Quantum algorithms for Gibbs sampling . . . . . . . . . . . . . . . 37 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.5 Appendix:Operator Fourier Transform: properties and error bounds . 61 2.6 Appendix:Nonasymptotic secular approximation . . . . . . . . . . . 64 2.7 Appendix:Proving approximate detailed balance . . . . . . . . . . . 77 2.8 Appendix:Discretization error for Lindbladians and discriminant proxies 89 2.9 Appendix:Implications for Lindbladians from system-bath interaction 101 2.10 Appendix:Spectral bounds and mixing times . . . . . . . . . . . . . 111 2.11 Appendix:Improved incoherent Lindbladian simulation . . . . . . . . 122 2.12 Appendix:Quantum simulated annealing . . . . . . . . . . . . . . . 131 2.13 Appendix:Impossibility of boosted shift-invariant in-place phase estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Chapter III: An efficient and exactly detailed-balanced quantum Gibbs sampler 138 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3.5 Appendix:Deriving time-domain represantations . . . . . . . . . . . 170 3.6 Appendix:Calculating the coherent term for the Metropolis-like weights178 3.7 Appendix:The discriminant gap and area law . . . . . . . . . . . . . 185 3.8 Appendix:Why Gaussians? . . . . . . . . . . . . . . . . . . . . . . 188 3.9 Appendix:Other notions of detailed balance . . . . . . . . . . . . . . 191 Chapter IV: Cooling is universal for quantum computation . . . . . . . . . . 197 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 viii 4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.5 Appendix:Notations and Preliminaries . . . . . . . . . . . . . . . . 210 4.6 Appendix:Local minima in quantum systems . . . . . . . . . . . . . 214 4.7 Appendix:Characterizing local minima under local unitary perturbations224 4.8 Appendix:Characterizing local minima under thermal perturbations . 228 4.9 Appendix:Complexity of finding a local minimum in quantum systems235 4.10 Appendix:Details of thermal Lindbladians . . . . . . . . . . . . . . 241 4.11 A polynomial-time quantum algorithm for finding a local minimum under thermal perturbations (Proof of Theorem 4.9.1) . . . . . . . . 250 4.12 Appendix:Characterizing energy gradients in low-temperature heat bath254 4.13 Appendix:Energy landscape of an Ising chain . . . . . . . . . . . . . 264 4.14 Appendix:All local minima are global in BQP-hard Hamiltonians (Proof of Theorem 4.9.2) . . . . . . . . . . . . . . . . . . . . . . . . 267 4.15 Appendix:Operator Fourier Transform . . . . . . . . . . . . . . . . 286 4.16 Appendix:Proving monotonicity of energy gradient under level splitting290 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 ix LIST OF ILLUSTRATIONS Number Page 2.1 The Metropolis-Hastings algorithm iterates a Markov chain to sample from the Gibbs distribution. Each step begins with a (random) jump: if the energy decreases, accept; if the energy increases, accept with a carefully chosen probability. Otherwise, reject the move. Remarkably, detailed balance can be enforced in a “lazy” manner via rejection sampling without storing the whole matrix. . . . . . . . . . . . . . . 12 2.2 (Up) Davies’ generator gives a continuous-time Markov generator on the energy spectrum (assuming the Hamiltonian is nondegenerate and the input state is diagonal in the energy basis.). The transitions are weighted by 𝛾(𝜔): the heating transitions (red) are suppressed by a Boltzmann factor relative to the cooling transitions (blue), entailing detailed balance. The operator 𝑨𝜈𝑎 contains the transitions with energy difference 𝜈, which requires an infinite-time Fourier Transform. (Down) Our Lindbladian Gibbs sampler can be considered a “semiclassical” random walk where nearby Bohr-frequencies 𝜔 ± O (𝜎𝑡−1 ) cannot be distinguished. The operator Fourier Transform 𝑨ˆ 𝑎 (𝜔) contains a band of transitions. This breaks the detailed balance condition; the fixed point deviates from the Gibbs state. . . . . . . . 14 2.3 Quantum circuit implementation of an approximate 𝛿-time step via a weak measurement scheme. The scheme can be extended to general Lindbladians that include the coherence term −i[𝑯, 𝝆] by applying O (𝛿2 )-precise Hamiltonian time-evolution for time 𝛿 on the system register before the above circuit is applied. For example, one could use Trotterized time-evolution. (In case ∥𝑯∥ > 1, the entire Lindbladian should be first scaled down by a factor of ∥𝑯∥.) . . . . . . . . . . . . 39 x 2.4 Circuit 𝑼 for block-encoding the Lindbladian. Practically, if we use the simpler weak-measurement-based simulation (Theorem 2.3.1), then by Corollary 2.3.1, we can use a single randomly chosen Lindblad operator 𝑨𝑎 at a time. Moreover, if 𝑨𝑎 is unitary, we can simply replace 𝑽 𝑗𝑢𝑚 𝑝 with 𝑨𝑎 , implying 𝑏 = 𝑐 = 0, i.e., the third and the forth registers can be omitted, thus 𝑛 + ⌈log(𝑁)⌉ + 2 qubits suffice to simulate the Lindbladian eL𝑡 . . . . . . . . . . . . . . . . . . . . . . 48 2.5 Circuit for operator Fourier Transform F for an operator 𝑶 acting on the system 𝝆. Of course, in our use, the operator may also act nontrivially on other ancillas. . . . . . . . . . . . . . . . . . . . . . 51 2.6 Circuit 𝑼′† 𝑹𝑼′ for block-encoding the discriminant in the fashion of Szegedy quantum walk. . . . . . . . . . . . . . . . . . . . . . . . . 55 2.7 The circuit for two-sided operator Fourier Transform. The gate 𝒑𝒓𝒆 𝒑 − prepares the flipped function 𝑓− (𝑡) = 𝑓 (−𝑡). Unlike the one-sided version (Figure 2.5), now we cannot understand the expression by Heisenberg evolution; this is, in spirit, more similar to doing two consecutive phase estimations. . . . . . . . . . . . . . . . . . . . . 57 2.8 Left: the weight as the step function. Right: An illustration for Í the Fourier Transformed amplitudes 𝑨𝜔¯ = 𝜈∈𝑆 𝜔 𝑨𝜈 𝑓 ( 𝜔¯ − 𝜈) = 0 Í ¯ −e −i(𝜈− 𝜔)𝑇 ¯ ei(𝜈− 𝜔)𝑇 𝑨 for Bohr frequency 𝜈 given the energy ¯ 0 −1 𝜈∈𝑆 𝜔0 𝜈 ei(𝜈− 𝜔)𝑡 label 𝜔. ¯ The expression coincides with the phase estimation profile. It peaks near energy 𝜈 = 𝜔¯ with a width ∼ 𝑇 −1 and decays polynomially. The profile in absolute value oscillates (blue), but we also display the norm bound to guide the eye (black). The secular approximation truncates the profile at an energy 𝜇¯ far in the tail 𝜇¯ ≫ 𝑇 −1 . . . . . . 65 2.9 Left: the weight function as (approximately) the Gaussian distribution but truncated at 𝑇. Right: An illustration for the Fourier Transformed amplitudes, which is also (approximately) Gaussian. It peaks near energy 𝜈 = 𝜔¯ with a width ∼ 𝑇 −1 and decays exponentially. The secular approximation truncates the profile at an energy 𝜇¯ in the tail 𝜇¯ ∼ 𝑇 −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.10 Alternative quantum circuit implementation of an approximate 𝛿-time step via a postselective weak measurement scheme. Let 𝑪 ′ be the circuit that we get by removing the two single qubit rotation gates † 𝒀𝛿 , 𝒀𝛿/4 from the first qubit. For our compression argument it is of paramount importance that 𝑪 ′ · |0𝑎+2 ⟩⟨0𝑎+2 | ⊗ 𝑰 = |0𝑎+2 ⟩⟨0𝑎+2 | ⊗ 𝑰. . 123 xi 2.11 𝑟 subsequent repetitions of the circuit 𝑪 ′ from Figure 2.10. The circuits 𝑪 ′ act on potentially nonadjacent qubits, which is indicated by the vertical curly connection between the visually split “halves” of the affected 𝑪 ′ circuits. . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.12 Alternative quantum circuit implementation of an approximate 𝛿time step via a postselective weak measurement scheme including the coherence term −𝑖[𝑯, 𝝆] for the block-encoded Hamiltonian 𝑯 = (⟨0𝑐 | ⊗ 𝑰)𝑽 (|0𝑐 ⟩ ⊗ 𝑰). . . . . . . . . . . . . . . . . . . . . . . 129 3.1 (Left) For the classical Gibbs distribution, the detailed balance condition is a pairwise relation between heating (red) and cooling (blue) transition rates, depending on the energy difference 𝜈 of states. (Right) For the quantum Gibbs state, the detailed balance condition refers to pairs of matrix elements of the density operator (expanded in the energy basis), where each matrix element is described by a pair of energies (of the basis elements in the ket and bra respectively), and therefore the relation depends on both of the respective energy differences 𝜈1 and 𝜈2 . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.2 (Left) For lattice Hamiltonians, our Lindbladian is a sum of quasilocal terms L 𝛽𝑎 localized around each jump 𝑨𝑎 with radius Õ (𝛽). Indeed, detailed balance is really about the energy difference, which can be diagnosed by Fourier Transforming the Heisenberg evolution 𝑨𝑎 (𝑡) = ei𝑯𝑡 𝑨𝑎 e−i𝑯𝑡 . Due to the Lieb-Robinson bounds, the localized Lindbladian terms effectively only depend on the local Hamiltonian patch nearby (up to exponentially decaying tail). (Right) This locality persists after purification, where two copies of the system are glued together. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.3 A plot of the filter functions 𝛾(𝜔) for Metropolis, Glauber and our filters arising from Gaussian linear combination(3.16)-(3.17) (with 𝜎𝐸 = 1𝛽 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.4 Circuit for block-encoding (3.3). The gate 𝒑𝒓𝒆 𝒑 is a shorthand for 𝑷𝒓𝒆 𝒑 √| 𝑓 | and 𝒑𝒓𝒆 𝒑′ for 𝑷𝒓𝒆 𝒑 𝑓 /√| 𝑓 | . . . . . . . . . . . . . . . . . 166 + + + xii 4.1 (a) Energy landscape with multiple local minima. Some Hamiltonians, such as magnets in a small external magnetic field, have multiple local minima. One is the ground state while the other local minima are suboptimal. (b) Energy landscape with one local minimum. For some local Hamiltonians, such as a family of BQP-hard Hamiltonians, the energy landscape over the entire 𝑛-qubit state space has a nice bowl shape, and the only local minimum is the global minimum. However, for QMA-hard Hamiltonians, the energy landscape necessarily contains many suboptimal local minima. (c) Energy landscape under local unitary perturbations. For any local Hamiltonian 𝑯, there are always doubly exponentially many local minima within the 𝑛-qubit state space that stems from a large barren plateau. Local unitary perturbations are reversible, while thermal perturbations are irreversible. . . . . . 198 4.2 The Bohr frequencies 𝜈 ∈ 𝐵(𝑯) = {𝐸𝑖 − 𝐸 𝑗 | 𝐸𝑖 , 𝐸 𝑗 ∈ Spec(𝑯)} are the differences of energy (eigenvalues of the Hamiltonian 𝑯). . . . . 212 4.3 The circuit-to-Hamiltonian mapping. For any 1D circuit, we can turn the circuit depth into an additional spatial dimension by introducing layers of swap gates; this particularly structured 2D circuit (now on 𝑛 qubits) is what we call 𝑼𝐶 . Then, we map the circuit 𝑼𝐶 to a 2DHamiltonian by introducing the clock qubits, which weave between the layers of system qubits in a zigzag order. This layout ensures that each clock qubit only controls nearby system qubits (in particular, the odd clock layers control only blue gates, and the even layers control only pink gates.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 4.4 The degenerate levels of 𝑯clock split under perturbations 𝑯prop and 𝑯in . In particular, the ground state splitting is tracked in blue shades. The careful choice of energy scales ensures that the levels can be identified with the original degenerate blocks. . . . . . . . . . . . . 271 4.5 The energy scales 1𝜏 ≪ ∥𝑽 ∥ ≪ Δ𝜈 in one plot. The Hamiltonian perturbation 𝑽 causes eigenvalues of 𝑯 to change by at most 𝛿𝜆 ≤ ∥𝑽 ∥, which splits the Bohr frequencies such that |𝜈 − 𝜈′ | ≤ 2𝛿𝜆 . Here 𝜇 is the cut-off frequency for the secular approximation 𝑺ˆ 𝑎𝑓 ,𝜇 (𝜔′) = Í Δ𝜈 −4𝛿𝜆 𝑎 ˆ ′ ′ ′ ′ 𝜈 ′ 𝑨𝜈 ′ 𝑓 (𝜔 − 𝜈 ) 𝑠ˆ 𝜇 (𝜔 − 𝜈 ). As long as we choose 𝜇 < 2 small enough, the secular approximation 𝑺ˆ 𝑎 (𝜔′) can only contain Bohr frequencies 𝑨𝜈′ from at most one block 𝜈, i.e., different blocks decohere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 xiii LIST OF TABLES Number Page 2.1 A comparison of existing thermal state preparation algorithms7 . . . . 137 xiv NOMENCLATURE Davies’ generator. A Lindbladian rigorously derived for a system interacting weakly with a thermal bath. It satisfies detailed balance exactly and has played the role of Glauber dynamics in finite-sizes and commuting systems but is unphysical for general noncommuting many-body systems. Today, there are a few modernized proposals of Davies’ like generators, but they all violate detailed balance in general. Detailed balance condition. A key condition in the design and analysis of Markov chains that prescribes a desired stationary state. This condition requires that the mass exchange between two configurations at stationarity cancels each other. Gibbs state. A quantum state that is a mixture of energy eigenstates with a Boltzmann weight. It is often assumed that a quantum system will eventually arrive at such a state at the same temperature as the bath. Glauber dynamics. A continuous-time classical Markov that satisfies detailed balance regarding the classical Gibbs distribution. For both numerical and analytic purposes, it is often the go-to dynamical model for classical systems interacting with a thermal bath. Ground states. The set of minimal energy eigenstates of a Hamiltonian matrix. A Gibbs state at infinitely low temperature would be a mixture of ground states. Lindbladian. A continuous-time Markovian process acting on quantum states, also known as Liouvillian, or master equation. Such effective description often arises from an open system coupled to a bath but may be considered in its own right. Markov chain. A discrete-time or continuous-time stochastic process where the future is only correlated to the past through the present. Often they can be implemented efficiently on a classical computer to sample from a desired distribution, and is also called Markov Chain Monte Carlo as an algorithmic method. Phase transition. A phenomenon describing a qualitative change in the physical properties of a system depending on the parameters such as magnetic field or temperature. Such transition can be static in terms of ground or thermal state properties (e.g., long-range correlations) or dynamic in terms of the behavior of an associated Linbladian (e.g., mixing times). Quantum algorithms. Algorithms that run on quantum computers, often described in terms of quantum gates. xv Quantum computational advantage. An advantage of quantum computers over classical computers for certain classes of problems. Currently, the quantum hardware is limited in quality and scale, so most evidence is theoretical. Quantum computers. Computers that can act quantum gates on qubits given classical instructions. In this thesis, we will assume them to be fault-tolerant so that we can execute quantum algorithms with high precision. Quantum Gibbs sampling. A quantum algorithm simulating a Quantum Markov chain (continuous or discrete time) whose stationary state is the Quantum Gibbs state. Quantum many-body physics. Quantum phenomena emerging from systems with many degrees of freedom, contrasting with few-body systems. Often, complex, unexpected behavior can occur, but it is very challenging to solve analytically or numerically. Quantum simulation. Simulation of a naturally appearing or artificial quantum system with a quantum computer that allows us to extract its physical properties in a programmable, controllable way. Thermal bath. A large quantum system with a characteristic temperature with Markovian properties. Thermal state. The state describing a quantum system in equilibrium with a thermal bath. Although broadly defined, this can sometimes mean the quantum Gibbs state. 1 Chapter 1 INTRODUCTION We often take for granted that a quantum system without isolation eventually arrives at the thermal state, or the Gibbs state; at low enough temperatures, the system must be in its ground state. This dogma dictates a static view we reason about the physical properties of quantum systems, from atoms to materials to chemicals. First, we write down a Hamiltonian operator; second, we solve the eigenvalue problem. This mathematical paradigm has led to the exact solution for the orbitals and spectrum of simple atomic and molecular systems. In the many-body case, while Schrodinger equations become too complex to solve analytically, an arsenal of classical simulation methods has been developed, such as density functional theory [89], tensor network (e.g., [145, 177]), and quantum Monte Carlo [34], that gives a remarkably detailed understanding of phase transitions, ground and thermal state properties, and reaction rates. However, typically, all these classical methods break down in their own ways in the presence of strong quantum correlation, from heavy nuclei, high-𝑇𝑐 superconductors, to catalysts. Recently, a new type of computer that runs on the laws of quantum mechanics, quantum computers, has gathered significant interest among physicists and computer scientists. Hopefully, future quantum computers could offer new lenses into an even broader array of quantum many-body problems that are otherwise classically intractable. Nevertheless, decades of search for particular physical systems where quantum computers can significantly outperform classical methods have been inconclusive. The challenge is that current devices are limited in their sizes and quality, leading the community to rely heavily on a theoretical proof of quantum computational advantage. From computational complexity theory, however, the standard many-body ground-state problem as formulated is generally intractable even for quantum computers (QMA-hard, as a quantum analog of NP-hard) [101]. Even though such worst-case pessimism may not apply to physically relevant quantum systems, it foreshadows that the search for provable quantum advantage in groundstate problems might not be clear-cut. Indeed, the efficacy of most existing quantum algorithm proposals for low-temperature physics (e.g., [14, 30, 36, 107]) relies on additional assumptions that are not always easy to verify and have been challenged in numerical studies [108]. Even though quantum simulation is commonly regarded as a 2 “killer application” where quantum computers may significantly accelerate scientific discovery, still today, there has not been a go-to quantum algorithm for preparing ground states or thermal states. Shor’s factoring algorithm from 1994 [159] remains a concrete, stand-alone example of an end-to-end, practical quantum computational advantage, while many other proposals [83] have later found efficient classical solutions [165]. A motivation for this thesis is to offer a fresh perspective on the quantum simulation problem. Conceptually, this requires confronting the fundamental “static” view of thermodynamics and shifting towards a more physically motivated “dynamical” formulation. This draws a parallel to the development of classical Markov chain Monte Carlo (MCMC) algorithms that allow for direct dynamical simulation of thermal fluctuations due to a bath, except that designing a truly quantum MCMC algorithm faced additional conceptual challenges rooted in quantum mechanics. These two physical and computational veins of pursuit and challenges come to a confluence: a new Nature-inspired quantum algorithmic principle for simulation of low-temperature physics—quantum Gibbs sampling. 1.1 Nature’s cooling process Take a step back; there is a living example of a large-scale quantum algorithm for low-temperature physics that we encounter every day for a wide range of systems: refrigerators. Our experimental colleagues routinely cool their devices by lowering the temperature of their dilution fridges and laser coolers, through which they observe Bose-Einstein condensation, superconducting phase transitions, and the emergence of nontrivial correlation in graphene. If any of the above quantum mechanical phenomena could be systematically and efficiently reproduced in programmable quantum computers, no one should doubt their practical usefulness as a quantum simulator. Therefore, a quantum algorithm that effectively simulates Nature’s cooling process should already be sufficient for quantum simulation purposes, in order to reproduce physical phenomena appearing in natural thermodynamic environments. Indeed, why should we be bothered that quantum computers cannot prepare QMAhard ground states if these high-complexity states never appear in Nature? From an algorithmic designer’s view, Nature’s cooling scheme has features rarely seen in usual quantum algorithms: it follows a macroscopic thermodynamic principle yet works universally for various systems and baths, and it is agnostic to initial state and noises, while standard quantum algorithms are often very structured and prone 3 to errors. It is tempting to speculate that extracting the essential working principle behind cooling could lead to a new quantum algorithmic paradigm that inherits and even improves its efficacy. Nevertheless, it has been nontrivial to obtain a mathematically simple but physically adequate description of the cooling process. From first principles, a quantum system interacting with a thermal environment is yet another many-body Schrodinger equation with substantial degrees of freedom; one can certainly simulate or solve the global Hamiltonian evolution [168], but that might be overkill for understanding the system alone, which is often much smaller than the bath. To minimize the complexity of such open system dynamics, physicists often introduce a master equation, or a Lindbladian, that aims to capture the effect of a bath by a much simpler continuoustime Markovian process acting upon the system (e.g., [29, 150]). However, the seminal Davies’ generator from 1974 [56, 58], albeit simple, is derived assuming the system to be coupled infinitely weakly to an infinitely large Markovian thermal bath. Shockingly, even in modern versions of Davies-like Lindbladians [121, 134] that adhere to realistic system-bath assumptions, there has not been a proof of convergence to the thermal state for general noncommuting Hamiltonians, or provably correct quantum algorithms that precisely simulate such Lindbladians. Even though there has been a great volume of open system literature, quantitative arguments beyond single-qubits appear relatively rare, and we will have to lay out algorithmic and analytic primitives ourselves (see chapter 2). 1.2 Markov Chain Monte Carlo algorithms Taking another step back, there is an algorithm that simulates classical thermodynamics in almost all the ways we wished for quantum: Markov Chain Monte Carlo (MCMC) algorithms, one of the most versatile yet simple algorithmic principles ever devised. For any target distribution — for example the Gibbs distribution of an Ising magnet — the Metropolis-Hastings algorithm [86, 130] prescribes a Markov Chain that samples from the distribution and gives estimates for statistical properties without explicitly storing the full probability distribution. The key ingredient of this recipe is the detailed balance condition, which simultaneously ensures that the target distribution is stationary and leads to a powerful framework to study their convergence time (the so-called mixing time), through a mathematically well-established quantity called the spectral gap (see, e.g., [110]). Remarkably, detailed balance admits very efficient algorithmic implementations, as simple as local spin updates in Ising models. Seventy years since its debut, today, Markov Chain Monte Carlo as an algorithmic 4 principle continues to apply to newly emerging problems beyond physics, from biology to artificial intelligence. For our purpose, we should not only view MCMC methods as a purely algorithmic construct but also as a succinct mathematical model of Nature’s cooling process. Most notably, Glauber dynamics [75] is a continuous-time classical Markov chain that can be traced to Davies’ generator (for classical Hamiltonians). Even though Glauber dynamics surely does not capture all nuances of open system physics (e.g., quantum mechanical and non-Markovian effects of the bath), the combination of desirable algorithmic and analytic features has nevertheless led to a comprehensive picture of dynamic phase transitions (in thermalization and in complexity) for classical spin systems, including magnets and spin glasses. If quantum computing will play a likewise revolutionary role in quantum thermodynamics, one naturally speculates that a suitable “quantum MCMC” algorithm will have an equally important and enduring impact as the classical case. Nonetheless, early attempts have been compounded by obstacles rooted in quantum mechanics. To see this, consider the following Metropolis-style procedure [167] on the energy eigenbasis: suppose we start with an energy eigenstate and apply a unitary update. If the energy decreases, we wish to “accept” this move; otherwise, we “reject” this move with a prescribed probability. However, this is in tension with (1) the no-cloning theorem (to return to the previous quantum state without keeping many copies) and (2) the energy-time uncertainty principle (to know confidently whether energy has increased or decreased). The first issue turns out to be a technical problem tied to discrete-time quantum Markov Chains and automatically resolves in continuous-time settings (i.e., Lindbladians) [149, 160, 168]. Nevertheless, the uncertainty in energy has remained a fundamental threat to quantum detailed balance, and existing work has failed to retain both the exactness of quantum detailed balance (the basis for modern Markov chain analysis) and finite energy resolution (inversely proportional to the time for Hamiltonian simulation, the dominant cost in most quantum simulation algorithms). Perhaps due to the lack of a simple “go-to” proposal that matches the features of classical Glauber dynamics, the potential impact of Quantum MCMC algorithms has mainly been under the shadows for the quantum computing community—see chapter 2 for a detailed account for literature. 5 1.3 Main result In this thesis, we propose a quantum Markov Chain Monte Carlo algorithm that can also be regarded as a succinct mathematical model of open system thermalization. For any Hamiltonian of interest, we construct a Lindbladian, a continuous-time quantum Markov chain with the following combination of desirable features akin to Glauber’s: • Exact Quantum Detailed balance. The Lindbladian satisfies quantum detailed balance exactly. Therefore, the quantum Gibbs is a stationary state of the Lindbladian. This allows for clean mathematical reasoning and systematically extends the formalism and techniques developed in classical and commuting cases for mixing time (e.g., [31, 96, 124]). From a quantum algorithm perspective, detailed balance is also naturally compatible with the quantum walk formalism [164], giving a coherent algorithm for the “purified” Gibbs state. • Efficient simulation. The Lindbladian dynamics can be efficiently simulated on a quantum computer, as long as the Hamiltonian dynamic can be simulated for a time depending on the temperature. We expect the simulation cost to be asymptotically optimal (up to polylogarithms). • Quasi-locality. For lattice Hamiltonians, the Lindbladian decomposes into a sum over (quasi)-local Lindbladians, with locality depending on the temperature (and polylogarithmically on other parameters, such as the precision and system size). Remarkably, each local Lindbladian is defined by local Hamiltonian patches, avoids global Hamiltonian simulation, and may lead to qualitatively better resource estimate. • Physical origins. The Lindbladian takes a similar form as Davies’-like generator and reduces to Davie’s generator (and hence Glauber dynamics for classical Hamiltonians) in certain parameter regimes. That is, our construction circumvents the aforementioned challenges and retains the celebrated features of classical MCMC algorithms that fueled its widespread impact. We expect our new construction to readily signify a multitude of new directions: • A provable quantum simulation algorithm. There has been a flourishing industry to study spectral gap and mixing time of classical Markov chains 6 (e.g., [110]). In some cases, a sharp theoretical proof of algorithmic efficiency can even be obtained. Our construction gives a well-defined, quasi-local, and detail-balanced map that enables such a study in noncommuting Hamiltonians. Indeed, recently, rapid mixing in lattice Hamiltonians at high temperatures has been proved using our Lindbladian [153]. In chapter 4, we also use the established framework to give implicit evidence of quantum advantage for the “coolable” states through a dedicated mixing time analysis. We hope that new theoretical proof enabled in the quantum case could “unstick” the current inconclusive status of quantum advantage in quantum simulation by unveiling new provably efficient instances. • A dynamics angle to quantum Gibbs states. The mixing time of the Lindbladian is deeply connected to the correlation and complexity of quantum Gibbs states. It was a celebrated result in classical Gibbs sampling that rapid mixing coincides with the decay of Gibbs correlation [124], and only recently has the equivalence been extended to some commuting quantum Hamiltonians [31, 96]. Our newly defined map gives a new dynamic angle to rigorously study intricate quantum correlation, such as the area-law, conditional mutual information, and topological order, in non-commuting Gibbs states. • A self-contained model of open system thermodynamics. As a quantum analog of Glauber dynamics, the physical origin of our construction gives a succinct model of open-system physics. Our Lindbladian enables a precise definition of dynamical thermal phase transitions associated with the mixing time. Traditionally, quantum statistical mechanics have largely focused on static properties (e.g., divergence of correlation length as an indicator of phase transitions), but we argue that a dynamical picture might yield new physics that is not visible in the statics. Related concepts include metastable states, the energy landscape, quantum spin glass, and self-correcting quantum memories, whose precise formulation for noncommuting Hamiltonian has also been lacking (see chapter 4). The explicit form of our Lindbladian also enables direct numerical studies regarding the above notions. To reiterate, given the celebrated theoretical and empirical triumph of Markov chain Monte Carlo methods and its successors over the past 70 years, quantum Gibbs sampling could serve similar roles in quantum computing. Especially given the current skepticism on the practical applicability of quantum computers, we aspire to 7 bring hope to the community by initiating a new wave of directions covering theory, experiment, numerics, and application. The remaining chapters of the thesis are organized as follows. In chapter 2, we introduce the algorithmic and theoretic foundation for the simulation and analysis of a family of Davies-like Lindbladians (not necessarily detailedbalanced). The main algorithmic result is a black-box simulation algorithm for Davies-like Lindbladians (in fact any Lindbladians with substantially large Kraus rank) and the operator Fourier transform, a subroutine that decomposes any operator according to the changes in energy (akin to how phase estimation decomposes the states by its energies). The main analytic result is that Nature’s cooling process, modeled by the existing Davies’-like Lindbladian, indeed converges to approximate Gibbs states; we introduce and prove a notion of approximate detailed balance that depends on the temperature, Hamiltonian simulation time, and the mixing time for the Lindbladian. In chapter 3, we give the advertised detailed balanced Lindbladian as a particular instance of the Davies’-like family introduced in chapter 2, whose analytic and algorithmic properties are largely inherited and improved. The new key insight is a recipe for a coherent term in addition to the dissipation part that perfectly cancels out deviations from quantum detailed balance. With exact detailed balance, the proof of Gibbs stationarity is immediate, and the Lindbladian naturally corresponds to a parent Hamiltonian (akin to the quantum walk operator for classical Markov chains) that allows for preparation for the purified Gibbs state. We do not know where there is a qualitative distinction from Davies’-like Lindbladians introduced in chapter 2. In chapter 4, we provide implicit evidence of quantum advantage in cooling by showing that Nature’s cooling process is just as powerful as universal quantum computation. In particular, we construct a family of 2D Hamiltonians whose ground states encode arbitrary polynomial-time quantum computations, and the associated Davies’-like Lindbladian converges to the ground state in polynomial time. Therefore, there are instances of ground state problems that are quantumly easy and classically hard unless 𝐵𝑄𝑃 = 𝐵𝑃𝑃 (roughly, unless classical computers can efficiently simulate all decision problems solvable with quantum computations). To study the mixing times, we introduce the idea of quantum local minima, which attempts to capture an energy landscape associated with the cooling process. Technically, the results are proven for physically derived Davies-like Lindbladian building upon the framework from chapter 2, but we expect an even faster convergence to hold with better 8 parameters for the Lindbladian with exact detailed balance from chapter 3. 9 Chapter 2 ALGORITHMIC PRIMITIVES FOR QUANTUM THERMAL STATE PREPARATION 2.1 Introduction How do we prepare quantum Gibbs or ground states on a quantum computer? This initial state preparation problem appears as the obstacle for simulating quantum systems [68, 116]– a popular candidate for practical quantum advantage. This mystery has its roots in the seemingly contradictory teachings of computer science and physics: computational complexity theory tells us that few-body Hamiltonian ground states are generally QMA-hard to prepare [4, 76, 101] and thus are likely intractable in general even for quantum computers; on the contrary, thermodynamics asserts that physical systems interacting with a thermal bath are naturally in the thermal states or ground states. How do we draw an appropriate boundary between the two cases? Practically, recent end-to-end industry resource estimates (e.g., [14, 30, 37, 107]) of quantum simulation rely on initial state preparation assumptions1, exposing our ignorance of the complexity of practically relevant states. Often, practitioners turn to heuristic algorithms such as the Variational Quantum Eigensolver (see, e.g., [169]) or the adiabatic algorithm (see, e.g., [7, 64]), yet each with concerns for practicality. The former suffers from the so-called Barren Plateau phenomena [128], and its scalability has been debated; the latter requires a gapped adiabatic path, which appears nontrivial in recent large-scale numerical studies for quantum chemistry applications [108]. So far, there is a thin consensus on a ‘go-to’ ground state or thermal state quantum algorithm that could work in practice. This work approaches the state preparation problem via Quantum Gibbs samplers. In physics language, this is closely related to open system dynamics where the system of interest is coupled to a thermal bath (see, e.g., [152]). Here, the conceptual boundary is blurred between the underlying physical process and the algorithm [168]. If a system thermalizes in nature and our physical model is accurate, we expect the associated quantum Gibbs sampler to converge quickly (i.e., the mixing time or the 1 More precisely, they assume the existence of trial states with good overlap with the ground state so that running phase estimation provably works [114]. See also [70]. 10 inverse-spectral gap is small); conversely, proving the latter also gives a rigorous formulation of open-system thermodynamics. This complements the mainstream formulation of closed-system thermodynamics via the Eigenstate Thermalization Hypothesis (see, e.g., [62]), where theoretical progress has been elusive. Practically, our general analysis for open system thermalization could be relevant to analog quantum simulators for Gibbs sampling, although our presentation mainly focuses on fault-tolerant quantum computers. In computer science language, quantum Gibbs samplers are the quantum analogs of classical Markov chain Monte Carlo (MCMC) algorithms, most notably Metropolis sampling (see, e.g., [110]). They proved to be an indispensable pillar in classical computer science, both theoretically and practically, for computational physics and, more recently, optimization problems and machine learning. In a nutshell, the simple yet general idea is a (discrete or continuous) Markov chain whose unique fixed point yields the target distribution 𝜋 (a vector with positive entries); given the energy 𝐸 𝑠 as a function of the configuration 𝑠, the Markov chain’s transition matrix 𝑴 satisfies 𝑴𝜋 = 𝜋 where e−𝛽𝐸 𝑠 𝜋 𝑠 := Í −𝛽𝐸 𝑠 𝑠e for each 𝑠 at temperature 1 . 𝛽 The algorithmic cost for preparing a sample from the Gibbs distribution 𝜋 scales directly with the mixing time, the number of iterations such that any initial conjugation converges to the stationary distribution 𝜋. The mixing time can be unpredictable and vary wildly depending on the specific problems (e.g., [110]). Theoretically, rapid mixing can sometimes be proven under suitable assumptions, most notably in lattice Ising models assuming exponential decay of Gibbs state correlation (see, e.g., [124]). Practically, even when mixing time estimates are elusive, MCMC algorithms often serve as a starting point for more sophisticated algorithms. Given the triumphant impact of classical Gibbs sampling, we argue that Quantum Gibbs samplers have been thus far underexplored in the community and will likely play a central role when more robust quantum computers become available. Indeed, in addition to quantum simulation, quantum Gibbs sampling has been identified as a key subroutine in solving semidefinite programs (SDPs) [11, 27] and quantum machine learning [9]. To clarify, we will focus on quantum Hamiltonians; quantum algorithms for classical Gibbs states are not in the scope of this work.2 2 Quantum Gibbs sampler for fast-forwardable Hamiltonian (including commuting Hamiltonians) is already well-defined since one can effectively apply phase estimation to exponential accuracy [167, 180]. The challenges we confront in this work are rooted in the noncommutativity. 11 To set the stage for quantum Gibbs sampling, we shall review the classical cousins, which we consider the seminal Metropolis-Hastings algorithm (see, e.g., [110]) as a representative. This algorithm iterates a discrete-time Markov chain as follows: apply a random “jump” (or “update,” “move”) 𝑨𝑎 with probability 𝑝(𝑎) (Figure 2.1). If the energy decreases, accept the move, otherwise accept only with probability e−𝛽𝜔 (i.e., rejecting the move with probability 1 − e−𝛽𝜔 ), where 𝜔 being the energy gain. This can be described as a stochastic matrix over pairs of configurations 𝑠′ 𝑠 ∑︁ 𝑴𝑠′ 𝑠 := 𝑝(𝑎)𝛾(𝐸 𝑠′ − 𝐸 𝑠 ) 𝑨𝑎𝑠′ 𝑠 + 𝑹𝑠′ 𝑠 𝛿 𝑠′ 𝑠 where 𝛾(𝜔) := min(1, e−𝛽𝜔 ), | {z } 𝑎∈𝐴 | {z } “Reject” “Accept” (2.1) and 𝑨𝑎𝑠′ 𝑠 are stochastic matrices corresponding to each move (e.g., flipping one of the spins). The matrix elements are weighted by the Metropolis factor 𝛾(𝜔) depending on the energy change. Importantly, the particular function satisfies a particular symmetry (Figure 2.1), known as the detailed balance condition 𝛾(𝜔)/𝛾(−𝜔) = e−𝛽𝜔 such that 𝑴𝑠 ′ 𝑠 𝜋 𝑠 = 𝜋 𝑠 ′ 𝑴𝑠 ′ 𝑠 for each 𝑠, 𝑠′ . (2.2) Detailed balance ensures that the Gibbs state 𝜋 is a fixed point of the Markov chain 𝑴𝜋 = 𝜋. The rejection part 𝑹𝑠′ 𝑠 is a diagonal matrix determined by the probability preserving constraints. Similarly, one may define a continuous-time Markov chain generator © ª ­ ® ∑︁ ­ ® 𝑎 𝑎 ­ 𝑳 𝑠′ 𝑠 := 𝑝(𝑎) ­𝛾(𝐸 𝑠′ − 𝐸 𝑠 ) 𝑨𝑠′ 𝑠 − 𝛿 𝑠′ 𝑠 𝛾(𝐸 𝑠′′ − 𝐸 𝑠 ) 𝑨𝑠′′ 𝑠 ®® for each {z } ­| ® 𝑠′′ 𝑎∈𝐴 ­ | {z }® “transition” “decay” « ¬ ∑︁ 𝑠, 𝑠′ . (2.3) The second term ensures that the generated semi-group e 𝑳𝑡 preserves probability. The operators 𝑨𝑎 can be arbitrary nonnegative matrices and need not be stochastic. While classical Markov chain Monte Carlo methods have been theoretically and practically mature, the quantum analogs are still in their infancy. The study of Quantum Gibbs sampling currently faces fundamental challenges; surprisingly, even a satisfactory map has not been appropriately defined for general noncommutative Hamiltonians. As the very first step, we need to algorithmically design a quantum 12 Markov chain M <latexit sha1_base64="w/dUxd60iQjel2jdp0h/ivQ0O2A=">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</latexit> Jump E decrease 1 ! s2 ) EPr(s increase =e Pr(s2 ! s1 ) … Accept <latexit sha1_base64="+Tf7HH9icwBrmzIuIrQyBjng8Ac=">AAAOfHichZdbbyM1FMdnl9tSaGnhkZcRoVIXaJVEy8JLpW2n6f2StrmUdkJkT5zEqucij+PdKhq+Fx+FJ17hSyA846Tj+gQ2L7HP739sz7H/c8EJo6moVv949vyDDz/6+JMXny599vnyyhera1920njCA9IOYhbzG4xSwmhE2oIKRm4STlCIGeniey/nXUl4SuOoJR4S0gvRKKJDGiChQv3Va3/IUTD1m3wj7ddcn9PRWCDO47du2q+/zGakbpHay8zddn3y63TTx0Qgd6OhNJtuIyf91Up1q1r8XNiozRoVZ/Zr9teW1/xBHExCEomAoTS9q1UT0ZsiLmjASLa05E9SkqDgHo3IFIVp+hDizF0PkRinNsuDi9jdRAx/7k1plEwEiQIlUWw4Ya6I3bwy7oByEgj2oBoo4FRN7QZjpKojVP0WzvJ0XThUgXXXnLLYH7U19+m24BPyg4sYK2LpNmYT0puO1cSck6HKtAbj6J6IbGndDEZIYIrzWdYRxpzIJZ+TiLwN4jBE0WDqSxKoHcsX4pthb0dF8xUHiE13MpvuGnQXUMQ8g3s2351TjCH09ozUPUAbBm2AgRvlwAB6+0bqPqAHBj0A9NCgh4AeGfQI0GODHgN6YtATQE8NegromUHPAD036Dko1XlZKgC9CyP1AtCmQZtg4GY5MIDepZF6CVIvy1QAvSsj9QqkXpWpAHrXRuo1oC2DtgBtG7QNaMegHUC7Bu0CemPQG0B/cQ2sOja/NfDtU/pd7uokVQKJuGpQFkf53eLJAHKnsD30tkQaIAB2NQB+l1gDDICnATC53NMA+Fs2NADulfsaAOPKAw2AZ+WhBsCu8kgD4FR5rAEwqTzRAPhTnmoArCnPNACulOcaAM/JCw2A3WRTA2AmeakBsIq80gAYQV5rADwgWxqA4y/bGoCTLzsagEMvuxqA8y5vNABHXRZHXRF4yuWtJrcApHQUzg6q6+uOJcGlBnElkgtVko/j+TB5M7Ofj/kOMzIUVl5e4uL1BhgLR5PZnL5qWWnjAo6RWATxPBGS8HHMEMLiKVFALz+I609pYw6Bp3AcktG8QL7uWJIBGo0y/Ud4Zl0py88A6xBuF4fngC8A6rVSUEqzu1pvqstahPx7wqPN6lb9R/LufVG3Mn/lhKL3hu3Li2IeFovx88W600otc/0F6060sN7TKdNKPevn4gXDJaaslsvqtowZOvZ/wmG5vlJW3PV5cScEG8IQTrUe09FI10tf1GPfXstjRllhnWFUbcHyjZyOnbOggCR7XDexr5JOIipKTm3O6X9C9aQTvKQtDnASs4dS0Mx7YISHpFTkHVvQMgWtBQLE1Z3l3UwTJ2rJoativ+VBn9GQitROoYOskKl/i1xPEmO6omfvM40G+ddYPL90rG5f6jCqj6ia/ckEG536Vu311qvLV5U3u7PPqRfO1843zoZTc35y3jiHTtNpO4Hzu/On85fz9/I/K9+ufL+yqaXPn81yvnKe/FZe/wviN4Fw</latexit> initial state (E2 E1 ) output Reject Unjump Figure 2.1: The Metropolis-Hastings algorithm iterates a Markov chain to sample from the Gibbs distribution. Each step begins with a (random) jump: if the energy decreases, accept; if the energy increases, accept with a carefully chosen probability. Otherwise, reject the move. Remarkably, detailed balance can be enforced in a “lazy” Also, (continuous-time) Glauberstoring dynamics [’63] eLtmix [⇢] ! manner via *rejection sampling without the whole matrix. <latexit sha1_base64="d8HZ5OYnczSGNDho/nRWdH7oFts=">AAAOZHichZfbbts2GMfV7tRlTZas2NWAQZgRYBi2wA7absBQoIlyPjqJ7WSJPIOUaZsIdQBFsQkE7V32NLvdLvcCe45Roh0x/LxVNyL5+38k9ZF/icIJo6loNv9+8vSDDz/6+JNnny589nxx6fPllS96aZzxgHSDmMX8CqOUMBqRrqCCkauEExRiRi7xrVfyS0l4SuOoI+4T0g/ROKIjGiChmgbLP/vk19wPkZgEiOVHhSsGeUjviuLGl3wS912f0/FEIM7jd64vUzoO0SD3MRGoGCw3mmvN6nJhoTUtNJzp1R6sLK74wzjIQhKJgKE0vWk1E9HPERc0YKRYWPCzlCQouEVjkqMwTe9DXLir5fRSm5WN89hNJkY/9XMaJZkgUaAkio0y5orYLTPgDikngWD3qoACTtXQbjBBHAVC5WnuKI/nhUPVsOqaQ1broJbgNn0jeEa+dxFjVVv6BrOM9POJGphzMlKRVmcc3RJRLKyajRESmOJylFWEMSdyweckIu+COAxRNMx9SYJCLYKaiG82extFvZYbhU03DboJKGKewT2bb84oxhB6W0boFqDbBt0GHW/XHQPo7RihO4DuGnQX0D2D7gG6b9B9QA8MegDooUEPAT0qTEfZ9Nigx4CeGPQEpOqkThWA3qkRegpo26Bt0HG77hhA78wIPQOhZ3UogN65EXoOQs/rUAC9CyP0AtCOQTuAdg3aBbRn0B6glwa9BPTKoFeA/uIaWFVsfm3g68f0u9LVSaoEEnFVoCyOyrfFow7kRmV76G2JNEAAbGoA/C6xBhgATwNgcrmlAfC33NYAuFfuaACMK3c1AJ6VexoAu8p9DYBT5YEGwKTyUAPgT3mkAbCmPNYAuFKeaAA8J081AHaTbQ2AmeSZBsAq8lwDYAR5oQHwgOxoALa/7GoAdr7saQA2vbzUAOx3eaUB2Oqy2uqKwF0urzW5BqA6PWjo+rpiSXCtQTyfHThAR+pkMuumLBb297FcYUZGwoorU1wdZ4CxcJRNx/RVyQqbVHCCxDyIZ4GQhA99hhBWX4kKeuVGXH1Mt2cQeArHIRnPEuTriiUZovG40DfCC+tJWbkHWI9wOzm8BHwOUMdHQSktblr9XKe1avJvCY9+aK6tvyJ372t1G63pQRKK3ttsP14U87CajF9O1s0brcL158w70cL1vg7JG+vFoBTP6S4xZa1Stm7LmKFj/ycc1fOrZdVbn1dvQrAgDOFU6zEdj3W+9EM91O25PETUGdYRRtbmTN+I6dkxcxJIiod5E/spaRZRUXNqc07/E6ovneA17XCAk5jd14J2WQM93Ce1oqzYgo4p6MwRIK7eLHdTTZyoKYeuavutbPQZDalI7RA6LCqZulvkIkuM4aqavc40GpZ/XfHs0bF6fanNqH6iWvYvEyz01tdar9denr1svN2c/k49c75yvnG+dVrOj85bZ89pO10ncH53/nD+dP5a/Gfp+dKLpS+19OmTacwL59G19PW/toh8VA==</latexit> analog of Markov chain generator3 L 𝛽 , a Lindbladian, [181] whose (unique) fixed point is the quantum Gibbs state 𝝆 𝛽 . More precisely, given a Hamiltonian 𝑯 and an inverse temperature 𝛽 design L𝛽 such that eL 𝛽 𝑡 [𝝆 𝛽 ] = 𝝆 𝛽 where 𝝆 𝛽 := e−𝛽𝑯 /Tr(e−𝛽𝑯 )(2.4) for any 𝑡 > 0. Subsequently, we may begin studying the properties of the proposed Lindbladian, especially the mixing time. This work aims to lay the foundation for the first challenge. The second challenge was partially addressed in Ref. [38] using more primitive Gibbs samplers. Unlike the classical case, the construction of quantum Gibbs samplers is nontrivial due to imprecise energy estimates for noncommuting Hamiltonians (i.e., the energy-time uncertainty principle); the fixed point would not be exactly the Gibbs state (2.4). Previous attempts [38, 167, 168, 180, 186] have their shortcomings, which we discuss in more detail in section 2.1 and Table 2.1. Our work, in parallel with the recent paper [149], provides the first implementable Lindbladian for Gibbs sampling, with provable guarantees and without unrealistic assumptions. To do so, we introduce a robust analytic framework, which additionally applies to physical Lindbladians derived in open systems and to coherent Gibbs samplers with Szegedy-type speedups. Our particular construction draws inspiration from thermalization in nature. As the starting point, a system in thermal contact with a bath can be effectively described by the so-called Davies generator in the Schrödinger Picture in a specific (weak3We focus on the infinitesimal generators for simplicity. One may alternatively consider discrete quantum channels, also known as completely-positive-trace-preserving (CPTP) maps. 13 coupling/infinite-time) limit ([56, 58, 152], and see [134] for a modern discussion) ª © ® ­ 1 ­ 𝑎 𝑎 † 𝑎 𝑎 † 𝑎 ® 𝑎 † LD𝑎𝑣𝑖𝑒𝑠 ( 𝝆) = 𝛾(𝜈) ­ 𝑨𝜈 𝝆( 𝑨𝜈 ) − (( 𝑨𝜈 ) 𝑨𝜈 𝝆 + 𝝆( 𝑨𝜈 ) 𝑨𝜈 ) ®, ® ­| {z } 2 𝑎∈𝐴 𝜈∈𝐵 ­ | {z }® “transition” “decay” ¬ « ∑︁ ∑︁ (2.5) where { 𝑨𝑎 } 𝑎∈𝐴 are the set of “quantum” jumps and 𝜈 ∈ 𝐵 := spec(𝑯) − spec(𝑯) are the Bohr frequencies, the set of energy differences of the Hamiltonian. This resembles its classical Markov chain cousin (2.3), featuring two terms: transition and decay rates. Since we work with density operators 𝝆 instead of probability vectors, the input 𝝆 must be formally sandwiched by operators on the left and right. However, if the input states are diagonal in the energy basis and the energy levels are nondegenerate, then the Davies’ generator can be faithfully represented as a continuous-time Markov chain (2.3) on the energy eigenstates by literally replacing 𝑠 → |𝜓𝑖 ⟩⟨𝜓𝑖 |, 𝑨𝑎𝑠′ 𝑠 → ⟨𝜓𝑖 | 𝑨𝑎 𝜓 𝑗 2 where 𝑯= ∑︁ 𝐸𝑖 |𝜓𝑖 ⟩⟨𝜓𝑖 |. (2.6) 𝑖 A concrete physical example is a geometrically local Hamiltonian on a lattice. The jump operators 𝑨𝑎 can be one-body Pauli operators on each lattice site, and the cardinality of jumps scales with the system size | 𝐴| ∝ 𝑛. Of course, Davies’ generator is merely an instance of many possible Lindbladian one can write down (see section 2.2), which generally may not come from thermodynamics. In (2.5), the quantum mechanical transition rate 𝑨𝜈𝑎 is defined as ∑︁ 𝑨𝜈𝑎 := 𝑷 𝐸𝑖 𝑨𝑎 𝑷 𝐸 𝑗 for each Bohr frequency 𝜈 ∈ 𝐵, 𝐸 𝑖 −𝐸 𝑗 =𝜈 where 𝑷 𝐸 denotes energy eigenspace projectors associated with energy 𝐸 (Figure 2.2). In general, the dynamics can be inherently quantum-mechanical when the energy subspaces are degenerate; the quantum transition can include coherent rotations within the subspaces 𝑷 𝐸 . The function 𝛾(𝜔) depends on the physical model of the bath. Under physical assumptions (thermal bath and Markovianity), the heating transitions are penalized by a Boltzmann factor relative to the cooling transitions4 𝛾(𝜔) = e−𝜔𝛽 𝛾(−𝜔). 4The sign convention we use (here and also for Fourier Transforms) might differ from that of other works in the open systems literature. 14 h i | Aa | j i Ej <latexit 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Aa (!) <latexit 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Aa ( !) ⌫ <latexit 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Aa (!) = <latexit 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Aa!¯ = <latexit 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Âa (!) Ei <latexit 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Âa ( !) <latexit 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1 Ej ± O( t ) 1 <latexit 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Ei ± O( t ) E <latexit 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<latexit 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! ⌫2 ± O( t 1 ) <latexit 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Figure 2.2: (Up) Davies’ generator gives a continuous-time Markov generator on the energy spectrum (assuming the Hamiltonian is nondegenerate and the input Ei ± O( t 1 ) state is diagonal in the energy basis.). The transitions are weighted by 𝛾(𝜔): the heating transitions (red) are suppressed by a Boltzmann factor relative to the cooling transitions (blue), entailing detailed balance. The operator 𝑨𝜈𝑎 contains the transitions with energy difference 𝜈, which requires an infinite-time Fourier Transform. (Down) Our Lindbladian Gibbs sampler can be considered a “semi-classical” random walk where nearby Bohr-frequencies 𝜔 ± O (𝜎𝑡−1 ) cannot be distinguished. The operator Fourier Transform 𝑨ˆ 𝑎 (𝜔) contains a band of transitions. This breaks the detailed balance condition; the fixed point deviates from the Gibbs state. <latexit sha1_base64="iHxVaSl8m8ucjsicyr1uCAZWnPM=">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</latexit> Mathematically, this ensures that the Davies’ generator LD𝑎𝑣𝑖𝑒𝑠 satisfies the quantum detailed balance condition (section 2.2) with respect to the Gibbs state 𝝆 𝛽 , implying that 𝝆 𝛽 is a fixed point (2.4). At first glance, the above properties of Davies generators (2.5) seem to qualify for a Quantum Gibbs sampler (2.4) - if we were able to simulate it efficiently. Unfortunately, the generator (2.5) is generally nonphysical because isolating an exact energy transition 𝜈 requires an infinite-time Fourier Transform5 over Heisenberg evolution ∫ ∞ 𝑎 e−i𝜔𝑡 ei𝑯𝑡 𝑨𝑎 e−i𝑯𝑡 d𝑡 for each 𝑎 ∈ 𝐴 and 𝜈 ∈ 𝐵. 𝑨𝜈 ∝ | {z } −∞ 𝑨 𝑎 (𝑡):= 5With the exception when the Hamiltonian spectrum takes well-separated discrete values with (roughly) known locations. 15 This allows the Davies generator to decohere arbitrary close Bohr frequencies 𝜔, 𝜔′. This contradicts the energy-time uncertainty principle, requiring the runtime to scale inversely with the level spacing, which generally can be exponentially small in the system size 𝑛. Unfortunately, at shorter times, the resulting fuzziness of energy resolution breaks detailed balance, which has been central to the analysis of (both classical and quantum) Markov chains. Many finite-time versions of the Davies generator have been proposed to capture more realistic physical settings [38, 121, 134, 150]. Yet, to the best of our knowledge, no Lindbladian arising from a finite-time Fourier Transform has been proven to have a fixed point close to the Gibbs state.6 Quantum Gibbs sampling algorithms face analogous technical challenges due to a limited algorithmic runtime. Here, the energy-time uncertainty principle incarnates as the statistical uncertainty of the energy measurement via quantum phase estimation. Several works [148, 180] evade this issue by imposing a convenient rounding promise on the Hamiltonian, requiring its spectrum to be disjoint from certain specific ranges of energy. Such a rounding promise is not physically motivated and does not hold in general but allows for rigorous performance guarantees for Gibbs samplers using boosted phase estimation. Very recently, Ref. [149] circumvents the need for a rounding promise by randomly alternating the phase estimation mesh, but this comes at a high additional algorithmic cost with potentially worsened mixing time due to forbidding certain transitions and seems distant from the physical origins of quantum Gibbs samplers. In this work, we present quantum Gibbs sampling algorithms inspired by thermalization in Nature. In particular, our construction is a “smoothed” version of the Davies’ generator (2.5). The first algorithm simulates a Lindbladian whose fixed point is approximately a quantum Gibbs state; the second algorithm further “quantizes” the Lindbladian to prepare the purified Gibbs state and features a quadratic Szegedy-type speedup. Our algorithms are efficient and comparatively simple to implement while having a provable performance guarantee. The key ingredient in our algorithmic design is to use the weighted operator Fourier Transform for the Lindblad operators ∫ ∞ 1 𝑎 ˆ 𝑨 (𝜔) = √ e−i𝜔𝑡 𝑓 (𝑡) 𝑨𝑎 (𝑡)d𝑡 for each 𝑎 ∈ 𝐴 and 𝜔 ∈ R. (2.7) 2𝜋 −∞ In practice, we use a discrete Fourier Transform (which will be denoted by 𝑨ˆ 𝑎 ( 𝜔) ¯ for discrete frequency label 𝜔). ¯ Still, for conceptual simplicity, we focus on 6 Ref. [38] gives a nonCPTP generator that does have approximately a Gibbs fixed point. 16 the continuous case in the introduction. Unlike in ordinary phase estimation where boosting usually adopts median-of-means tricks [138], we weigh the Fourier Transform by a Gaussian distribution 𝑓 (𝑡) with a tunable width 𝜎𝑡 . Consequently, the Fourier Transform remains a Gaussian, achieving an analog of a boosted phase estimation with uncertainty 𝜔 ± O (𝜎𝑡−1 ). To give performance guarantees for our construction, our main analytic insight is to define a notion of approximate detailed balance (see section 2.7) −1/4 −1/4 1/4 −1/4 † 𝝆 1/4 𝑶 𝝆 −1/4 ] 𝝆 1/4 L [𝝆 1/4 𝛽 L [𝝆 𝛽 𝛽 𝛽 ≈ 𝝆𝛽 𝛽 𝑶 𝝆𝛽 ] 𝝆𝛽 for each operator 𝑶 to handle energy uncertainty. In the exact case, this is the quantum generalization of classical detailed balance, where the distribution becomes an operator, and the Markov transition matrix becomes a superoperator. Our construction and analysis of the Gibbs sampler are physically inspired insofar as it closely resembles the Davies generator of Eqn. (2.5), but we do not know whether it can be derived in some physical limit from a weak system-bath coupling. Incidentally, starting from a microscopic system-bath interaction, a recent proposal [121, 134] specifically derives from first principles a Lindbladian with Lindblad operators 𝑨ˆ 𝑎 (𝜔) :∝ ∫ 𝑇/2 e−i𝜔𝑡 𝑨𝑎 (𝑡)d𝑡 for each 𝑎∈𝐴 and 𝜔 ∈ R. −𝑇/2 Here, the Fourier Transform time-scale 𝑇 sets the energy uncertainty 𝜔 ± O (𝑇 −1 ); the fixed point will not be exactly the Gibbs state. Using our new analytic framework, we show that this Lindbladian derived in Refs. [121, 134] has a stationary state close to the Gibbs state. To the best of our knowledge, our work completes the first general proof of many-body Gibbs states in the open system setting (assuming a reasonably short mixing time). The runtime of both our algorithms has simple dependence on the mixing time or spectral gap of the Lindbladian. In general, the gap will depend sensitively on the details of the physical system, and its calculation for specific Hamiltonians is beyond the scope of this work. We refer to Refs. [31, 38, 96, 97] and references therein for a more detailed discussion of mixing times (and spectral gaps) for various Hamiltonian and Lindbladians. At an intuitive level, we expect the Lindbladian for lattice systems with jump operators on each site to have a constant local Lindbladian gap7 independent of the volume at high enough temperatures or within the same 7The local gap, in our normalization, is defined as 𝑛 · 𝜆 g𝑎 𝑝 (L), where 𝜆 g𝑎 𝑝 (L) is the Lindbladian 17 phase. In practice, we believe that quantum Gibbs sampling algorithms will be employed on a case-by-case basis in combination with various heuristics, as is the case with classical Monte Carlo algorithms. Existing work The first attempt at designing a quantum algorithm for Gibbs sampling with per-update efficiency guarantees is the quantum Metropolis algorithm [167], as a quantum analog of (2.1). The main guiding principle is to “do Metropolis sampling over the energy spectrum” in the spirit of (2.6). To do so, we need two quantitative changes due to quantum mechanics, one algorithmic and one analytic: algorithmically, one needs a subroutine to “reject” a quantum state back to the same energy. Classically, one can clone the configuration 𝑠 before the update. Quantumly, however, we cannot clone the (unknown) quantum state and must be careful not to collapse the quantum state due to energy measurement. Ref. [167] handles this issue using the Mariott-Watrous [123] algorithm (or the “rewinding” technique in quantum Cryptography). Crucially, this algorithmic subroutine preserves probability, drawing a distinction from imaginary time evolution or post-selection, but this comes with its limitation and significantly complicates the algorithm. Second, energy measurements based on quantum phase estimation have a finite resolution inversely proportional to the runtime 𝛿𝐸 ∼ 1/𝑇. Consequently, the detailed balance condition may not generally hold, and one needs to prove that the fixed point remains approximately the Gibbs state. The technical results of [167] contain three approaches based on different phase estimation subroutines: (i) assuming perfect phase estimation (with performance guarantee but with an exponential Hamiltonian simulation time); (ii) un-boosted phase estimation (without performance guarantees); (iii) boosted shift-invariant phase estimation (with performance guarantees).8 Unfortunately, we recently realized that such a boosted, shift-invariant phase estimation (see section 2.13) is provably impossible;9 we do not know whether Quantum Metropolis sampling [167], as stated explicitly, actually works in practice. The quantum metropolis is regarded as an important theoretical milestone, but due to its complicated form (especially due to the rejection subroutine), the particular algorithm largely serves as a high-level inspiration. eigenvalue gap and 𝑛 is the system size. A parallel version of the algorithm could, in principle, gain this 𝑛 factor in the circuit depth. Also, the Lindbladian gap should not be confused with the energy gap of the Hamiltonian, which is not directly relevant to Gibbs sampling at nonvanishing temperatures. 8The approximate detailed balance argument was later completed in [38]. 9The authors [167] communicated with us that there might be ways to fix their algorithm. 18 From a physical point of view, one may implement Nature’s quantum algorithm [168] by emulating the global system-bath interaction. However, this black-box approach is a double-edged sword: indeed, this method should work as well as Nature, but as we know from open system thermodynamics, nonasymptotic results are extremely challenging without liberal use of approximations [152, 168], rendering the result qualitative but not quantitative. (For example, it is elusive how big of a bath is needed for the desired accuracy.) Recently, Ref. [38, 134, 160] took the physics inspirations seriously and quantitatively studied a system-bath interaction from scratch. Ref. [134] revisits the text-book open system derivation and extracts a nonasymptotic version of Davies’ generator with explicit error bounds and without unphysical limits. However, it was not known whether the derived Lindbladian has the Gibbs state as the stationary state; ref. [38] was the first provably polynomial-time algorithm for Gibbs state assuming a reasonably short mixing time, although it assumes good control of the bath and its error bounds are large polynomials and impractical to apply; ref. [160] is conceptually similar to [38] but focuses more on near-term feasibility. Technically, its accuracy guarantees require the Eigenstate Thermalization Hypothesis, which significantly simplifies the analysis. Unfortunately, both cases [38, 160] failed to extract a Lindbladian (the generators are not completely positive). In some sense, this motivates us to give a unifying conceptual and analytic perspective on this subject. Coherent quantum Metropolis sampling [186] is a natural generalization of quantum Metropolis sampling [167] that further gives a quadratic runtime speedup by invoking Szegedy’s quantum walk strategy [164]. Since the dissipative map is quantized as a Hermitian operator, one cannot evolve a semi-group but requires an additional quantum simulated annealing step [161] (a particular adiabatic state preparation along temperatures) to prepare the purified Gibbs state; see section 2.12. Unfortunately, Ref. [186] assumes perfect phase estimation and a nondegenerate Hamiltonian spectrum, and it was unclear how one incorporates imperfect phase estimation in such a coherent algorithm. Ref. [180] improves and generalizes their result but still makes an unphysical rounding promise assumption: the Hamiltonian spectrum has periodic gaps to amplify phase estimation. Perhaps inspired by the rounding promise, the recent related work [149] proposes an algorithm that implements a Lindbladian that also provably has approximately Gibbs fixed point using randomized rounding. Their approach is quite different in nature from ours, and they are not known to enjoy the quadratic speedup. Randomized rounding seems to incur large resource overhead (See Table 2.17 ) and substantially 19 departs from the physical origin of these ideas. Our Lindbladian Gibbs samplers build upon the literature for open system simulation [49] as well as the coherent Gibbs sampler of [180]. In both cases, we remove the need for any unphysical assumption (e.g., especially the rounding promise) yet still maintain a simple error bound. This is made possible by identifying the right choice of jump operators in terms of discrete Fourier Transform and refining the analytic technical tool introduced in [38] (nonasymptotic bounds for secular approximation)10. In a nutshell, it seems the “right” approach to quantum Gibbs sampling is to simulate a (continuous-time) Lindbladian, which Nature implements by default, instead of a (discrete-time) Metropolis-Hastings style quantum channel. The rejection step is handled automatically for any Lindbladians. We leave for future work to simplify the rejection step in quantum Metropolis sampling [167] or to design a discrete-time channel with provable guarantees.11 We further note a general distinction between coherent Gibbs samplers and Lindbladian Gibbs samplers regarding obtaining the fixed point. The former relies on the gap of the Lindbladian staying open along the entire adiabatic path from high to low temperature 0 → 𝛽, while the latter (e.g., our construction or [149]) does not. Thus, the two algorithmic costs are not directly comparable. Empirical intuition from classical Gibbs sampling suggests that for “simple problems” where the gaps remain largely open throughout the phase of interest, adiabatic and direct sampling methods perform similarly. However, for strongly frustrated systems like spin glasses, adiabatic heuristics are the go-to Monte Carlo method. It is tempting to speculate that the same will be true in the quantum Gibbs sampling case, which lends itself well to our approach. In addition to Monte Carlo style algorithms, other thermal state preparation algorithms based on quite different principles also exist; we briefly summarize their gate complexities in Table 2.1. We only discuss methods where quantitative arguments are possible and pay less attention to heuristic approaches such as variational circuits [51, 125, 158, 184, 188]), energy filtering assuming good initial states, and heuristic quantum assisted Monte Carlo [119, 157]. 10 Some preliminary version of approximate detailed balance was discussed in an earlier version of [38] regarding quantum Metropolis sampling. That part was completely removed after the authors realized the phase estimation assumption was impossible as stated (section 2.13). 11 For Lindbladians, the designer has the freedom to choose arbitrary Lindblad operators, and the decay part automatically guarantees trace-preserving. However, it is more challenging to design quantum channels as the trace-preserving condition seems less flexible. Indeed, Quantum Metropolis Sampling [167] had to invoke Mariot-Wattrous-style rewinding [123] multiple times to ensure probability is preserved, which unfortunately increases the complexity of the algorithm. 20 Outline and main results Our discussion features the following Lindbladian in the Schrödinger picture   ∑︁ ∫ ∞ 1  ˆ𝑎 𝑎 † ˆ𝑎 𝑎 † ˆ ˆ L 𝛽 [𝝆] := 𝛾(𝜔) 𝑨 (𝜔) 𝝆 𝑨 (𝜔) − 𝑨 (𝜔) 𝑨 (𝜔), 𝝆 d𝜔 (2.8) 2 −∞ 𝑎∈𝐴 with the anti-commutator { 𝑨, 𝑩} = 𝑨𝑩 + 𝑩 𝑨. We can read out the set of Lindblad operators ∫ ∞ √︁ 1 𝑎 𝑎 ˆ ˆ e−i𝜔𝑡 𝑓 (𝑡) 𝑨𝑎 (𝑡)d𝑡.(2.9) { 𝛾(𝜔) 𝑨 (𝜔)} 𝑎∈𝐴,𝜔∈R where 𝑨 (𝜔) := √ 2𝜋 −∞ In particular, all Lindbladians we consider in this work, natural or algorithmic, satisfies the following symmetry and normalization conditions: • The set of jump operators 𝑨𝑎 , which “drives” the transition, can be arbitrary (and often depends on the Hamiltonian) as long as the set contains their adjoints ∑︁ { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴} and ∥ 𝑨𝑎† 𝑨𝑎 ∥ ≤ 1. (2.10) 𝑎∈𝐴 Indeed, classical Metropolis sampling often starts with a reversible Markov chain to algorithmic impose the detailed balance condition; the quantum analog is the adjoint condition. Single-site Pauli operators (which are individually selfadjoint) are handy choices, but few-body operators with arbitrary connectivity are certainly permissible12. The normalization is natural for block-encoding the set of jump operators (Definition 2.1.2 and (2.27)). For example, choosing √︁ single-site Paulis as jump operators requires dividing them by | 𝐴| (where | 𝐴| is the cardinality of the set) to fulfill the normalization requirement.13 • The Fourier Transform in the time domain is weighted by a filter function 𝑓 (𝑡) that is real and ℓ2 -normalized ∫ ∞ ∗ 2 | 𝑓 (𝑡)| 2 d𝑡 = 1. (2.11) 𝑓 (𝑡) = 𝑓 (𝑡) for 𝑡 ∈ R and ∥ 𝑓 ∥ 2 := −∞ Sometimes, we drop the subscript by ∥ 𝑓 ∥ = ∥ 𝑓 ∥ 2 . The real constraint serves a purpose similar to reversibility in classical Gibbs sampling. When considering discrete Fourier Transforms (which is necessary for implementation), we adapt Í the corresponding (discrete) normalization 𝑡¯∈𝑆𝑡 | 𝑓 ( 𝑡¯)| 2 = 1. 0 12 In fact, the ability to perform carefully chosen (often not natural) jumps is what empowers classical Gibbs sampling algorithms, e.g., cluster updates. 13This is slightly different from the physical setting where each jump has operator norm ∥ 𝑨 𝑎 ∥ = 1. There, the “strength” of the Lindbladian (2.13) scales with the number of jumps | 𝐴|. 21 • The transition weight 𝛾(𝜔) can be any function satisfying the KMS condition and the bound 𝛾(𝜔)/𝛾(−𝜔) = e−𝛽𝜔 and 0 ≤ 𝛾(𝜔) ≤ 1 for fixed 𝛽 and each 𝜔 ∈ R. (2.12) This coincides with the classical recipe for detailed balance (2.2). Natural choices include the Metropolis weight 𝛾(𝜔) = min(1, e−𝛽𝜔 ) or the (smoother) Glauber dynamics weight 𝛾(𝜔) = (e 𝛽𝜔 + 1) −1 . To summarize, the above list of symmetry conditions is the key to ensuring (approximate) detailed balance; the above normalization choices are not only natural for implementation but also conveniently ensures that the “strength” of the Lindbladian is normalized ∥L 𝛽 ∥ 1−1 ≤ 2 (2.13) in the superoperator 1-1 norm. Lindbladians from Nature Before we discuss Gibbs sampling algorithms, we first address the fundamental question: why do Gibbs states faithfully capture physical systems in thermal equilibrium? In physics, the quantum Gibbs state is often imposed without rigorous justification. As a mathematical physics result, we complete the first proof of open system thermodynamics: the Gibbs state is indeed approximately the fixed point of Lindbladians governing open system dynamics. Of course, this further requires a rigorous derivation of Lindbladian from reasonable open system assumptions; this is not the intention of this work, but thankfully, this has been worked out under a Markovian, weak-coupling assumption [134]. All we need as a black box is that it indeed satisfies the constraints we imposed (2.10),(2.11), and (2.12). For simplicity, we have omitted the Hamiltonian part of the Lindbladian and focus only on the dissipative part; see section 2.9 for the complete results. Theorem 2.1.1 (Gibbs state is thermodynamic). Any L 𝛽 satisfying the symmetry and normalization conditions (2.10),(2.11), and (2.12) with the particular weight function 𝑓 (𝑡) = 1 (𝑡 ≤ |𝑇 |/2) √ 𝑇 22 has an approximate Gibbs fixed point √︂ ∥ 𝝆 𝑓 𝑖𝑥 (L 𝛽 ) − 𝝆 𝛽 ∥ 1 ≤ O ! 𝛽 𝑡 𝑚𝑖𝑥 (L 𝛽 ) . 𝑇 (2.14) In particular, (dropping the Hamiltonian term and under suitable normalization) such a Lindbladian can arise from a system (with Hamiltonian 𝑯) interacting weakly with a Markovian bath (with inverse temperature 𝛽) through jump operators 𝑨𝑎 . The above introduces the notion of mixing time for Linbladians: the time scale for which any pair of initial states become indistinguishable. Of course, the physical interpretation of this time scale depends on how the Lindbladian is normalized; for our cases, we conveniently have that ∥L 𝛽 ∥ 1−1 ≤ 2 (2.13). Definition 2.1.1 (Lindbladian mixing time). For any Lindbladian L, we define the mixing time 𝑡 𝑚𝑖𝑥 (L) in the Schrödinger picture to be the shortest time for which eL𝑡 𝑚𝑖 𝑥 [𝝆 − 𝝆′] 1 ≤ 1 ∥ 𝝆 − 𝝆′ ∥ 1 2 for any states 𝝆, 𝝆′ . Theorem 2.1.1 states that the approximation of Gibbs state degrades at a low temperature, a poor energy resolution (i.e., a short Fourier Transform time 𝑇), or a long mixing time. While the parameters 𝛽, 𝑇 are tunable parameters; the mixing time is generally Hamiltonian dependent. Still, one may obtain a bound using additional assumptions (such as the decay of correlation of commuting Hamiltonian Gibbs states [16, 96] or the Eigenstate Thermalization Hypothesis [38, 160]), empirical intuition, or conversion from a numerically-obtained spectral gap (Proposition 2.2.3). Assuming some grasp of the mixing time, how large should the time scale 𝑇 get to obtain a Gibbs sample? Roughly, according to the error bound (2.14), the time 𝑇 should scale with the mixing time by 2 𝑇 ∼ 𝛽𝑡 𝑚𝑖𝑥 /𝜖 2 . More carefully, the RHS (2.14) does not obviously permit a solution for the desired accuracy 𝜖 (similarly for Theorem 2.1.3 and Theorem 2.1.4). Indeed, the mixing time 𝑡 𝑚𝑖𝑥 can generally depend on other parameters, especially the width 𝑇; heuristically, one may guess that the mixing time 𝑡 𝑚𝑖𝑥 depends mildly on the width 𝑇, but we leave a careful analysis for future works. An optimistic instance is when the Eigenstate Thermalization Hypothesis holds, and the mixing time at finite energy resolution can be related to the infinite resolution (𝑇 → ∞) case [38, 160], which can be calculated. 23 The main analytic challenge to prove Theorem 2.1.1 is how to control the convergence and fixed point of Lindbladians without exact detailed balance. Our technical contribution is to formulate an approximate detailed balance condition using the appropriate superoperator norm −1/4 −1/4 1/4 −1/4 † 𝝆 1/4 · 𝝆 −1/4 ] 𝝆 1/4 L [𝝆 1/4 𝛽 L [𝝆 𝛽 𝛽 𝛽 − 𝝆𝛽 𝛽 · 𝝆𝛽 ] 𝝆𝛽 . 2−2 (2.15) The above two superoperators are each other’s adjoints, and thus, the above quantifies the magnitude of certain anti-Hermitian component of the Lindbladian under similarity transformation. Indeed, traditionally, the detailed balance condition is convenient as it effectively reduces the mixing time of superoperators to the spectral theory of Hermitian operators, which is conceptually and technically more transparent. Our observation is that the consequences of detailed balance, including Gibbs fixed point and spectral bounds on mixing time, are remarkably stable against perturbation. Much ink is devoted to bound (2.15), which is yet another technical challenge. Indeed, we are inverting the Gibbs state, which has exponentially small weights. The energy uncertainty in the Fourier Transforms 𝑨ˆ 𝑎 (𝜔) could potentially blow up (2.15). In response, we further introduce an intermediate Lindbladian by applying a rigorous secular approximation (related to the rotating wave approximation), such that transitions with large energy deviation are truncated. 𝑨ˆ 𝑎 (𝜔) ≈ 𝑺ˆ 𝑎 (𝜔) such that ⟨𝜓𝑖 | 𝑺ˆ 𝑎 (𝜔) 𝜓 𝑗 = 0 whenever (𝐸𝑖 − 𝐸 𝑗 ) − 𝜔 ≫ 0. The secular approximation interplays nicely with the operator Fourier Transform and should be widely applicable in the rigorous, nonasymptotic analysis of open-system Lindbladians. Conceptually, there are two opposite ways to understand Theorem 2.1.1: pessimistically, the Gibbs state may not be physical if the mixing time is too long, and we might have to simulate the natural Lindbladian to understand its fixed point; optimistically, if the Gibbs state is indeed physical, we might ignore its physical origin and take a short-cut to design even more efficient Gibbs sampling algorithms. The two perspectives are individually addressed in the following sections. Simulating Nature Taking a step back from Gibbs sampling, how do we simulate open system dynamics in nature? This boils down to the task of Lindbladian simulation, which has been 24 studied largely restricted to the black-box setting [49]. However, we have an explicit Lindbladian in mind to simulate. As a result, we had to modify existing black-box input models to capture our Lindbladian (2.9); this also inspires us to design even more efficient Lindbladian simulation algorithm for our access model. First, we define how we want our Lindbladian to be block-encoded. Definition 2.1.2 (Block-encoding of a Lindbladian). Given a purely irreversible Lindbladian  ∑︁  1 † 1 † † L [𝝆] := 𝑳 𝑗 𝝆𝑳 𝑗 − 𝑳 𝑗 𝑳 𝑗 𝝆 − 𝝆𝑳 𝑗 𝑳 𝑗 , 2 2 𝑗 ∈𝐽 we say that a matrix14 𝑼 is a block-encoding of the Lindblad operators {𝑳 𝑗 } 𝑗 ∈𝐽 if 15 ∑︁ | 𝑗⟩ ⊗ 𝑳 𝑗 for 𝑏, 𝑐 ∈ N. ( 0𝑏 ⊗ 𝑰) · 𝑼 · (|0𝑐 ⟩ ⊗ 𝑰) = 𝑗 ∈𝐽 Indeed, because of the many Lindblad operators 𝑗 ∈ 𝐽, other access models are certainly valid (e.g., given block-encoding for each Lindblad operator 𝑳 𝑗 [49, 112]). Nevertheless, Definition 2.1.2 interplays nicely with our Lindbladian (especially the operator Fourier Transform) and the following efficient Lindbladian simulation algorithm. Theorem 2.1.2 (Linear-time Lindbladian simulation, simplified). Suppose 𝑼 is a unitary block-encoding of the Lindbladian L as in Definition 2.1.2. Let 𝑡 > 1 and 𝜖 ≤ 1/2, then we can simulate the map e𝑡L to error 𝜖 in diamond norm using  𝑡  𝑡  O 𝑐 + log( ) log( ) (resettable) ancilla qubits, 𝜖 𝜖 e(𝑡) O (controlled) uses of 𝑼 and 𝑼 † , e(𝑡 (𝑐 + 1)) and O other two-qubit gates. See Theorem 2.3.2 for the complete result and Theorem 2.3.1 for a simpler algorithm with suboptimal asymptotic scaling. Compared with the best existing results as Í sum-of-norm 𝑗 ∈𝐽 ∥ 𝑳 †𝑗 𝑳 𝑗 ∥, we achieve a strictly better scaling with the norm-ofÍ sum ∥ 𝑗 ∈𝐽 𝑳 †𝑗 𝑳 𝑗 ∥ when the Lindblad operators are altogether block-encoded as 14 For implementation purposes 𝑼 will be a unitary quantum circuit, but we also consider nonunitary block-encodings for the sake of analysis. 15 In the first register, we could use any orthonormal basis, sticking to computational basis elements | 𝑗⟩ is just for ease of presentation. Intuitively, one can think about 𝑏 as the number of ancilla qubits used for implementing the operators 𝑳 𝑗 , while typically 𝑎 − 𝑏 ≈ log |𝐽 |. 25 in Definition 2.1.2; if we are only given block-encodings for each Lindblad operators, we can always convert them to our input model (Definition 2.1.2) and recover the Í existing scaling 𝑗 ∈𝐽 ∥ 𝑳 †𝑗 𝑳 𝑗 ∥. It remains to create a unitary block-encoding (Definition 2.1.2) for our particular Lindblad operators (2.9). Since our algorithms run on discrete qubits, our implementation requires discretizing the operator Fourier Transform, with a change of the notation ∑︁ 𝑨ˆ 𝑎 (𝜔) → 𝑨ˆ 𝑎 ( 𝜔) ¯ := e−i𝜔¯ 𝑡¯ 𝑓 ( 𝑡¯) 𝑨𝑎 ( 𝑡¯) for each 𝑎 ∈ 𝐴 and 𝜔¯ ∈ 𝑆𝜔0 . 𝑡¯∈𝑆𝑡0 (2.16) The discretized frequency and time labels 𝜔 → 𝜔¯ ∈ 𝑆𝜔0 = {0, ±𝜔0 , · · · } and 𝑡 → 𝑡¯ ∈ 𝑆𝑡0 = {0, ±𝑡0 , · · · } corresponds to the discrete Fourier Transform (section 2.5) using a finite grid of size 𝑁 = 𝑆𝜔0 = 𝑆𝑡0 , which can be stored using ⌈log(𝑁)⌉ additional ancillas.16 While the discretization parameters are needed for explicit algorithmic implementation, conceptually, they merely incur logarithmic overhead in the runtime and ancillas (section 2.8). We may now concretely present our algorithmic goal: efficiently construct a block-encoding in the form ∑︁ √︁ ( 0𝑏 ⊗ 𝐼)𝑼(|0𝑐 ⟩ ⊗ 𝐼) = 𝛾( 𝜔)| ¯ 𝜔, ¯ 𝑎⟩ ⊗ 𝑨ˆ 𝑎 ( 𝜔). ¯ (2.17) 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 Lemma 2.1.1 (Efficient block-encoding). In the setting of Theorem 2.1.1, a unitary block-encoding 𝑼 for the (discretized) Lindblad operators (2.17) can be created using one query of 𝑷𝒓𝒆 𝒑, 𝑾, 𝑽 𝑗𝑢𝑚 𝑝 , 𝑸𝑭𝑻, and (controlled) Ham. sim. time for 𝑯 O (𝑇) 𝑛 + 1 + ⌈log2 (| 𝐴|)⌉ + ⌈log2 (𝑁)⌉ (resettable) qubits. See Lemma 2.3.1 for the explicit construction. In perceivable usage, we expect the number of Fourier labels registers 𝑁 to scale polynomially with all other parameters (section 2.8) 𝑁 ∼ Poly(𝑛, 𝛽, ∥𝑯∥, 𝑇, 𝜖 −1 , 𝑡, | 𝐴|) for a good approximation for the continuous Fourier Transform (2.9). Morally, our algorithm extracts the essential functionality of a Markovian bath (which naively 16We require 𝑁𝜔 0 ≥ 4∥𝑯∥ + 2/𝛽 to store all possible energy transitions. 26 may require a substantial number of qubits to implement [38, 160]) by merely polylogarithmic resettable ancillas. To make the simulation cost transparent, we list the main circuit components required for implementation: the controlled Hamiltonian simulation ∑︁ | 𝑡¯⟩⟨𝑡¯| ⊗ 𝑒 ±i𝑡¯𝑯 , 𝑡¯∈𝑆𝑡0 the unitary for preparing the filter function in superposition ∑︁ 𝑷𝒓𝒆 𝒑 : 0̄ → | 𝑓 ⟩ := 𝑓 ( 𝑡¯)| 𝑡¯⟩, 𝑡¯∈𝑆𝑡0 the controlled rotation for transition weights ! √︁ √︁ ∑︁ 𝛾( 𝜔) ¯ − 1 − 𝛾( 𝜔) ¯ √︁ √︁ 𝑾 := ⊗ | 𝜔⟩⟨ ¯ 𝜔|, ¯ 1 − 𝛾( 𝜔) ¯ 𝛾( 𝜔) ¯ 𝜔∈𝑆 ¯ 𝜔 0 the quantum Fourier Transform 𝑸𝑭𝑻, and the block-encoding 𝑽 𝑗𝑢𝑚 𝑝 of the jump Í operators 𝑎∈𝐴 |𝑎⟩ ⊗ 𝑨𝑎 . In practice, synthesizing the above incurs additional overhead but should be treated as an independent subroutine to study (see section 2.3); we expect Hamiltonian simulation to be the dominant source of cost, which we present17 by the accumulated time for the (controlled) unitaries ei𝑯𝑡¯. The key idea behind implement the unitary block-encoding (2.17) is the operator Fourier Transform (section 2.3) as an alternative to phase estimation (Figure 2.5) ∑︁ | 𝜔⟩ F [·] : | 𝑓 ⟩ ⊗ 𝑶 → ¯ ⊗ 𝑶 𝜔¯ 𝜔¯ which is physically motivated, compatible with our analytic framework, and leads to simple explicit circuits. Improving Nature Suppose our goal is to prepare the Gibbs state, then according to Theorem 2.1.1 and Lemma 2.1.1 we may algorithmically simulate the physical Lindbladian till the mixing time. However, the Fourier Transform occurring in Nature (Theorem 2.1.1), in fact, has a “heavy tail” in the frequency domain; this uncertainty in energy may significantly contribute to the Gibbs state error. With full algorithmic freedom, can 17To obtain the end-to-end gate complexity, one should specify a Hamiltonian simulation subroutine (e.g.,[44, 116, 117]). 27 we do better? In this section, we simply tweak the Lindbladian by considering a nicer Fourier Transform weight 2 − 𝑡 2 𝑓 (𝑡) ∝ e 4𝜎𝑡 with Gaussian width 𝜎𝑡 . The width sets the Hamiltonian simulation time scale ∼ 𝜎𝑡 and the energy resolution ∼ 𝜎𝑡−1 . The Gaussian distribution is particularly nice as it enjoys sharp concentration in both time and frequency domains; in principle, other normalized functions are also permissible, such as Kaiser-window functions [20, 127], as long as they can be Í efficiently generated in superposition 𝑡¯∈𝑆𝑡 | 𝑓 ( 𝑡¯)⟩, but we will stick to Gaussians for 0 simplicity. Theorem 2.1.3 (Approximate Gibbs fixed point). Any Lindbladian L 𝛽 (2.8) satisfying the symmetry and normalization conditions (2.10),(2.11), and (2.12) with the normalized Gaussian weight 2 − 𝑡 2 𝑓 (𝑡) ∝ e 4𝜎𝑡 has an approximate Gibbs fixed point  𝛽 e 𝑡 𝑚𝑖𝑥 (L 𝛽 ) . ∥ 𝝆 𝑓 𝑖𝑥 (L 𝛽 ) − 𝝆 𝛽 ∥ 1 = O 𝜎𝑡  See section 2.2 for the proof. We briefly present the analogous block-encoding costs; the Gaussian width 𝜎𝑡 plays a similar role as the time scale 𝑇, and Gaussians exhibit a better scaling than (2.14) due to its sharp concentration. Lemma 2.1.2 (Efficient block-encoding). In the setting of Theorem 2.1.3, a unitary block-encoding 𝑼 for the (suitably discretized) Lindblad operators (2.17) can be constructed using one query each to 𝑷𝒓𝒆 𝒑, 𝑾, 𝑽 𝑗𝑢𝑚 𝑝 , 𝑸𝑭𝑻, and Õ (𝜎𝑡 ) (controlled) Ham. sim. time for 𝑯 𝑛 + 1 + ⌈log2 (| 𝐴|)⌉ + ⌈log2 (𝑁)⌉ (resettable) qubits. See Lemma 2.3.1 for the explicit construction (which is essentially the same circuit leading to Lemma 2.1.1) and the required 𝑁 to ensure a good discretization error (section 2.8). Therefore, the Gaussian width merely needs to scale as 𝜎𝑡 ∼ 𝛽𝑡 𝑚𝑖𝑥 /𝜖 to prepare a Gibbs sample; see Table 2.1 for the total cost. 28 Quantum-walk speedup With full algorithmic freedom, we may further depart from physics and seek a Szegedytype speedup by considering a coherent representation of the Lindbladian [164, 180].18 Inheriting the notation of (2.8), we consider the following discriminant proxy: ∑︁ ∫ √︁ D 𝛽 := 𝛾(𝜔)𝛾(−𝜔) 𝑨ˆ 𝑎 (𝜔) ⊗ 𝑨ˆ 𝑎∗ (𝜔) 𝑎∈𝐴  𝛾(𝜔)  𝑎 † ˆ𝑎 𝑎 †∗ ˆ 𝑎 ∗ ˆ − 𝑨 (𝜔) 𝑨 (𝜔) ⊗ 𝑰 + 𝑰 ⊗ 𝑨 (𝜔) 𝑨 (𝜔) d𝜔. (2.18) 2 The superscript 𝑶 ∗ denotes the entry-wise complex conjugation. Indeed, as required by the quantum walk formalism, this operator is Hermitian (Corollary 2.3.2) (D 𝛽 ) † = ( D 𝛽 ) assuming symmetries (2.10), (2.11). Analogously to the discriminant of classical Markov chains, the discriminant proxy is approximately the Davies-type Lindbladian (2.8) superoperator conjugated by the Gibbs state (section 2.3) but vectorized into an operator (section 2.3). This construction comes at the cost of duplicating the Hilbert space but, as a bonus, provides access to the following canonical purification of the Gibbs state ∑︁ ∑︁ √ 𝝆𝛽 ∝ e−𝛽𝐸𝑖 /2 |𝜓𝑖 ⟩ ⊗ 𝜓𝑖∗ where 𝑯 = 𝐸𝑖 |𝜓𝑖 ⟩⟨𝜓𝑖 | 𝑖 𝑖 as the (approximate) top eigenvector of D 𝛽 . The superscript 𝜓𝑖∗ denotes entrywise complex conjugate in the computational basis19. If the Hamiltonian is diagonal in the computation basis, this is essentially equivalent to the purified classical distribution. √ For general Hamiltonians, the state 𝝆 𝛽 is also known as the thermofield double state in quantum gravity (see, e.g., [122]). Taking a partial trace recovers the Gibbs state, but the purification can sometimes be more useful, e.g., for efficient verification by a swap test or faster evaluation of observables (see, e.g., [102]). Theorem 2.1.4 (Approximate purified Gibbs state). Instantiate the Lindbladian parameters of Theorem 2.1.3 with Gaussian width 𝜎𝑡 for the corresponding discriminant proxy D 𝛽 (2.18). Then, the leading eigenvector 𝜆 1 ( D 𝛽 ) well-approximates the Gibbs state   𝛽 1 e ∥ 𝜆 1 (D 𝛽 ) − 𝝆 𝛽 ∥ = O . 𝜎𝑡 𝜆 𝑔𝑎 𝑝 ( D 𝛽 ) The quantity 𝜆 𝑔𝑎 𝑝 ( D 𝛽 ) := 𝜆 1 (D 𝛽 ) − 𝜆 2 ( D 𝛽 ) is the spectral gap. 18 For convenience, we define the discriminant such that it is shifted by the identity matrix compared to definitions in earlier work. 19The above purified state is independent of which basis one applies complex conjugation to. 29 See section 2.3 for the proof of Theorem 2.1.4. The top-eigenvector error resembles the Lindbladian case (Theorem 2.1.3) with the mixing time 𝑡 𝑚𝑖𝑥 (L 𝛽 ) replaced by the inverse spectral gap 𝜆−1 𝑔𝑎 𝑝 ( D 𝛽 ). Though, unlike its Lindbladian cousin L 𝛽 , the discriminant proxy D 𝛽 does not generate a semi-group and does not by itself prepare its gapped eigenvector; we need to additionally perform quantum simulated annealing (section 2.12), which is basically adiabatic state preparation supplemented with a natural adiabatic path from high to low temperature 0 < 𝛽′ < 𝛽. This additional step uses √︃ ∼ (adiabatic path length) 𝛽∥𝑯∥ × (worst root-inverse-gap) 𝜆−1 𝑔𝑎 𝑝 queries to block-encodings of 𝑰 + D 𝛽′ across values of 𝛽′. In other words, the quantum-walk speedup√︃against Lindbladians boils down to replacing the mixing time by 𝑡 𝑚𝑖𝑥 → 𝛽∥𝑯∥ 𝜆−1 𝑔𝑎 𝑝 ; this speedup comes with the cost of doubling the number of qubits 𝑛 → 2𝑛. Of course, to perform adiabatic state preparation, we must algorithmically construct the block-encoding for 𝑰 + D 𝛽 . Lemma 2.1.3 (Efficient block-encoding). In the setting of Theorem 2.1.4, a unitary block-encoding of 𝑰 + D 𝛽 can be constructed up to 𝜖 spectral norm error using O (1) query each to 𝑷𝒓𝒆 𝒑, 𝑾, 𝑽 𝑗𝑢𝑚 𝑝 , 𝑸𝑭𝑻, 𝑭, 𝑷, and Õ (𝜎𝑡 ) (controlled) Ham. sim. time for 2𝑛 + ⌈log2 (| 𝐴|)⌉ + Õ (1) 𝑯 (resettable) qubits. See Proposition 2.3.5 for the circuit for Lemma 2.1.3. In the above, we have implicitly chosen the appropriate discretization 𝑁 (section 2.8). Note that to obtain a quantum-walk speedup, we made use of two additional (low-cost) circuit components (see section 2.3): the reflection on energy ∑︁ 𝑭 := |−𝜔⟩⟨ ¯ 𝜔| ¯ 𝜔∈𝑆 ¯ 𝜔0 and a permutation of the jump operator labels ∑︁ ′ 𝑷 := |𝑎′⟩⟨𝑎| where 𝑨𝑎 = ( 𝑨𝑎 ) † for each 𝑎 ∈ 𝐴. 𝑎∈𝐴 To grasp the algorithmic cost, we roughly expect the width along the adiabatic path to scale as 𝛽′ 𝜎𝑡 (𝛽 ) ∼ ′ 𝜖 𝜆 𝑔𝑎 𝑝 ( D 𝛽′ ) ′ for 𝜖 ′-approximated √ 𝜌 𝛽′ ; 30 see Table 2.1 and section 2.12 for an quantitative accumulated cost for the adiabatic algorithm. The remainder of the main text is organized by the analytic (section 2.2) and the algorithmic parts (section 2.3). We begin the analytic exposition by reviewing basic facts circling detailed balance and mixing time, and then introduce consequences of approximate detailed balance. The algorithmic arguments include black-box Lindbladian simulation and a general recipe to quantize a Lindbladian. These abstract algorithms can be understood assuming merely block-encodings for the Lindblad operator, whose explicit circuit construction is laid out in section 2.3. We conclude the main text by highlighting plausible future directions in section 2.4. The appendices are organized as follows. We begin with the supporting details for our key analytic and algorithmic argument: section 2.5 discusses properties of the operator Fourier Transform and the secular approximation; section 2.7 proves approximate detailed balance for the constructed Lindbladians L 𝛽 and discriminant proxies D 𝛽 . The rest of the appendices consist of isolated topics. section 2.8 discusses the relation between continuous Fourier Transforms, which is conceptually simple, and the discrete Fourier Transform, which we implement. Fortunately, the rule of thumb is that the Fourier Transform register merely needs to be poly-logarithmic for a small discretization error. section 2.9 discusses Lindbladians arising from a microscopic open system derivation and prove their fixed point accuracy; this requires a moderate generalization of the main analytic framework. section 2.10 is devoted to supporting approximate detailed balance (section 2.2), especially on perturbation theory for nonHermitian matrices; these facts tend to be intuitively akin to the Hermitian case but we include the (nonstandard) proofs for completeness. section 2.12 reviews quantum simulated annealing in a modern quantum algorithm language, which we largely employ as a black box. 2.2 Approximate stationarity of the Gibbs state We begin our analysis of the generator in Eqn. (2.8) by recalling some general properties of detailed balance Lindbladians before introducing the key notion of approximate detailed balance. At the heart of classical Markov chain Monte Carlo algorithms is a rapid mixing Markov chain whose fixed point yields the desired distribution. In the quantum setting, central to our discussion is the generator of a quantum dynamical semi-group [29, 182], the Lindbladian in the Schrödinger 31 Picture L [𝝆] = −i[𝑯, 𝝆] + ∑︁  𝑗 ∈𝐽 1 𝑳 𝑗 𝝆 𝑳 †𝑗 − {𝑳 †𝑗 𝑳 𝑗 , 𝝆} 2  parameterized by a set of Lindblad operators {𝑳 𝑗 } 𝑗 ∈𝐽 and a Hermitian matrix 𝑯. Mathematically, the above elegant form encompasses all possible Lindbladians, including, but not restricted to, those arising from a microscopic system-bath derivation. In particular, from an algorithmic perspective, we enjoy the additional freedom of choosing favorable Lindblad operators 𝑳 𝑗 with the hope that 1. the Lindbladian can be implemented efficiently, 2. the fixed point is unique and yields the desired state 𝝆 𝛽 , and 3. the Lindbladian converges rapidly. The above summarizes the desirable criteria for a quantum Gibbs sampler.20 The convergence depends on the particular Hamiltonian of interest and is generally nontrivial to analyze. Fortunately, the detailed balance condition enables systematic analysis of quantum dynamical semi-groups, similarly to how detailed balance is central in analyzing classical Markov chains (see, e.g., [110]). Definition 2.2.1 (Detailed balance condition). For a normalized, full-rank state 𝝆 ≻ 0, we say that an endomorphism L satisfies 𝝆-detailed balance whenever the associated discriminant is self-adjoint with respect to 𝝆, i.e., D ( 𝝆, L) := 𝝆 −1/4 L [𝝆 1/4 · 𝝆 1/4 ] 𝝆 −1/4 . = 𝝆 1/4 L † [𝝆 −1/4 · 𝝆 −1/4 ] 𝝆 1/4 = D ( 𝝆, L) † . In the above definition (and in the rest of the paper), we define the adjoint of a superoperator with respect to the Hilbert-Schmidt inner product ⟨𝑿, 𝒀⟩ 𝐻𝑆 = Tr( 𝑿 †𝒀). Explicitly, ∑︁ ∑︁ C [·] = 𝛼 𝑗 𝑨 𝑗 [·] 𝑩 𝑗 implies C † [·] = 𝛼∗𝑗 𝑨†𝑗 [·] 𝑩†𝑗 for any 𝑨 𝑗 , 𝑩 𝑗 𝑗 since Tr( 𝑿 † C [𝒀]) = Tr( 𝑿 † 𝑗 ∑︁ 𝑗 𝛼 𝑗 𝑨 𝑗 𝒀 𝑩 𝑗 ) = Tr(( ∑︁  † 𝛼∗𝑗 𝑨†𝑗 𝑿 𝑩†𝑗 ) †𝒀) = Tr[ C † ( 𝑿) 𝒀]. 𝑗 (2.19) 20 One can certainly consider discrete-time quantum channels [167] as Gibbs sampler candidates. However, the continuous-time Lindbladian, inspired by physics, appears technically nicer for our purposes. 32 This will be revisited when defining the vectorization (section 2.3). In particular, the superoperator adjoint for Lindbladians coincides with converting between the Heisenberg and Schrödinger pictures Tr[𝝆L † [𝑶]] = Tr[L [𝝆]𝑶] for each 𝝆⪰0 and 𝑶 using that 𝝆 † = 𝝆 and that Lindbladians L [𝝆] = (L [𝝆]) † preserves Hermiticity. Analogously to the classical case, the detailed balance condition considers a similarity transformation according to the target distribution weights 𝝆.21 The detailed balance condition brings about two desirable properties. First, it ensures that the state 𝝆 is a fixed point (Point 2). Proposition 2.2.1 (Gibbs fixed point [166]). If a superoperator L generates a trace-preserving map and satisfies 𝝆-detailed balance, then it annihilates the state √ L [𝝆] = 0, or equivalently, D ( 𝝆, L) [ 𝝆] = 0. Proof. We know that the infiniestimal exponential map e𝜀L [·] = [·] + 𝜀L [·] +
O 𝜀 2 [·] is trace-preserving, thus the leading order term must satisfy L † [𝑰] = 0. Therefore, √ √ 0 = 𝝆 1/4 L † [𝑰] 𝝆 1/4 = D ( 𝝆, L) † [ 𝝆] = D ( 𝝆, L) [ 𝝆] using 𝝆-detailed balance in the last equality. ■ Second, it relates the Lindbladian mixing time to the spectral gap (Point 3). We only state the following result here but later prove a qualitatively more robust statement in Proposition 2.2.3 applicable to the approximate case. Proposition 2.2.2 (Mixing time from spectral gap [97]). If a Lindbladian L satisfies 𝝆-detailed balance, then 𝑡 𝑚𝑖𝑥 (L) ≤ ln(2∥ 𝝆 −1/2 ∥) , 𝜆 𝑔𝑎 𝑝 (L) where 𝜆 𝑔𝑎 𝑝 (L) is the eigenvalue gap of the Lindbladian, and the mixing time 𝑡 𝑚𝑖𝑥 is the smallest time for which eL𝑡 𝑚𝑖 𝑥 [𝝆1 − 𝝆2 ] 1 ≤ 1 ∥ 𝝆1 − 𝝆2 ∥ 1 2 for any states 𝝆1 , 𝝆2 . 21Technically, an alternative definition of detailed balance may distribute the power somewhat arbitrarily 𝝆 𝑠 [·] 𝝆 1/2−𝑠 , but we stick to the symmetric case 𝝆 1/4 [·] 𝝆 1/4 for simplicity. 33 The analysis of a superoperator gap 𝜆 𝑔𝑎 𝑝 (L) is perhaps more tractable than the mixing time 𝑡 𝑚𝑖𝑥 but still nontrivial and instance specific.22 Otherwise, we see that the detailed balance condition readily addresses two criteria (Point 2 and Point 3) for Gibbs samplers. Approximate detailed balance Unfortunately, we do not know of general efficient constructions of quantum Gibbs samplers satisfying the detailed balance condition exactly (Point 1)23; this is rooted in the energy-time uncertainty principle where quantum algorithms only access the energies of a quantum system approximately. As our main technical contribution, we formulate the 𝜖-approximate detailed balance condition that addresses all three requirements for a quantum Gibbs sampler. Definition 2.2.2 (Approximate detailed balance condition). For any Lindbladian L and full-rank state 𝝆, take a similarity transformation and decompose into the Hermitian and the anti-Hermitian parts D ( 𝝆, L) = 𝝆 −1/4 L [𝝆 1/4 · 𝝆 1/4 ] 𝝆 −1/4 = H ( 𝝆, L) + A ( 𝝆, L). D ( 𝝆, L) † = 𝝆 1/4 L † [𝝆 −1/4 · 𝝆 −1/4 ] 𝝆 1/4 = H ( 𝝆, L) − A ( 𝝆, L) We say the Lindbladian L satisfies the 𝜖-approximate 𝝆-detailed balance condition if the anti-Hermitian part A is small 1 D ( 𝝆, L) − D ( 𝝆, L) † 2−2 = ∥A ( 𝝆, L)∥ 2−2 ≤ 𝜖 . 2 If the anti-Hermitian part vanishes, we recover the exact detailed balance condition D ( 𝝆, L) † = D ( 𝝆, L). If not, we show that the fixed point still approximates the state 𝝆 (Point 2). Corollary 2.2.1 (Fixed point accuracy). If a Lindbladian L satisfies the 𝜖-approximate 𝝆-detailed balance condition, then its fixed point 𝝆 𝑓 𝑖𝑥 (L) deviates from 𝝆 by at most ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆∥ 1 ≤ 20𝑡 𝑚𝑖𝑥 (L)𝜖 . See section 2.10 for the proof. We see that the error bound deteriorates if the map has a large anti-Hermitian component A or if the Lindbladian mixes slowly.24 The 22 For the experts, the gap may not give the tightest possible mixing time bounds; techniques beyond gap-based bounds typically require proving a Log-Sobolev inequality, which can be very challenging in the noncommuting cases. 23This problem is resolved in a follow-up work [42]. 24We actually prove a stronger statement in section 2.10 that gives a bound in terms of the gap 𝜆 𝑔𝑎 𝑝 (H ) of the Hermitian part. 34 anti-Hermitian component involves the inverse Gibbs state 𝝆 −1 , and might be difficult to bound directly. As a remedy, it is helpful to introduce an intermediate Lindbladian L ′ for which approximate detailed balance is easier to show. In that case, we can write ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆∥ 1 ≤ ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆 𝑓 𝑖𝑥 (L ′)∥ 1 + ∥ 𝝆 𝑓 𝑖𝑥 (L ′) − 𝝆∥ 1 . The first term on the RHS does not directly involve the inverse 𝝆 −1 and can be controlled by a Lindbladian perturbation bound as follows. Lemma 2.2.1 (Fixed point difference). For any two Lindbladians L1 and L2 , the difference of their fixed points (in the Schrödinger picture) is bounded by ∥ 𝝆 𝑓 𝑖𝑥 (L1 ) − 𝝆 𝑓 𝑖𝑥 (L2 )∥ 1 ≤ 4∥L1 − L2 ∥ 1−1 · 𝑡 𝑚𝑖𝑥 (L1 ). See section 2.10 for the proof. Conveniently, even without detailed balance, the mixing time 𝑡 𝑚𝑖𝑥 remains controlled by spectral properties of the Hermitian part (addressing Point 3): Proposition 2.2.3 (Mixing time from Hermitian gap). For any Lindbladian L and a full-rank state 𝝆, suppose the self-adjoint component H = H ( 𝝆, L) satisfies 1 𝜆 1 (H ) ≤ , 𝜆 𝑔𝑎 𝑝 (H ) 100 then 𝑡 𝑚𝑖𝑥 (L) ≤ 3 ln(3∥ 𝝆 −1/2 ∥) . 𝜆 𝑔𝑎 𝑝 (H ) See section 2.10 for the proof. In particular, the top eigenvalue can be bounded by the anti-Hermitian part 𝜆 1 (H ) ≤ ∥A ∥ 2−2 for any Lindbladian (2.76); therefore, it remains to provide an efficient construction of the Lindbladian (section 2.3) and prove approximate detailed balance (section 2.7). Proof of fixed point correctness (Theorem 2.1.3) We are now in a position to prove our first main theorem: the proximity of the stationary state 𝝆 𝑓 𝑖𝑥 to the Gibbs state 𝝆 𝛽 . Most of the technical definitions and lemmata are relegated to Appendix 2.6. Here, we address the essential features of the proof together with some essential tools. The main technical argument introduces an intermediate Lindbladian L 𝑠𝑒𝑐 ∥ 𝝆 𝑓 𝑖𝑥 (L 𝛽 ) − 𝝆 𝛽 ∥ 1 ≤ ∥ 𝝆 𝑓 𝑖𝑥 (L 𝛽 ) − 𝝆 𝑓 𝑖𝑥 (L 𝑠𝑒𝑐 )∥ 1 + ∥ 𝝆 𝑓 𝑖𝑥 (L 𝑠𝑒𝑐 ) − 𝝆 𝛽 ∥ 1 and uses the fixed point error bounds (Lemma 2.2.1) for the first term and (Proposition 2.10.3) for the second term. 35 The first error arises from the secular approximation (section 2.6), defined by truncating the transitions in the frequency domain 𝑨ˆ 𝑎 ( 𝜔) ¯ → 𝑺ˆ 𝑎 ( 𝜔) ¯ such that ⟨𝜓𝑖 | 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝜓𝑗 = 0 whenever (𝐸𝑖 − 𝐸 𝑗 ) − 𝜔¯ > 𝜇¯ for a tunable truncation parameter 𝜇. ¯ See (2.41), (2.45) for the precise definition of ¯ The purpose of this truncation is the secular approximated jump operators 𝑺ˆ 𝑎 ( 𝜔). to ensure approximate detailed balance: conjugating L 𝑠𝑒𝑐 with the Gibbs state 𝝆, as required in comparing with the similarity transformation, remains well-behaved (section 2.6). The truncation parameter 𝜇¯ is not physical but rather a proof artifact. Intuitively, our choice of Gaussian weight ensures its Fourier Transform to remain (approximately) another Gaussian (see section 2.6), which has a rapidly decaying tail. Thus, we expect the error from truncating the Gaussian tail to be small whenever 𝜇¯ ≳ 𝜎𝑡−1 . Thus, with the Gaussian weight, the secular approximation incurs a mild error; this error becomes more severe with the step-function weights given by nature, whose Fourier Transform has a heavy tail (Proposition 2.6.6). The second error is the most technical part, showing that the secular-approximated operator L 𝑠𝑒𝑐 satisfies approximate detailed balance (See section 2.7). We highlight the full technical statement as follows. Lemma 2.2.2 (Approximate detailed balance). Consider a Lindbladian in the following form L= ∑︁ 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 𝛾( 𝜔) ¯ ˆ𝑎 𝛾( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ [·] 𝑺ˆ 𝑎 ( 𝜔) ¯ †− { 𝑺 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔), ¯ ·}, 2 where 𝛾( 𝜔)/𝛾(− ¯ 𝜔) ¯ = e−𝛽𝜔¯ for each 𝜔¯ ∈ 𝑆𝜔0 . Suppose there exists 𝜇¯ ≤ 𝛽−1 such that the operators satisfy ⟨𝜓𝑖 | 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝜓𝑗 = 0 whenever (𝐸𝑖 − 𝐸 𝑗 ) − 𝜔¯ > 𝜇¯ Í for the eigenvalue decomposition of 𝑯 = 𝑗 𝐸 𝑗 |𝜓 𝑗 ⟩⟨𝜓 𝑗 |, and there is a permutation ′ 𝑷 : 𝑎 → 𝑎′ such that 𝑺ˆ 𝑎 ( 𝜔) ¯ † = 𝑺ˆ 𝑎 (−𝜔) ¯ for each 𝑎, 𝜔. ¯ Then, for the Gibbs state 𝝆 = e−𝛽𝑯 /Tr[e−𝛽𝑯 ] we have ∑︁ ∑︁ 1 © ª D ( 𝝆, L) − D ( 𝝆, L) † 2−2 ≤ O ­ 𝛽 𝜇∥ ¯ 𝛾( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)∥ ¯ ®. 2 𝑎∈𝐴 𝜔∈𝑆 ¯ 𝜔0 « ¬ 36 The above is a simplified version of Lemma 2.7.3, which we prove in section 2.7. Our normalization further simplifies the RHS to O (𝛽 𝜇). ¯ We now combine the above estimates to prove Theorem 2.1.3. Proof of Theorem 2.1.3. While Theorem 2.1.3 is stated in the continuum limit 𝑁 → ∞ (2.8), we give general error bounds at finite 𝑁 (2.21) and then take the 𝑁 → ∞ limit (2.8). Introduce the secular-approximated Lindblad operator L 𝑠𝑒𝑐 to bound the fixed point error ∥ 𝝆 𝑓 𝑖𝑥 (L 𝛽 ) − 𝝆 𝛽 ∥ 1 ≤ ∥ 𝝆 𝑓 𝑖𝑥 (L 𝛽 ) − 𝝆 𝑓 𝑖𝑥 (L 𝑠𝑒𝑐 )∥ 1 + ∥ 𝝆 𝑓 𝑖𝑥 (L 𝑠𝑒𝑐 ) − 𝝆 𝛽 ∥ 1 ≤ 2∥L 𝛽 − L 𝑠𝑒𝑐 ∥ 1−1 𝑡 𝑚𝑖𝑥 (L 𝛽 ) + 10 D ( 𝝆, L 𝑠𝑒𝑐 ) − D ( 𝝆, L 𝑠𝑒𝑐 ) † 𝑡 𝑚𝑖𝑥 (L 𝑠𝑒𝑐 )    ≤ O ∥L 𝛽 − L 𝑠𝑒𝑐 ∥ 1−1 + D ( 𝝆, L 𝑠𝑒𝑐 ) − D ( 𝝆, L 𝑠𝑒𝑐 ) † 𝑡 𝑚𝑖𝑥 (L 𝛽 ) . (2.20) The second inequality uses Lemma 2.2.1 for the first term and l Corollary 2.2.1 for the m ln(1/2) last term. The third inequality is that 𝑡 𝑚𝑖𝑥 (L 𝑠𝑒𝑐 ) ≤ 𝑡 𝑚𝑖𝑥 (L 𝛽 ) ln(1/2+𝑡 𝑚𝑖 𝑥 (L 𝛽 ) ∥L 𝛽 −L 𝑠𝑒𝑐 ∥ 1−1 ) (Proposition 2.10.4), which further simplifies to 𝑡 𝑚𝑖𝑥 (L 𝑠𝑒𝑐 ) = O (𝑡 𝑚𝑖𝑥 (L 𝛽 )) since we must have ∥L 𝛽 − L 𝑠𝑒𝑐 ∥ 1−1 𝑡 𝑚𝑖𝑥 (L 𝛽 ) = O (1) otherwise the trace distance bound becomes vacuous. Now, we evaluate approximate detailed balance (Lemma 2.2.2) and the secular approximation error using Lemma 2.6.2, Proposition 2.6.7, and that the Gaussian tail √︃Í √︁ −1 −𝑇 2 /4𝜎 2 2 𝑡 ) in the time domain is bounded directly by | 𝑡¯|≥𝑇 | 𝑓 ( 𝑡¯)| = O ( 𝑇/𝜎𝑡 e    2 2 2 2 2 2 2 2 2 2 (2.20) ≤ O 𝑇 𝜔¯ 0 + e−𝑇 /4𝜎𝑡 + e−𝑁 𝑡0 /16𝜎𝑡 + e−𝑁 𝜔0 𝜎𝑡 /2 + e− 𝜇¯ 𝜎𝑡 + 𝛽 𝜇¯ · 𝑡 𝑚𝑖𝑥 (L 𝛽 )   √︁ 𝛽 √︁ −𝑁 2 𝜔20 𝜎𝑡2 /2 ≤ O (𝜎𝑡 𝜔0 log(1/(𝜎𝑡 𝜔0 )) + log(𝜎𝑡 /𝛽) + e ) · 𝑡 𝑚𝑖𝑥 (L 𝛽 ) . 𝜎𝑡 √︁ The second inequality chooses the free parameter 𝑇 = 2𝜎𝑡 ln(1/(𝜎𝑡 𝜔0 )) and √︁ √︁ 2 2 2 2 2 2 𝜇¯ = 𝜎𝛽𝑡 ln(𝜎𝑡 /𝛽) and uses that e−𝑁 𝑡0 /16𝜎𝑡 = e−𝜋 /4𝜔0 𝜎𝑡 = O (𝜎𝑡 𝜔0 log(1/(𝜎𝑡 𝜔0 ))) to simplify the expression. For the continuum case (2.21), we have the simpler bound   𝛽 √︁ ∥ 𝝆 𝑓 𝑖𝑥 (L 𝛽 ) − 𝝆 𝛽 ∥ 1 = O log(𝜎𝑡 /𝛽) · 𝑡 𝑚𝑖𝑥 (L 𝛽 ) if 𝑁𝜔0 → ∞, 𝜔0 → 0, 𝜎𝑡 where discretization parameter 𝜔0 and 𝑁 disappears in the continuum limit. ■ 37 2.3 Quantum algorithms for Gibbs sampling In this section, we present two algorithms for approximately preparing the Gibbs state 𝝆 𝛽 , both of which are inspired by the dynamical semi-group generated by the Lindbladian L 𝛽 . This first algorithm, which we call the incoherent Gibbs sampling algorithm, directly simulates the time evolution eL 𝛽 𝑡 by introducing ancillas. The second, which we call the coherent Gibbs sampling algorithm, is a Szegedy-type quantum walk algorithm. It enables implementing an orthogonal projector onto the √ √ coherent Gibbs state | 𝝆 𝛽 ⟩⟨ 𝝆 𝛽 | with a quadratic speedup with respect to the real spectral gap of the generator L 𝛽 . This projector can then be used in conjunction with simulated annealing (section 2.12) to prepare the purified Gibbs state. In the circuit constructions, we will extensively use the following rotation gates ! √ √ ! √ 0 −i 1 − 𝜃 − 𝜃 √ √ 𝒀𝜃 := e−i arcsin 𝜃𝒀 = with the Pauli-Y matrix 𝒀 = . 𝜃 1−𝜃 i 0 Our quantum Gibbs sampling algorithms We describe two Lindbladian simulation algorithms: the first exhibits Trotter-like scaling and repeatedly uses a simple (randomized) and weak-measurement gadget (Theorem 2.3.1, Corollary 2.3.1); the second is inspired by [49] and has asymptotically almost optimal scaling with time and error Theorem 2.3.2 but requiring a more involved circuit and slightly more ancilla qubits. Both arguments are general as they assume merely a block-encoding of the Lindbladian (Definition 2.1.2); the particular block-encoding for our proposed Gibbs sampler is constructed explicitly in another section (section 2.3). Further, we “quantize” the Lindbladians and present coherent Gibbs sampling algorithms that prepare the (canonical) purification of an approximate Gibbs state via simulated annealing (section 2.12). The procedure assumes that we have a block-encoding of the discriminant matrix of our Lindbladian, which then enables a Szegedy-type quadratic speedup in the simulation time. However, the total speedup is only sub-quadratic on the gap dependence because of the cost to block-encode the discriminant matrix. Incoherent Lindbladian simulation algorithms Guided by [49], we propose two different implementation methods for incoherent (trajectory-based) simulation of the Lindbladians that describe our Gibbs sampler. The first method is based on a product formula and repeatedly uses a weak mea- 38 surement scheme25 for implementing a small time step. The resulting scaling is analogous to the performance of “vanilla” Trotter-based Hamiltonian simulation: the complexity for an 𝜖-accurate-time-𝑡 Lindbladian evolution scales as 𝑡 2 /𝜖. Our weak measurement scheme gives rise to simple and low-depth circuits for simulating Lindbladians given block-encoding access. The usefulness of weak measurements should come as no surprise, as they are also extremely helpful in other noncommutative state preparation tasks as well (see, e.g., [72]), and the very recent independent work of [53]. The common theme in these applications is the exploitation of some quantum Zeno-like effect,26 but on a higher level, these applications also show some conceptual differences. We leave it for future work to explore whether there is a more fundamental connection between our weak measurement scheme and that of [53, 72]. The second method is based on the algorithm of [49], which achieves a closeto-optimal scaling with respect to time and accuracy. Although the asymptotical complexity is much improved, the corresponding circuits are more complicated as they use a linear combination of unitaries (LCU), oblivious amplitude amplification, and advanced “compression” techniques. We leave it to future work to determine how the two schemes perform in practice. For both algorithms, it suffices to assume that a purely irreversible Lindbladian Í without the Hamiltonian term L [·] = 𝑗 ∈𝐽 𝑳 𝑗 [·] 𝑳 †𝑗 − 12 {𝑳 †𝑗 𝑳 𝑗 , ·} is provided in the form of a “block-encoding” (i.e., dilation) as Definition 2.1.2.27 In particular, recall 25 Our weak measurement scheme is very similar to the short-time evolution by the auxiliary Hamiltonian 𝐽 utilized in [49]; however, our approach is a bit more direct and made it clear that a block-encoding of the jump operators suffices as input. 26 In our case, the quantum Zeno-like effect is manifest in the quadratically reduced amplitude of 𝑐 |0 ⊥⟩ in (2.22). 27 Recent work [112, 113] assumes the Lindbladian jumps are individually block-encoded while we assume the entire set of jumps is encoded in a single unitary. We give strictly better complexity for simulating Lindbladians under this input model, which holds for our Gibbs sampling algorithm and that of [149] (leading to direct improvement for the latter). Remarkably, even if the jumps are individually block-encoded [112, 113], these can be converted to our input model. Still, even accounting for the conversion overhead, we recover (up to polylogarithmic factors) their complexity for Lindbladian simulation. The main innovation here seems to be the generalization of the input model, as the earlier Lindbladian simulation algorithms also seem to work [176] under this more general input assumption. 39 our proposed Lindbladian Gibbs sampler (as discretization of (2.8))   ∑︁ 1 ˆ𝑎 † ˆ𝑎 𝑎 𝑎 † ˆ ˆ L 𝛽 := ¯ 𝑨 ( 𝜔), ¯ ·} 𝛾( 𝜔) ¯ 𝑨 ( 𝜔) ¯ [·] 𝑨 ( 𝜔) ¯ − { 𝑨 ( 𝜔) 2 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 √︁ with Lindblad operators { 𝛾( 𝜔) ¯ 𝑨ˆ 𝑎 ( 𝜔)} ¯ 𝑎∈𝐴,𝜔∈R ¯ , (2.21) and its block-encoding can be found in Eqn. 2.41 in section 2.3. However, working with abstract block encodings makes our simulation results general and also simplifies our presentation and proofs, as the operator Fourier Transform naturally fits this definition (Figure 2.4). Our weak-measurement scheme is not only simple but also improves, e.g., the sparse Lindbladian simulation algorithm of [43, Theorem 9].28 Also, the lower bound on the “total evolution time” for simple iterative circuits in [49] suggests that the performance of similar schemes may be optimal. 𝒀𝛿 |0⟩ 0𝑏 0𝑐−𝑏 discard / reset 𝑼† 𝑼 𝝆 ≈ e𝛿L[𝝆] Figure 2.3: Quantum circuit implementation of an approximate 𝛿-time step via a weak measurement scheme. The scheme can be extended to general Lindbladians that include the coherence term −i[𝑯, 𝝆] by applying O (𝛿2 )-precise Hamiltonian time-evolution for time 𝛿 on the system register before the above circuit is applied. For example, one could use Trotterized time-evolution. (In case ∥𝑯∥ > 1, the entire Lindbladian should be first scaled down by a factor of ∥𝑯∥.) Theorem 2.3.1 (Weak-measurement for incoherent Lindbladian simulation). Suppose 𝑼 is a block-encoding of the purely irreversible Lindbladian L as in Definition 2.1.2. We can simulate the action of the superoperator e𝑡L to precision 𝜖 in diamond norm using 𝑐+1  2 O 𝑡 /𝜖   O (𝑏 + 1)𝑡 2 /𝜖  and (resettable) ancilla qubits, (controlled) uses of 𝑼, 𝑼 † , other two-qubit gates. 28 Indeed, the complexity is improved by about a factor of 𝑘 4 , where 𝑘 is the sparsity. 40 Proof. We can simulate an approximate 𝛿-time step by L using the following weak-measurement scheme displayed in Figure 2.3. 1. Apply 𝑼. √ 2. Append an ancilla qubit in state |0⟩ and rotate it with angle arcsin 𝛿 controlled on the 0𝑏 state (indicating the successful application of a jump). 3. Apply 𝑼 † controlled on the ancilla qubit being 0. 4. Measure and discard all but the system register. Assuming the system register is in the pure state |𝜓⟩, this circuit 𝑪 acts as follows: |0⟩ · |0𝑐 ⟩|𝜓⟩ (1) → |0⟩ · 𝑼|0𝑐 ⟩|𝜓⟩     √ (2) √ → 1 − 𝛿|0⟩ + 𝛿|1⟩ · |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ + |0⟩ · ( 𝑰 − |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰)𝑼|0𝑐 ⟩|𝜓⟩ √ = |0⟩ · 𝑼|0𝑐 ⟩|𝜓⟩ + 𝛿|1⟩ · 0𝑏 ( 0𝑏 ⊗ 𝑰)𝑼|0𝑐 ⟩|𝜓⟩ | {z } ′ | 𝜓0 ⟩ :=   √ − (1 − 1 − 𝛿)|0⟩ · |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩   √ √ (3) → |0⟩ · |0𝑐 ⟩|𝜓⟩ + 𝛿|1⟩ · 0𝑏 𝜓0′ − (1 − 1 − 𝛿)|0⟩ · 𝑼 † |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ √ = |0⟩ · |0𝑐 ⟩|𝜓⟩ + 𝛿|1⟩ · 0𝑏 𝜓0′ √ − (1 − 1 − 𝛿)|0⟩ · |0𝑐 ⟩(⟨0𝑐 | ⊗ 𝑰)𝑼 † ( 0𝑏 ⊗ 𝑰) · ( 0𝑏 ⊗ 𝑰)𝑼|0𝑐 ⟩|𝜓⟩   √ − (1 − 1 − 𝛿)|0⟩ · ( 𝑰 − |0𝑐 ⟩⟨0𝑐 | ⊗ 𝑰)𝑼 † |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ © ª ∑︁ √ ­ ® † 𝑳 𝑗 𝑳 𝑗 ®® |𝜓⟩ = |0⟩ · |0 ⟩ ­ 𝑰 − (1 − 1 − 𝛿) | {z } 𝑗 ∈𝐽 ­ ® 𝛿 2 +O 𝛿 ( ) « ∑︁ 2 √ ¬ √ 𝑏 | 𝑗⟩ 𝑳 𝑗 |𝜓⟩ − (1 − 1 − 𝛿) |0⟩ · |0𝑐 ⊥⟩, + 𝛿|1⟩ · 0 | {z } 𝑗 ∈𝐽 𝛿 2 2 +O ( 𝛿 ) 𝑐 ­ (2.22) where |0𝑐 ⊥⟩ is some quantum state such that ∥|0𝑐 ⊥⟩∥ ≤ 1 and (⟨0𝑐 | ⊗ 𝑰) · |0𝑐 ⊥⟩ = 0.
Tracing out the first 𝑎 + 1 qubits, we get that the resulting state is O 𝛿2 -close to the desired state. Indeed, let |𝜓 ′⟩ denote the final state above in (2.22); we now show that   ′ ′ ∥(I + 𝛿L) [|𝜓⟩⟨𝜓|] − Tr𝑐+1 [|𝜓 ⟩⟨𝜓 |] ∥ 1 = O 𝛿2 (2.23) 41 by observing that Tr𝑐+1 [|𝜓 ′⟩⟨𝜓 ′ |]     ′ ′ ′ ′ = Tr𝑐 (⟨0| ⊗ 𝑰) · |𝜓 ⟩⟨𝜓 | · (|0⟩ ⊗ 𝑰) + Tr𝑐 (⟨1| ⊗ 𝑰) · |𝜓 ⟩⟨𝜓 | · (|1⟩ ⊗ 𝑰) = ( 0𝑐+1 ⊗ 𝑰) · |𝜓 ′⟩⟨𝜓 ′ | · ( 0𝑐+1 ⊗ 𝑰)   𝑐 𝑐 ′ ′ 𝑐 𝑐 + Tr𝑐 (⟨0| ⊗ 𝑰 − |0 ⟩⟨0 |) · |𝜓 ⟩⟨𝜓 | · (|0⟩ ⊗ 𝑰 − |0 ⟩⟨0 |) ∑︁ +𝛿 𝑳 𝑗 |𝜓⟩⟨𝜓| 𝑳 †𝑗 𝑗 ∈𝐽 ! !     ∑︁ 𝛿 ∑︁ † 𝛿 = 𝑰− 𝑳 𝑳 𝑗 + O 𝛿2 |𝜓⟩⟨𝜓| 𝑰 − 𝑳 † 𝑳 𝑗 + O 𝛿2 2 𝑗 ∈𝐽 𝑗 2 𝑗 ∈𝐽 𝑗   ∑︁ + O 𝛿2 Tr𝑐 |0𝑐 ⊥⟩⟨0𝑐 ⊥| + 𝛿 𝑳 𝑗 |𝜓⟩⟨𝜓| 𝑳 †𝑗 𝑗 ∈𝐽 = |𝜓⟩⟨𝜓| + 𝛿 ∑︁ 𝑳 𝑗 |𝜓⟩⟨𝜓| 𝑳 †𝑗 − 𝑗 ∈𝐽 o   𝛿 n ∑︁ † 𝑳 𝑗 𝑳 𝑗 , |𝜓⟩⟨𝜓| + O 𝛿2 2 𝑗 ∈𝐽   = (I + 𝛿L) [|𝜓⟩⟨𝜓|] + O 𝛿2 . Convexity implies (2.23) also holds for mixed input states. To extend to the diamond norm, observe that L [·] ⊗ 𝑰[·] 𝑰 has Lindblad operators 𝑳 𝑗 ⊗ 𝑰 and therefore 𝑼 ⊗ 𝑰 is a block-encoding for L [·] ⊗ 𝑰[·] 𝑰. This implies that the trace-norm bound of (2.23) holds with respect to L [·] ⊗ 𝑰[·] 𝑰 as well, and so we can conclude that     (I + 𝛿L) [·] − Tr𝑐+1 𝑪 |0𝑐+1 ⟩⟨0𝑐+1 | ⊗ · 𝑪 † ^ = O 𝛿2 . (2.24)
The triangle inequality then implies that the implemented map is O 𝛿2 -close in
diamond distance to e𝛿L , since ∥(I + 𝛿L) − e𝛿L ∥ ^ = O 𝛿2 as shown by, e.g., [49, Appendix B].29  2 𝜖 Choosing 𝛿 = Θ( 𝑡 ) ensures that the error in a single time-step is bounded by O 𝜖𝑡 2 , 2 and repeating the process Θ( 𝑡𝜖 )-times induces an error that is bounded by 𝜖 for the 2 entire time-𝑡 evolution. The complexity is then Θ( 𝑡𝜖 )-times the cost of implementing the circuit in Figure 2.3. ■ In addition to purely irreversible Lindbladian, as noted below (Figure 2.3), the above weak measurement scheme can be amended with the Hamiltonian evolution term. 29 Here we implicitly used the fact that a block-encoded Lindbladian has norm at most one. This follows from the observation that ∥ Proposition 2.3.1. Í † 𝑗∈𝐽 𝑳 𝑗 𝑳 𝑗 ∥ ≤ 1, which is a direct consequence of 42 In our Gibbs sampling algorithm, the original random process typically proceeds by a random “jump” operator 𝑨𝑎 for a uniformly random 𝑎 ∈ 𝐴. We then obtain the final generators by applying the operator Fourier Transform to these “jump” operators. Naïvely applying our weak measurement scheme to such a Lindbladian would require us to use all the “jump” operators in each iteration. However, we show in the next corollary that it suffices to randomly pick a single “jump” operator in each iteration. In some situations, we could hope for further improvement by parallelization of these jumps if their operator Fourier Transform remains localized. Corollary 2.3.1 (Improved randomized simulation for convex combinations of Í Lindbladians). Suppose that a purely irreversible Lindbladian L [·] = 𝑖 𝑝𝑖 L𝑖 [·] is a convex combination of the purely irreversible Lindbladians L𝑖 [·] which are given by their respective block-encodings. In Theorem 2.3.1, we can replace each weak-measurement gadget for L [·] by an independently sampled weak-measurement gadget for L𝑖 [·] according to the distribution 𝑝𝑖 while keeping the same asymptotic
iteration count O 𝑡 2 /𝜖 . Proof. It suffices to show that     ∑︁ † 𝛿L 𝑐+1 𝑐+1 ∥e [·] − 𝑝𝑖 Tr𝑐+1 𝑪𝑖 |0 ⟩⟨0 | ⊗ [·] 𝑪𝑖 ∥ ^= O 𝛿2 . (2.25) 𝑖 From (2.24) we know that the weak measurement gadget 𝑪𝑖 in Figure 2.3 for L𝑖† [·] satisfies     ∥(I + 𝛿L𝑖 ) [·] − Tr𝑐+1 𝑪𝑖 |0𝑐+1 ⟩⟨0𝑐+1 | ⊗ [·] 𝑪𝑖† ∥ ^ = O 𝛿2 . By linearity and the triangle inequality, it follows that     ∑︁ ∑︁ ∥(I + 𝛿 𝑝𝑖 L𝑖 ) [·] − 𝑝𝑖 Tr𝑐+1 𝑪𝑖 |0𝑐+1 ⟩⟨0𝑐+1 | ⊗ [·] 𝑪𝑖† ∥ ^ = O 𝛿2 . 𝑖 𝑖
Since ∥(I + 𝛿L) − e𝛿L ∥ ^ = O 𝛿2 , by the triangle inequality, we get the sought inequality in (2.25). ■ Now, we turn to our second incoherent simulation result that is roughly based on the algorithm of [49] but contains further improvements and fixes. We obtain improved complexity because we assume that the Lindbladian is provided via a block-encoding, while effectively [49] construct a (potentially suboptimal) block-encoding within their Í algorithm. Their complexity depends on 𝑗 ∈𝐽 ∥ 𝑳 †𝑗 𝑳 𝑗 ∥, while our algorithm can in 43 principle achieve a dependence like ∥ 𝑗 ∈𝐽 𝑳 †𝑗 𝑳 𝑗 ∥ when an efficient block-encoding is provided — which is the case for our explicit block-encodings outlined in the next section (section 2.3). To our knowledge, this is the first Lindbladian simulation algorithm that achieves both near-linear time dependence and a complexity that Í scales with ∥ 𝑗 ∈𝐽 𝑳 †𝑗 𝑳 𝑗 ∥. Note that this improvement looks similar to how [12] improved over [92] on the complexity of estimating multiple expectation values, but the techniques are very different. Here, the improvement stems from the following efficient block-encoding construction. Í Proposition 2.3.1. Given a block-encoding of a Lindbladian (Definition 2.1.2), we get a block-encoding of   ∑︁ 𝑳 †𝑗 𝑳 𝑗 via 𝑽 := (𝒀 1 ⊗ 𝑼 † ) · 2|0𝑏+1 ⟩⟨0𝑏+1 | ⊗ 𝑰 − 𝑰 · (𝒀 1 ⊗ 𝑼), 2 𝑗 ∈𝐽 2 √ where |±⟩ := (|0⟩ ± |1⟩)/ 2. Proof. We calculate ( 0𝑐+1 ⊗ 𝑰) · 𝑽 · ( 0𝑐+1 ⊗ 𝑰)       𝑐 † 𝑏+1 𝑏+1 𝑐 = ⟨−| ⊗ (⟨0 | ⊗ 𝑰)𝑼 · 2|0 ⟩⟨0 | ⊗ 𝑰 − 𝑰 · |+⟩ ⊗ 𝑼(|0 ⟩ ⊗ 𝑰)       𝑐 𝑐 † 𝑏+1 𝑏+1 = ⟨−| ⊗ (⟨0 | ⊗ 𝑰)𝑼 · 2|0 ⟩⟨0 | ⊗ 𝑰 · |+⟩ ⊗ 𝑼(|0 ⟩ ⊗ 𝑰) = (⟨0𝑐 | ⊗ 𝑰) · 𝑼 † · (|0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰) · 𝑼 · (|0𝑐 ⟩ ⊗ 𝑰) ! ! ∑︁ ∑︁ ∑︁ † ′ ′ ⟩ | ⟨ | 𝑳 †𝑗 𝑳 𝑗 . 𝑗 ⊗ 𝑳𝑗 = = 𝑗 ⊗ 𝑳𝑗 𝑗 ∈𝐽 𝑗 ′ ∈𝐽 ■ 𝑗 ∈𝐽 This block-encoding construction and the following generic Lindbladian simulation algorithm answers an open question30 recently posed by Rall, Wang, and Wocjan [149], and can significantly improve their complexity. Although we do not use the above block-encoding explicitly, this observation is implicitly used in our weak measurement schemes (Figure 2.3-Figure 2.10) that enable us to prove the following result, whose proof is presented in section 2.11. Theorem 2.3.2 (Compressed incoherent Lindbladian simulation algorithm). Suppose 𝑼 is a block-encoding of the Lindblad operators of a purely irreversible Lindbladian 30 See [149, Section 7] “That one special Kraus operator involves all the 𝐿 ’s. Does there exist any 𝑗 specialÍtreatment of this special Kraus operator so that we can leverage the special structure of the oracle 𝑗 | 𝑗⟩ ⊗ 𝐿 𝑗 to get rid of the O (𝑚) dependence?” 44 L as in Definition 2.1.2. Let 𝜖 ≤ 1/2, then we can simulate the action of the superoperator e𝑡L to precision 𝜖 in diamond norm using O ((𝑐 + log((𝑡 + 1)/𝜖)) log((𝑡 + 1)/𝜖))   log((𝑡 + 1)/𝜖) O (𝑡 + 1) log log((𝑡 + 1)/𝜖) and O ((𝑡 + 1)(𝑐 + 1)polylog((𝑡 + 1)/𝜖)) (resettable) ancilla qubits, (controlled) uses of 𝑼 and 𝑼 † , other two-qubit gates. If the Lindbladian has a coherent part −i[𝑯, 𝝆], and we have access  to a block log((𝑡+1)/𝜖) 𝑐 𝑐 𝑡L encoding of 𝑯 = (⟨0 |⊗𝑰)𝑽 (|0 ⟩⊗𝑰), then we can simulate e with O (𝑡 + 1) log log((𝑡+1)/𝜖) additional (controlled) uses of 𝑽 and 𝑽 † . Crucially, the complexity scales almost linearly with time 𝑡 and poly-logarithmic with the precision 𝜖 while using very few ancillas, representing a large asymptotic speedup compared to the 𝑡 2 /𝜖 complexity of the weak-measurement scheme (Theorem 2.3.1). Coherent Lindbladian simulation algorithms With a quantum computer, we further ask for a coherent Gibbs sampler that outputs Í √ the purified distribution ( 𝝆 𝛽 ∝ 𝑖 e−𝛽𝐸𝑖 /2 |𝜓𝑖 ⟩ ⊗ 𝜓𝑖∗ ) on two copies of the Hilbert space. A desirable coherent Gibbs sampler should satisfy the following conditions: 1. A Hermitian operator C can be efficiently block-encoded on the duplicated Hilbert space, 2. its top-eigenvector is unique and yields the purified state √ 𝝆 , and 3. there exists an adiabatic path of operators C(𝑠) whose top-eigenvalue-gap remains open. A general coherent Gibbs sampler may not refer to an existing Lindbladian. However, a natural candidate of the operator C is to take the vectorized discriminant D( 𝝆, L) † associated with a detailed balance Lindbladian (Definition 2.2.1), as how one quantizes classical Markov chains [166]. Formally, we define vectorization of a superoperator by31 ∑︁ ∑︁ C [·] = 𝛼 𝑗 𝑨 𝑗 [·] 𝑩 𝑗 → C = 𝛼 𝑗 𝑨 𝑗 ⊗ 𝑩𝑇𝑗 (vectorization), 𝑗 𝑗 31 One might be tempted to use 𝑩† instead in the vectorization, but that definition leads to inconsistencies. Indeed, if we would use 𝑩† for vectorization then the two different representations of the scalar 1 ⊗ 1 = 1 = i ⊗ −i would lead to different vectorizations ±1. 45 where 𝑩𝑇𝑗 denotes the transpose of the matrix 𝑩 𝑗 in the computational basis |𝑖⟩. We use curly fonts C for superoperators and bold fonts C for the vectorized superoperators (which is, a matrix).32 For a matrix 𝑨, let us denote its vectorized (or purified) version by | 𝑨⟩ := (𝐼 ⊗ 𝑇 −1 ) 𝑨 (purification) using the “transpose” map 𝑇 |𝑖⟩ = ⟨𝑖|. This automatically ensures the correctness of the fixed point (Point 2). Proposition 2.3.2. For any full-rank state 𝝆 and any Lindbladian L, we have that √ D( 𝝆, L) † 𝝆 = 0. Further, if L satisfies 𝝆-detailed balance, we also have that √ D( 𝝆, L) 𝝆 = 0. The above follows from a direct calculation using that any Lindbladian is tracepreserving L † [𝑰] = 0. However, to turn the above into the advertised efficient algorithm (Theorem 2.1.4), we need two key components reminiscent of the incoherent case. First, we need a proxy for the discriminant with efficient block-encoding. As we mentioned (2.18), we consider ∑︁ √︁ D𝛽 = 𝛾( 𝜔)𝛾(− ¯ 𝜔) ¯ 𝑨ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑨ˆ 𝑎 ( 𝜔) ¯ ∗ (2.26) 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0  𝛾( 𝜔) ¯  ˆ𝑎 𝑨 ( 𝜔) ¯ † 𝑨ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑰 + 𝑰 ⊗ 𝑨ˆ 𝑎 ( 𝜔) ¯ ∗† 𝑨ˆ 𝑎 ( 𝜔) ¯ ∗ 2 as a proxy for D( 𝝆, L 𝛽 ) † , − where the 𝑨ˆ 𝑎 ( 𝜔) ¯ are the same operator Fourier Transforms as in the incoherent case (2.7); the block-encoding for D can be obtained given the block-encoding for 𝑨ˆ 𝑎 ( 𝜔) ¯ (section 2.3). The map is self-adjoint D 𝛽 = D†𝛽 due to Hermiticity 𝑨𝑎 = 𝑨𝑎† and properties of weighted Fourier Transform (section 2.5). More carefully, we do not implement exactly the discriminant D 𝛽 , but merely an approximation Di𝑚 𝑝𝑙 ≈ D 𝛽 due to additional implementation errors for the Gaussian weight and truncation errors for the Gaussian tail. Second, we need to formulate a notion of approximate detailed balance for the above discriminant proxy. ∗ † ∗ 𝑗 𝛼𝑗 𝑨𝑗 ⊗ 𝑩𝑗. † † ∗ 𝑗 𝛼 𝑗 𝑨 𝑗 [·] 𝑩 𝑗 (2.19), whose vectorization is 32 Note that C † is well defined. The (matrix) adjoint of the vectorized operator is Í On the other hand, the superoperator adjoint C † [·] is Í then the same 𝑗 𝛼∗𝑗 𝑨†𝑗 ⊗ 𝑩∗𝑗 . Í 46 Definition 2.3.1 (𝜖-Discriminant proxy). We say a Hermitian matrix D is an 𝜖-discriminant proxy for Lindbladian L and a full-rank state 𝝆 if ∥ D − D( 𝝆, L) † ∥ ≤ 𝜖 . Indeed, this implies approximate detailed balance D ( 𝝆, L) † ≈ D ( 𝝆, L) for the Lindbladian L by taking the adjoints. We can think of Definition 2.3.1 as a different form of the approximate detailed balance condition (Definition 2.2.2) that controls the top eigenvector error up to the spectral gap (Point 2). Proposition 2.3.3 (Fixed point error). Suppose a gapped Hermitian operator D is an 𝜖-discriminant proxy for a Lindbladian L and a full-rank state 𝝆. Then, its top √ eigenvector is approximately the purified state 𝝆 |𝜆 1 ( D)⟩ − √ 𝝆 √ ≤4 2 𝜖 . 𝜆 𝑔𝑎 𝑝 ( D) Here, the gap dependence naturally arises from eigenvalue (Corollary 2.10.1) and eigenvector perturbation (Proposition 2.10.2) arguments. Unlike Lindbladians, the cost for preparing the coherent Gibbs state scales directly with the gap via quantum simulated annealing; the mixing time of the original Lindbladian is not linked directly to the algorithmic cost. Thirdly, the block-encoding by itself does not prepare the desired top eigenvector, unlike a Lindbladian; this additionally requires a standard subroutine called quantum simulated annealing [179, 186]: adiabatically change the inverse temperature from 𝛽′ = 0 → 𝛽′ = 𝛽. The algorithmic cost is associated with the gaps along the adiabatic path [24]; see section 2.12. Explicit block-encodings In this section, we lay out the circuit ingredients to construct the advertised Lindbladians L 𝛽 and discriminants D 𝛽 . First, we show how to construct a blockencoding of the discretized Lindbladian with Lindblad operators (2.16) from a block-encoding of the jump operators 𝑨𝑎 . Then, we further construct a blockencoding of the corresponding discriminant proxy - with the additional assumption that the set of jump operators is self-adjoint ({ 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴}) and the Fourier weight function 𝑓 is real. It is not surprising that implementing the discriminant proxy requires more symmetry constraints, as its definition already implicitly draws from these symmetries. 47 Block-encoding L 𝛽 For both incoherent algorithms (Theorem 2.3.1,Theorem 2.3.2), we have assumed Í that a purely-irreversible Lindbladian L [·] = 𝑗 ∈𝐽 𝑳 𝑗 [·] 𝑳 †𝑗 + {𝑳 †𝑗 𝑳 𝑗 , ·} is given by a unitary block-encoding 𝑼 (Definition 2.1.2). Here, we construct a block-encoding √︁ unitary 𝑼 for the advertised Lindblad operators 𝛾( 𝜔) ¯ 𝑨ˆ 𝑎 ( 𝜔) ¯ labeled by 𝑎, 𝜔: ¯ ∑︁ √︁ 𝛾( 𝜔)| ¯ 𝜔⟩ ¯ ⊗ |𝑎⟩ ⊗ 𝑨ˆ 𝑎 ( 𝜔) ¯ 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 for the advertised Lindbladian (discretization of (2.8), recap of (2.21)) .   ∑︁ 1 ˆ𝑎 † ˆ𝑎 𝑎 𝑎 † ˆ ˆ L 𝛽 := 𝛾( 𝜔) ¯ 𝑨 ( 𝜔) ¯ 𝑨 ( 𝜔), ¯ ·} . ¯ [·] 𝑨 ( 𝜔) ¯ − { 𝑨 ( 𝜔) 2 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 We begin by laying out the registers explicitly, including the additional ancillae for block-encoding. |0⟩ |{z} registers: ⊗ Boltz. weight | 𝜔⟩ ¯ ⊗ |{z} Bohr freq. 0𝑏 |{z} block. enc. anc. ⊗ |𝑎⟩ ⊗ 𝑨ˆ 𝑎 ( 𝜔) ¯ |{z} | {z } jump system From right to left, the registers individually correspond to: the physical system of interest; the jump labels |𝑎⟩ and additional ancillae to accommodate block-encoding access (indicating successful application by the all-zero state 0𝑏 ); the frequency register is dedicated to the operator Fourier Transform, storing the weight | 𝑓 ⟩ or the Bohr frequencies | 𝜔⟩; ¯ finally, an ancilla qubit for storing the Bohr-frequency √︁ √︁ dependent Boltzmann weights in the amplitudes 𝛾( 𝜔)|0⟩ ¯ + 1 − 𝛾( 𝜔)|1⟩. ¯ We specify the discrete Fourier Transform parameters that determine the dimension of the Bohr frequency register as follows. The Fourier frequencies 𝜔¯ and times 𝑡¯ are integer multiples of 𝜔0 and 𝑡0 respectively such that   2𝜋 ⌈𝑁⌋ 𝜔0 𝑡 0 = , and 𝑆 := − ⌈(𝑁 − 1)/2⌉, . . . , −1, 0, 1, . . . , ⌊(𝑁 − 1)/2⌋ , 𝑁 and 𝑆𝜔⌈𝑁⌋ := 𝜔0 · 𝑆 ⌈𝑁⌋ , 0 𝑆𝑡⌈𝑁⌋ := 𝑡0 · 𝑆 ⌈𝑁⌋ . 0 We use a “bar” to denote discretized variables; the (Bohr) frequency register takes values 𝜔¯ ∈ 𝑆𝜔0 . To implement the Fourier Transform when 𝑁 = 2𝑛 , we specify the signed binary representation for the integers 𝑆 ⌈𝑁⌋ as follows: 10𝑛−1 , . . . , 1𝑛 for each 0𝑛 , . . . , 01𝑛−1 for each 0, 1, . . . , 𝑁/2 − 1. − 𝑁/2, . . . , −1, 48 𝒀1−𝛾(𝜔) ¯ |0⟩ 0̄ 𝑷𝒓𝒆 𝒑 |0⟩ | 𝜔⟩ ¯ QFT 0𝑏 0𝑏 0𝑐−𝑏 |𝑎⟩ 𝑽𝑗 𝑝 𝝆 e−i𝑯𝑡¯ ei𝑯𝑡¯ 𝛾( 𝜔) ¯ 𝑨ˆ 𝑎 ( 𝜔) ¯ 𝝆 𝑨ˆ 𝑎 ( 𝜔) ¯ † Figure 2.4: Circuit 𝑼 for block-encoding the Lindbladian. Practically, if we use the simpler weak-measurement-based simulation (Theorem 2.3.1), then by Corollary 2.3.1, we can use a single randomly chosen Lindblad operator 𝑨𝑎 at a time. Moreover, if 𝑨𝑎 is unitary, we can simply replace 𝑽 𝑗𝑢𝑚 𝑝 with 𝑨𝑎 , implying 𝑏 = 𝑐 = 0, i.e., the third and the forth registers can be omitted, thus 𝑛 + ⌈log(𝑁)⌉ + 2 qubits suffice to simulate the Lindbladian eL𝑡 . Where it does not cause confusion we will drop ⌈𝑁⌋ from the superscript and will simply write 𝜔¯ ∈ 𝑆𝜔0 and 𝑡¯ ∈ 𝑆𝑡0 . We will set the value of 𝑁 and 𝜔0 such that the Bohr frequencies 𝐵 = spec(𝑯) − spec(𝑯) are contained within the range of energies ∥𝑯∥ ≤ 𝑁 𝜔0 . 2 The only “physical” energy scale in the above is the Hamiltonian strength ∥𝑯∥. Indeed, setting the scale to 𝑁 requires only log(𝑁) qubits in the readout register. The circuit consists of the following ingredients, each acting on some appropriate subset of the registers: • Block-encoding 𝑽 𝑗𝑢𝑚 𝑝 of the jump operators 𝑨𝑎 in the form of Definition 2.1.2: ∑︁ |𝑎⟩ ⊗ 𝑨𝑎 . ( 0𝑏 ⊗ 𝑰𝑎 ⊗ 𝑰 𝑠𝑦𝑠 )] · 𝑽 𝑗𝑢𝑚 𝑝 · (|0𝑐 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 ) = (2.27) 𝑎∈𝐴 The operators 𝑨𝑎 need not be self-adjoint nor proportional to a unitary. Still, √︁ one may conveniently choose | 𝐴| 𝑨𝑎 to be unitary for all 𝑎 ∈ 𝐴, (e.g., few-body unitary operators). Then, we can set 𝑏 = 0 and choose ! ∑︁ ∑︁ |𝑎⟩ 𝑽 𝑗𝑢𝑚 𝑝 = |𝑎⟩⟨𝑎| ⊗ 𝑨𝑎 · (𝑩 ⊗ 𝑰 𝑠𝑦𝑠 ) where 𝑩|0𝑐 ⟩ = . √︁ (2.28) | 𝐴| 𝑎∈𝐴 𝑎∈𝐴 49 Note that implementing the Lindbladian does not require the set of jump operators to contain the adjoints { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴}; this assumption is only used for approximate detailed balance and the fixed point correctness (Theorem 2.1.3). • Controlled Hamiltonian simulation ∑︁ | 𝑡¯⟩⟨𝑡¯| ⊗ 𝑒 ±i𝑡¯𝑯 . 𝑡¯∈𝑆𝑡0 • Quantum Fourier Transform 1 ∑︁ −i𝜔¯ 𝑡¯ | 𝜔⟩. 𝑸𝑭𝑻𝑁 : | 𝑡¯⟩ → √ e ¯ 𝑁 𝜔∈𝑆 ¯ 𝜔 0 • State preparation unitary for the Fourier Transform weights, acting on the frequency register 𝑷𝒓𝒆 𝒑 𝑓 such that 𝑷𝒓𝒆 𝒑 𝑓 0̄ = | 𝑓 ⟩. Naturally, the weight 𝑓 ( 𝑡¯) as amplitudes of a state is normalized ∑︁ | 𝑓 ( 𝑡¯)| 2 = 1. 𝑡¯∈𝑆𝑡0 It could be, e.g., an easily preparable step function or a Gaussian whose tail decays rapidly. Gaussian states are attractive because they are relatively easy to prepare [127], but as a matter of fact, any other so-called window function could be used, such as the Kaiser-window [20, 127] potentially providing further overhead improvements. • Controlled filter for the Boltzmann factors acting on the frequency register and the Boltzmann weight register ∑︁ 𝑾 := 𝒀1−𝛾(𝜔) ¯ 𝜔| ¯ where 0 ≤ 𝛾( 𝜔) ¯ ≤ 1 and 𝛾( 𝜔) ¯ = 𝛾(−𝜔)e ¯ −𝛽𝜔¯ . ¯ ⊗ | 𝜔⟩⟨ 𝜔∈𝑆 ¯ 𝜔0 The constraint 0 ≤ 𝛾( 𝜔) ¯ ≤ 1 ensures the matrix 𝒀1−𝛾(𝜔) ¯ is unitary; the −𝛽 𝜔 ¯ symmetry (i.e., the KMS condition) 𝛾( 𝜔) ¯ = 𝛾(−𝜔)e ¯ gives lower weights for “heating” transitions and is closely related to the detailed balance condition. Important examples of weight functions are (Metropolis) 𝛾( 𝜔) ¯ = min(1, e−𝛽𝜔¯ ) and (Glauber) 𝛾( 𝜔) ¯ = 1 , e 𝛽𝜔¯ + 1 50 which both reduce to the step function in the 𝛽 → ∞ limit. Note that the range of energy labels 𝜔¯ is finite; we choose a large energy readout range 𝑁𝜔0 ≥ 4∥𝑯∥ + 2𝛽 to ensure all possible transitions are covered by the discretization range (after secular approximation 𝜇 ≤ 1𝛽 ). We may generally synthesize the controlled filter from elementary gates at cost33 (polynomial degree of 𝛾) × Poly(log(𝑁), log(1/𝜖)), which is Õ (1 + 𝛽∥𝑯∥) for the Glauber weight. For the Metropolis weight, one can achieve the same scaling by manually switching between 1 and e−𝛽𝜔¯ at 𝜔¯ = 0. In principle, since we are merely controlling a qubit, we may directly implement any efficient computable function (perhaps with terrible overhead). Further, combining the controlled Hamiltonian simulation and Quantum Fourier Transform yields the advertised operator Fourier Transform (Figure 2.5) acting on the frequency and system register 34 ∑︁ | 𝜔⟩ F [·] : | 𝑓 ( 𝑡¯)⟩ ⊗ 𝑶 → ¯ ⊗ 𝑶ˆ 𝑓 ( 𝜔) ¯ 𝜔∈𝑆 ¯ 𝜔0 where 1 ∑︁ −i𝜔¯ 𝑡¯ 𝑶ˆ 𝑓 ( 𝜔) ¯ := √ e 𝑓 ( 𝑡¯)𝑶 ( 𝑡¯) 𝑁 𝑡¯∈𝑆𝑡 and 𝑶 (𝑡) := ei𝑯𝑡 𝑶e−i𝑯𝑡 . 0 See section 2.5 for basic properties of the operator Fourier Transform. Our implementation is inspired by, but differs from [180]; they sandwich the jump operators with phase estimation and its inverse. The operator picture, inspired by physics, is more natural and tangible. Our construction allows for flexibility in the choice of the weight function 𝑓 ( 𝑡¯). As shown in Figure 2.4, we assemble the above ingredients to obtain the unitary 𝑼 such that
𝑼( 𝑰 ⊗ 0̄ ⊗ 𝑰𝑽 ) = (𝑾 ⊗ 𝑰𝑽 ) · 𝑰 ⊗ F [𝑷𝒓𝒆 𝒑 𝑓 0̄ ⊗ 𝑽 𝑗𝑢𝑚 𝑝 ] , (2.29) where 𝑰 is the single qubit identity and 𝑰𝑽 is the identity on the registers on which 𝑽 𝑗𝑢𝑚 𝑝 acts. As described in Footnote 34, intuitively speaking, we only apply the operator Fourier Transform on the “system” register; indeed, in the corresponding 51 𝑷𝒓𝒆 𝒑 0̄ 𝑸𝑭𝑻 e−i𝑯𝑡¯ 𝝆 𝑶 ei𝑯𝑡¯ | 𝜔⟩ ¯ 𝑶 𝜔¯ 𝝆(𝑶 𝜔¯ ) † Figure 2.5: Circuit for operator Fourier Transform F for an operator 𝑶 acting on the system 𝝆. Of course, in our use, the operator may also act nontrivially on other ancillas. circuit (Figure 2.5), the jump label register |𝑎⟩ and the block-encoding ancillae 0𝑏 is only affected by 𝑽 𝑗𝑢𝑚 𝑝 . Lemma 2.3.1 (Explicit Block-encoding). The untiary 𝑼 in (2.29) gives a blockencoding for our Lindbladian (2.8)  
𝑏 ⟨0| ⊗ 𝑰𝜔¯ ⊗ 0 ⊗ 𝑰𝑎 ⊗ 𝑰 𝑠𝑦𝑠 · 𝑼 · |0⟩ ⊗ 0̄ ⊗ |0𝑐 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 ∑︁ √︁ 𝛾( 𝜔)| ¯ 𝜔⟩ ¯ ⊗ |𝑎⟩ ⊗ 𝑨ˆ 𝑎𝑓 ( 𝜔). = ¯ (2.30) 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 Proof. (LHS of (2.30))  

= ⟨0| ⊗ 𝑰𝜔¯ ⊗ 0𝑏 ⊗ 𝑰𝑎 ⊗ 𝑰 𝑠𝑦𝑠 · (𝑾 ⊗ 𝑰𝑽 ) · |0⟩ ⊗ F [| 𝑓 ⟩ ⊗ 𝑽 𝑗𝑢𝑚 𝑝 ] |0𝑐 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 (by (2.29)) = ((⟨0| ⊗ 𝑰𝜔¯ )𝑾 (|0⟩ ⊗ 𝑰𝜔¯ ) ⊗ 𝑰𝑎 ⊗ 𝑰 𝑠𝑦𝑠 ) · = ∑︁   𝑰𝜔¯ ⊗ 0𝑏 ⊗ 𝑰𝑎 ⊗ 𝑰 𝑠𝑦𝑠 F [| 𝑓 ⟩ ⊗ 𝑽 𝑗𝑢𝑚 𝑝 ] |0𝑐 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 (⟨0|𝒀1−𝛾(𝜔) ¯ 𝜔| ¯ ⊗ 𝑰𝑎 ⊗ 𝑰 𝑠𝑦𝑠 ) · (F [| 𝑓 ⟩ ⊗ |𝑎⟩ ⊗ 𝑨𝑎 ]) ¯ |0⟩ ⊗ | 𝜔⟩⟨ 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 (by (2.27) and Figure 2.5) = ∑︁ √︁ ¯ 𝛾( 𝜔)| ¯ 𝜔⟩ ¯ ⊗ |𝑎⟩ ⊗ 𝑨ˆ 𝑎𝑓 ( 𝜔). (by OFT and controlled filter) ■ 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 33 By first applying controlled Hamiltonian simulation for 𝒀 rotation Í √︁ 𝜔¯ e i( 𝜔/2∥ ¯ 𝑯 ∥ )𝒀 ⊗ | 𝜔⟩⟨ ¯ 𝜔| ¯ and then apply QSVT to map 𝜔/2∥𝑯∥ ¯ → arcsin( 1 − 𝛾( 𝜔)). ¯ This is reminiscent of [79]. 34 If the operator 𝑶 maps between larger Hilbert spaces than 𝑯, we formally extend its action trivially so that 𝑶 (𝑡) = ( 𝑰 ⊗ ei𝑯 𝑡 )𝑶 ( 𝑰 ′ ⊗ e−i𝑯 𝑡 ). Of course, this formal extension does not incur any additional cost.
 52 Block-encoding D 𝛽 We now describe the explicit and efficient circuit that implements the advertised vectorized discriminant proxy (discretization of (2.18), recap of (2.26)) ∑︁ √︁ D𝛽 = 𝛾( 𝜔)𝛾(− ¯ 𝜔) ¯ 𝑨ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑨ˆ 𝑎 ( 𝜔) ¯ ∗ 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0  𝛾( 𝜔) ¯  ˆ𝑎 † ˆ𝑎 𝑎 ∗† ˆ 𝑎 ∗ ˆ − 𝑨 ( 𝜔) ¯ 𝑨 ( 𝜔) ¯ ⊗ 𝑰 + 𝑰 ⊗ 𝑨 ( 𝜔) ¯ 𝑨 ( 𝜔) ¯ , 2 assuming that the set of jump operators is self-adjoint in the sense that { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴} and the Fourier weight function 𝑓 is real. Combining the circuit with simulated annealing then leads to the advertised quadratic speedup. Our discriminant proxy and its block-encoding is an instantiation of the following general construct that is self-adjoint as a superoperator (and hence Hermitian after vectorization)35. Proposition 2.3.4 (Self-adjoint discriminant proxies). Given a purely irreversible Lindbladian with Lindblad operators 𝑳 𝑗 for 𝑗 ∈ 𝐽, and a permutation 𝑷 : 𝑗 → 𝑗 ′ on the set 𝐽, the following superoperator (and hence its vectorization) is self-adjoint: 1 ∑︁ 𝑳 𝑗 [·] 𝑳 †𝑗 ′ + 𝑳 †𝑗 [·] 𝑳 𝑗 ′ − {𝑳 †𝑗 𝑳 𝑗 , ·}, 2 𝑗 ∈𝐽 1 ∑︁ ∗† ∗ † † ∗ D(𝑷, {𝑳 𝑗 }) = 𝑳 𝑗 ⊗ 𝑳 ∗† 𝑗′ + 𝑳 𝑗 ⊗ 𝑳 𝑗′ − 𝑳 𝑗 𝑳 𝑗 ⊗ 𝑰 − 𝑰 ⊗ 𝑳 𝑗 𝑳 𝑗 . 2 𝑗 ∈𝐽 D (𝑷, {𝑳 𝑗 }) := Proof. D (𝑷, {𝑳 𝑗 }) † = 1 ∑︁ † 𝑳 [·] 𝑳 𝑗 ′ + 𝑳 𝑗 [·] 𝑳 †𝑗 ′ − {𝑳 †𝑗 𝑳 𝑗 , ·} = D (𝑷, {𝑳 𝑗 }). 2 𝑗 ∈𝐽 𝑗 ■ The vectorization is hence also self-adjoint. Corollary 2.3.2. If 𝑓 is real (2.11) and the set of jump operators is self-adjoint (2.10), then the discriminant proxy (2.26) is Hermitian. Proof. Due to footnote 32, it suffices to verify that the superoperator is self-adjoint: This follows from Proposition 2.3.4 by setting the permutation 𝑷 : ( 𝜔, ¯ 𝑎) → (−𝜔, ¯ 𝑎′)  Í Í  −1 ∗† 1 1 † ∗ ∗ −1 we additionally have 𝑗 ∈ 𝐽 2 𝑳 𝑗 ⊗ 𝑳 𝑗′ + 2 𝑳 𝑗′ ⊗ 𝑳 𝑗 ≈ 𝑗∈𝐽 𝝆 4 ⊗ (𝝆 ) 4 𝑳 𝑗 ⊗  1  Í Í 1 1 1 𝑳 ∗𝑗 𝝆 4 ⊗ ( 𝝆 ∗ ) 4 and 𝑗 ∈ 𝐽 𝑳 †𝑗 𝑳 𝑗 ≈ 𝑗 ∈ 𝐽 𝝆 4 𝑳 †𝑗 𝑳 𝑗 𝝆 − 4 , then the resulting discriminant proxy 35 If D is close to the discriminant D( 𝝆, L) † . This is exactly what we show in 2.6 for the discriminant proxy (2.26). 53 ′ ′ such that 𝑨𝑎† = 𝑨𝑎 and using the operator Fourier Transform property 𝑨ˆ 𝑎 (−𝜔) ¯ †= ′ ′ 𝑨ˆ𝑎 † ( 𝜔) ¯ for real weight 𝑓 (2.39), implying that 𝑨ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑨ˆ 𝑎 (−𝜔) ¯ †∗ = 𝑨ˆ 𝑎 ( 𝜔) ¯ ⊗ ′ 𝑎 ∗ 𝑎 † 𝑎 ∗ 𝑨ˆ ( 𝜔) ¯ = 𝑨ˆ (−𝜔) ¯ ⊗ 𝑨ˆ ( 𝜔) ¯ . ■ Now that we have verified the symmetries of the desired discriminant proxy, we move on to our explicit construction. By the standard quantum walk recipe [164], we design an isometry and a reflection such that ′ 𝑰 + D(𝑷, {𝑳 𝑗 }) = 𝑻 † 𝑹𝑻 ′, which is block-encoded as in Figure 2.6. Proposition 2.3.5 (A block-encoding for discriminant proxies). Using the notation of Proposition 2.3.4, let ′ ′ 𝑹 := 𝑰 − ( 𝑰 ⊗ 𝚷) + 𝒁 ⊗ |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑷 ⊗ 𝑰 𝑠𝑦𝑠 ⊗ 𝑰 𝑠𝑦𝑠′ | {z } =:𝑹0 where 𝑏′ 𝑏′ 𝚷 := |0 ⟩⟨0 | ⊗ 𝑰 𝐽 ⊗ 𝑰 𝑠𝑦𝑠 ⊗ 𝑰 𝑠𝑦𝑠′ , √ and 𝒁 is the Pauli-Z operator such that 𝒁|±⟩ = |∓⟩ for |±⟩ := (|0⟩ ± |1⟩)/ 2. If 𝑼 is a unitary block-encoding of the Lindbladian such that  ′   ′  ∑︁ 𝑏 𝑐 | 𝑗⟩ ⊗ 𝑳 𝑗 , ⟨0 | ⊗ 𝑰 𝐽 ⊗ 𝑰 𝑠𝑦𝑠 · 𝑼 · |0 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 = 𝑗 ∈𝐽 then, we obtain a block encoding for the (shifted) discriminant proxy  ′   ′  ⟨0𝑐 +1 | ⊗ 𝑰 𝑠𝑦𝑠 ⊗ 𝑰 𝑠𝑦𝑠′ · 𝑼 D(𝑷,{𝑳 𝑗 }) · |0𝑐 +1 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 ⊗ 𝑰 𝑠𝑦𝑠′ = 𝑰 + D(𝑷, {𝑳 𝑗 }). using ′† 𝑼 D(𝑷,{𝑳 𝑗 }) := 𝑼 · 𝑹 · 𝑼 ′ ′  where 𝑼 = |+⟩⟨+| ⊗ 𝑼 ⊗ 𝑰 𝑠𝑦𝑠′ + |−⟩⟨−| ⊗ 𝑰 𝑠𝑦𝑠 ⊗ 𝑼 ∗  and the unitary 𝑼 ∗ is the conjugate of 𝑼 but acting on a copy of the system register (𝑠𝑦𝑠′). ′ ′ Proof. Consider the isometries 𝑻 := 𝑼(|0𝑐 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 ) and 𝑻 ∗ := 𝑼 ∗ (|0𝑐 ⟩ ⊗ 𝑰 𝑠𝑦𝑠′ ), and ′ ′  𝑻 := 𝑼 |0 𝑐 ′ +1 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 ⊗ 𝑰  𝑠𝑦𝑠′
1 = √ |+⟩ ⊗ 𝑻 ⊗ 𝑰 𝑠𝑦𝑠′ + |−⟩ ⊗ 𝑰 𝑠𝑦𝑠 ⊗ 𝑻 ∗ . 2 54 ′ To understand the product 𝑻 † 𝑹𝑻 ′, we first calculate ( 𝑰 ⊗ 𝚷)𝑻 ′ and 𝑹0𝑻 ′: ′ 1 ∑︁ |+⟩ ⊗ |0𝑏 ⟩ ⊗ | 𝑗⟩ ⊗ 𝑳 𝑗 ⊗ 𝑰 𝑠𝑦𝑠′ ( 𝑰 ⊗ 𝚷)𝑻 ′ = √ 2 𝑗 ∈𝐽 ′ 1 ∑︁ |−⟩ ⊗ |0𝑏 ⟩ ⊗ | 𝑗⟩ ⊗ 𝑰 𝑠𝑦𝑠 ⊗ 𝑳 ∗𝑗 , +√ 2 𝑗 ∈𝐽 ′ 1 ∑︁ |−⟩ ⊗ |0𝑏 ⟩ ⊗ | 𝑗 ′⟩ ⊗ 𝑳 𝑗 ⊗ 𝑰 𝑠𝑦𝑠′ 𝑹0 𝑻 ′ = √ 2 𝑗 ∈𝐽 ′ 1 ∑︁ |+⟩ ⊗ |0𝑏 ⟩ ⊗ | 𝑗 ′⟩ ⊗ 𝑰 𝑠𝑦𝑠 ⊗ 𝑳 ∗𝑗 , +√ 2 𝑗 ∈𝐽 (2.31) (2.32) using the bit-flip 𝒁 (+ ↔ −), and the permutation 𝑷 ( 𝑗 ↔ 𝑗 ′). Finally, we get that ′ ′ ′ ′ 𝑻 † 𝑹𝑻 ′ = 𝑻 † 𝑰𝑻 ′ − 𝑻 † ( 𝑰 ⊗ 𝚷)𝑻 ′ + 𝑻 † 𝑹0𝑻 ′ ′ ′ = 𝑰 − 𝑻 † ( 𝑰 ⊗ 𝚷) · ( 𝑰 ⊗ 𝚷)𝑻 ′ + 𝑻 † ( 𝑰 ⊗ 𝚷) · 𝑹0𝑻 ′ (since ( 𝑰 ⊗ 𝚷) 𝑹0 = 𝑹0 ) 1 ∑︁ 1 ∑︁ † † ∗ ∗ 𝑳 + 𝑳 𝑗 𝑳 𝑗 ⊗ 𝑰 𝑠𝑦𝑠′ + 𝑰 𝑠𝑦𝑠 ⊗ 𝑳 ∗† 𝑳 𝑗 ⊗ 𝑳 ∗† =𝑰− ′ + 𝑳 𝑗′ ⊗ 𝑳 𝑗 𝑗 𝑗 𝑗 2 𝑗 ∈𝐽 2 𝑗 ∈𝐽 (by (2.31)-(2.32)) = 𝑰 + D(𝑷, {𝑳 𝑗 }). ■ Specializing the above recipe for (2.26) yields a block-encoding of D 𝛽 using the following ingredients: • A unitary block-encoding for the Lindbladian  ′   ′  ⟨0𝑏 | ⊗ 𝑰𝜔¯ ⊗ 𝑰𝑎 ⊗ 𝑰 𝑠𝑦𝑠 · 𝑼 · |0𝑐 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 = ∑︁ √︁ 𝛾( 𝜔)| ¯ 𝜔⟩ ¯ ⊗ |𝑎⟩ ⊗ 𝑨ˆ 𝑎 ( 𝜔). ¯ 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 An example would be the block-encoding (2.30) instantiating the parameters ′ 𝑏′ = 𝑏 + 1 and |0𝑐 ⟩ = 0𝑐+1 0̄ after appropriately rearranging the registers. • Negation on the Bohr frequency register ∑︁ 𝑭 := |−𝜔⟩⟨ ¯ 𝜔| ¯ such that 𝑭 2 = 𝑰𝜔¯ . 𝜔∈𝑆 ¯ 𝜔0 • Permutation (involution) of the jump operator labels ∑︁ ′ 𝑷 := |𝑎′⟩⟨𝑎| where 𝑨𝑎 = ( 𝑨𝑎 ) † for each 𝑎∈𝐴 𝑎 ∈ 𝐴. (2.33) 55 𝒁 |0⟩ 0𝑏+1 𝑼 0𝑐−𝑏 ⟨0| ′ 𝑼 𝑷 0𝑏+1 ′† 0𝑐−𝑏 0̄ 0̄ sys sys’ Figure 2.6: Circuit 𝑼′† 𝑹𝑼′ for block-encoding the discriminant in the fashion of Szegedy quantum walk. Note that if the jump operators are Hermitian, e.g., Pauli matrices, then we can simply take the permutation to be the identity 𝑷 = 𝑰𝑎 . To reiterate, compared to the incoherent case, the discriminant proxy can be implemented with two extra unitaries 𝑭 and 𝑷, an additional copy of the system register (which we denoted by 𝑠𝑦𝑠𝑡𝑒𝑚′), and an additional ancilla qubit. registers: |±⟩ |{z} ⊗ 𝑻/𝑻 ∗ selector ′ |0𝑏 ⟩ |{z} block. enc. anc. ⊗ | 𝜔⟩ ¯ ⊗ |𝑎⟩ ⊗ 𝑨ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑨𝑎 ( 𝜔) ¯ ∗ |{z} |{z} | {z } | {z } Bohr freq. jump system system’ If we combine the constructions of section 2.3-section 2.3, we can see that the number of qubits is #𝑞𝑢𝑏𝑖𝑡𝑠 = 2𝑛 + ⌈log2 (𝑁)⌉ + 𝑐 + 2, coming from the two copies of the system register, the frequency register, the ancillae for the block-encoding of the jumps, and one additional ancillae introduced in each of √︁ section 2.3-section 2.3. When the normalized jump operators | 𝐴| 𝑨𝑎 are unitaries, we can have 𝑐 as small as ⌈log2 (| 𝐴|)⌉; see for example (2.28). Proof of coherent Gibbs sampler (Theorem 2.1.4) In this section, we prove guarantees for our coherent Gibbs sampler (Theorem 2.1.4) in a similar vein as the Lindbladian case (Theorem 2.1.3). The required lemmas and propositions are analogous but refer to the spectral gap instead of the mixing time. 56 Proof of Theorem 2.1.4. We present general bounds on finite 𝑁 (2.26) and then take the large 𝑁 limit for the continuum (2.18). We bound the eigenvector distance by the operator norm bounds (Proposition 2.10.1, Proposition 2.10.2 and Proposition 2.3.3): secular approximation error (Lemma 2.6.2 and Proposition 2.6.7), and discriminant † proxy (Lemma 2.7.3 and D𝑠𝑒𝑐 = D𝑠𝑒𝑐 )36 ∥ 𝜆1 ( D 𝛽 ) − √ 6∥ D 𝛽 − D( 𝝆, L 𝑠𝑒𝑐 ) † ∥ 𝜆 𝑔𝑎 𝑝 ( D 𝛽 )   6 † ≤ ∥ D 𝛽 − D 𝑠𝑒𝑐 ∥ + ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 ) ∥ 𝜆 𝑔𝑎 𝑝 (D 𝛽 ) 𝝆 ∥≤ 2 2 2 2 2 2 2 2 2 2 (e−𝑁 𝜔0 𝜎𝑡 /2 + e−𝑁 𝑡0 /16𝜎𝑡 + e− 𝜇¯ 𝜎𝑡 + e−𝑇 /4𝜎𝑡 + 𝜔0𝑇) + 𝛽 𝜇¯ ≤O 𝜆 𝑔𝑎 𝑝 (D 𝛽 ) ! √︁ √︁ 2 2 2 𝜎𝑡 𝜔0 log(1/(𝜎𝑡 𝜔0 )) + (𝛽/𝜎𝑡 ) log(𝜎𝑡 /𝛽) + e−𝑁 𝜔0 𝜎𝑡 /2 ≤O . 𝜆 𝑔𝑎 𝑝 ( D 𝛽 ) √︁ The fourth inequality chooses the free parameter 𝑇 = 2𝜎 ln(1/(𝜎𝑡 𝜔0 )) and 𝑡 √︃ √︁ 2 2 2 2 2 2 𝛽 𝜇¯ = 𝜎𝑡 ln( 𝜎𝛽𝑡 ) and uses that e−𝑁 𝑡0 /16𝜎𝑡 = e−𝜋 /4𝜔0 𝜎𝑡 = O (𝜎𝑡 𝜔0 log(1/(𝜎𝑡 𝜔0 ))) to reduce the expression. In the continuum limit (2.18), the discretization parameters 𝜔0 and 𝑁 disappear, and the RHS becomes ! √︁ √ 𝛽 log(𝜎𝑡 /𝛽) ∥ 𝜆1 ( D 𝛽 ) − 𝝆 ∥ = O 𝜎𝑡 𝜆 𝑔𝑎 𝑝 ( D 𝛽 ) by taking the limit 𝑁𝜔0 → ∞, 𝜔0 → 0. ■ Note the user only chooses the time limit 𝑇, Gaussian width 𝜎𝑡 , and the Discrete Fourier Transform resolution 𝜔0 and the number of points 𝑁; the truncation parameter 𝜇¯ only appears implicitly in the analysis of secular approximation. Compared with the fixed point error for Lindbladians (section 2.2), the Hermiticity and gap substantially simplifies the analysis. Metropolis sampling with arbitrary spectral target weights Í Looking beyond sampling Gibbs states 𝝆 ∝ 𝑖 e−𝛽𝐸𝑖 |𝜓𝑖 ⟩⟨𝜓𝑖 |, we may modify the Í circuit to sample from arbitrary weight function 𝝆 ∝ 𝑖 𝑝(𝐸𝑖 )|𝜓𝑖 ⟩⟨𝜓𝑖 | that could be 36 Here, we implicitly assume that D block-encoding is implemented exactly. 𝑖𝑚 𝑝𝑙 is Hermitian, which holds for example if 𝑹 in the ! 57 0̄ 𝑷𝒓𝒆 𝒑 − 𝑸𝑭𝑻 𝐸¯ 2 0̄ 𝑷𝒓𝒆 𝒑 𝑸𝑭𝑻 𝐸¯ 1 𝝆 e−i𝑯𝑡¯ ei𝑯𝑡¯ 𝑶 𝑶 𝑓 ,𝐸¯2 ,𝐸¯1 𝝆(𝑶 𝑓 ,𝐸¯2 ,𝐸¯1 ) † Figure 2.7: The circuit for two-sided operator Fourier Transform. The gate 𝒑𝒓𝒆 𝒑 − prepares the flipped function 𝑓− (𝑡) = 𝑓 (−𝑡). Unlike the one-sided version (Figure 2.5), now we cannot understand the expression by Heisenberg evolution; this is, in spirit, more similar to doing two consecutive phase estimations. useful in other contexts. We reserve this section to point out the required ingredients and adaptations; we will stick to Gibbs sampling for the rest of the paper, but our coherent and incoherent algorithms should also apply in the general setting. Instead of applying the operator Fourier Transform for the Bohr frequencies (the energy differences), sampling from arbitrary weight requires accessing both energies before and after the jump, resembling [149, 167, 180]. The appropriate Lindbladian takes the following form:   ∑︁ 1 𝑎 𝑎 † 𝑎 † 𝑎 L𝑔𝑒𝑛𝑒𝑟𝑎𝑙 [𝝆] := 𝛾( 𝐸¯ 2 , 𝐸¯ 1 ) 𝑨ˆ 𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ) 𝝆 𝑨ˆ 𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ) − { 𝑨ˆ 𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ) 𝑨ˆ 𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ), 𝝆} . 2 ¯ ¯ 𝑎∈𝐴, 𝐸 2 , 𝐸 1 ∈𝑆 𝜔0 (2.34) To implement a block-encoding of the above Lindbladian, we need different Fourier Transform components and a controlled filter for the two-argument Metropolis weight. • Phase estimation isometry (in the Schrödinger picture) 1 ∑︁ ∑︁ ¯ Φ 𝑓 := √ 𝑓 ( 𝑡¯) 𝐸¯ ⊗ e−i( 𝐸−𝑯) 𝑡¯. 𝑁 𝐸¯ ∈𝑆 𝑡¯∈𝑆𝑡¯ 𝜔0 0 This circumvents the impossibility results as it does not take the shift-invariant form (section 2.13). 58 • Two-sided operator Fourier Transform (section 2.3) F [| 𝑓2 ⟩ ⊗ | 𝑓1 ⟩ ⊗ 𝑶] := (Φ 𝑓2 ⊗ 𝑰 𝐸¯1 ) ◦ ( 𝑰 𝐸¯1 ⊗ 𝑶) ◦ Φ 𝑓1 ∑︁ = 𝐸¯ 2 ⊗ 𝐸¯ 1 ⊗ 𝑶ˆ 𝑓1 , 𝑓2 ( 𝐸¯ 2 , 𝐸¯ 1 ), 𝐸¯ 2 , 𝐸¯ 1 ∈𝑆 𝜔0 where 1 ∑︁ ¯ ¯ 𝑶ˆ 𝑓1 , 𝑓2 ( 𝐸¯ 2 , 𝐸¯ 1 ) := 𝑓2 ( 𝑡¯2 ) 𝑓1 ( 𝑡¯1 )e−i( 𝐸2 −𝑯) 𝑡¯2 𝑶e−i( 𝐸1 −𝑯) 𝑡¯1 . 𝑁¯ ¯ 𝑡2 ,𝑡 1 ∈𝑆𝑡¯0 In our particular case, we set 𝑓1 ( 𝑡¯) = 𝑓 ( 𝑡¯) and 𝑓2 ( 𝑡¯) = 𝑓 (−𝑡¯) =: 𝑓− ( 𝑡¯) for a normalized function 𝑓 to get 1 ∑︁ ¯ ¯ 𝑶 𝑓 , 𝑓− ( 𝐸¯ 2 , 𝐸¯ 1 ) = 𝑓 (−𝑡¯2 ) 𝑓 ( 𝑡¯1 )e−i( 𝐸2 −𝑯) 𝑡¯2 𝑶e−i( 𝐸1 −𝑯) 𝑡¯1 (2.35) 𝑁¯ ¯ 𝑡 2 ,𝑡1 ∈𝑆𝑡¯0 which will be short-handed by 𝑶 𝑓 ,𝐸¯2 ,𝐸¯1 . This construction is reminiscent of [180], but it does not require a rounding promise. • Controlled filter for the Metropolis weight ∑︁ 𝑾 := 𝒀1−𝛾( 𝐸¯2 ,𝐸¯1 ) ⊗ | 𝐸¯ 2 ⟩⟨𝐸¯ 2 | ⊗ | 𝐸¯ 1 ⟩⟨𝐸¯ 1 | 𝐸¯ 2 , 𝐸¯ 1 ∈𝑆 𝜔0 such that 0 ≤ 𝛾( 𝐸¯ 2 , 𝐸¯ 1 ) ≤ 1 and 𝛾( 𝐸¯ 2 , 𝐸¯ 1 ) 𝑝( 𝐸¯ 2 ) = . 𝛾( 𝐸¯ 1 , 𝐸¯ 2 ) 𝑝( 𝐸¯ 1 ) The ratio constraint ensures approximate detailed balance. If the set of jump operators is self-adjoint and 𝑓 is real, we may construct a valid Hermitian discriminant proxy as outlined in Proposition 2.3.4 — the corresponding involution is SWAP ⊗ 𝑷, where SWAP acts on the two phase estimation registers, and 𝑷 is a permutation as in (2.33). This construction yields a discriminant proxy analogous to (2.26) due to the following “skew symmetry” of the two-sided operator Fourier Transform. We also include the two-index version of operator Parseval’s identity. Proposition 2.3.6 (Parseval’s identity). For a set of matrices 𝑨𝑎 ∈ C𝑛×𝑛 , consider their two-sided discrete Fourier Transform weighted by a complex-valued function 59 𝑓 : 𝑆𝑡0 → C as in (2.35). Then, 𝑨ˆ 𝑎𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ) † = ( 𝑨ˆ𝑎† ) 𝑓 ∗ ( 𝐸¯ 2 , 𝐸¯ 1 ), and ∑︁ ∑︁ 𝑨ˆ 𝑎𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ) † 𝑨ˆ 𝑎𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ) = ∥ 𝑓 ∥ 22 𝑎∈𝐴 𝐸¯ 1 , 𝐸¯ 2 ∈𝑆 𝜔 ∑︁ | 𝑓 ( 𝑡¯1 )| 2 e−i𝑯𝑡¯1 𝑨𝑎† 𝑨𝑎 ei𝑯𝑡¯1 𝑡¯1 ∈𝑆𝑡0 0 ⪯ ∥ 𝑓 ∥ 42 ∥ ∑︁ 𝑨𝑎† 𝑨𝑎 ∥ · 𝑰 and 𝑎∈𝐴 ∑︁ ∑︁ 𝑨ˆ 𝑎𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ) 𝑨ˆ 𝑎𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ) † = ∥ 𝑓 ∥ 22 𝑎∈𝐴 𝐸¯ 1 , 𝐸¯ 2 ∈𝑆 𝜔 ∑︁ ∑︁ | 𝑓 ( 𝑡¯2 )| 2 ei𝑯𝑡¯2 𝑨𝑎 𝑨𝑎† e−i𝑯𝑡¯2 𝑎∈𝐴 𝑡¯2 ∈𝑆𝑡0 0 ⪯ ∥ 𝑓 ∥ 42 ∥ ∑︁ 𝑨𝑎 𝑨𝑎† ∥ · 𝑰. 𝑎∈𝐴 Proof. For each 𝑎 ∈ 𝐴, we have † ∑︁ 1 ¯ ¯ ª © 𝑓 (−𝑡¯2 ) 𝑓 ( 𝑡¯1 )e−i( 𝐸2 −𝑯) 𝑡¯2 𝑨𝑎 e−i( 𝐸1 −𝑯) 𝑡¯1 ® 𝑨ˆ 𝑎𝑓 ( 𝐸¯ 2 , 𝐸¯ 1 ) † = ­ 𝑁¯ ¯ ¬ « 𝑡2 ,𝑡1 ∈𝑆𝑡¯0 † ¯ ¯ © 1 ∑︁ ª =­ 𝑓 ( 𝑡¯2 ) 𝑓 ( 𝑡¯1 )ei( 𝐸2 −𝑯) 𝑡¯2 𝑨𝑎 e−i( 𝐸1 −𝑯) 𝑡¯1 ® 𝑁¯ ¯ « 𝑡2 ,𝑡1 ∈𝑆𝑡¯0 ¬ 1 ∑︁ ¯ ¯ = 𝑓 ( 𝑡¯2 ) ∗ 𝑓 ( 𝑡¯1 ) ∗ ei( 𝐸1 −𝑯) 𝑡¯1 𝑨𝑎† e−i( 𝐸2 −𝑯) 𝑡¯2 = ( 𝑨ˆ𝑎† ) 𝑓 ∗ ( 𝐸¯ 2 , 𝐸¯ 1 ). 𝑁¯ ¯ 𝑡2 ,𝑡1 ∈𝑆𝑡¯0 ■ Our analytic arguments (section 2.5, section 2.7) can be adapted to the above Lindbladian (2.34) and discriminant variant, but we will stick to the operator Fourier Transform for simplicity throughout the paper. 2.4 Discussion Our work aimed to lay the algorithmic and analytic foundation for Monte Carlo-style Quantum Gibbs samplers. We have presented families of efficiently implementable algorithms for Gibbs sampling in terms of Lindbladians with guarantees for fixedpoint accuracy. We have confronted technical issues from energy uncertainty (which have haunted quantum Gibbs sampling algorithms for a decade) by highlighting a key algorithmic component, the operator Fourier Transform, and introducing a general analytic framework: the secular approximation and approximate detailed balance. These arguments are compatible with the Szegedy-type speedup and preparation of the purified Gibbs state. Our construction is conceptually simple as it draws inspiration from the physical mechanism of open-system thermalization, especially the Davies’ generator. Conversely, 60 our argument completes the first proof of Gibbs state stationarity for physically derived Lindbladians, especially the coarse-grained master equations [121, 134]. Potentially, this could lead to proposals of quantum Gibbs samplers on analog quantum simulators. Still, our analysis is restricted to the open system setting where the bath is Markovian by assumption; we hope the precise statement in the open system settings inspires further insight toward closed-system thermodynamics. Would quantum Gibbs samplers be the ultimate solution to the ground state preparation problem by setting 𝛽 ≫ 1? Our work only answers the first half of this problem by writing down some candidate Lindbladians with a provably accurate Gibbs fixed point and efficient simulation algorithm. Still, the missing piece of the puzzle is the mixing time (or the spectral gap) of the proposed Lindbladians. A scientifically informative first step is to directly benchmark the performance of quantum Gibbs samplers numerically for viable system sizes. This would give concrete estimates of the realistic costs of quantum simulation. From a mathematical physics perspective, there have been efforts to prove rapid mixing [16, 31] (convergence at a logarithmic depth log(𝑛)) of Quantum Gibbs samplers for lattice Hamiltonian in the spirit of classical Ising models [124]. There, most results have been restricted to commuting Hamiltonians due to the lack of a satisfactory formulation of noncommuting Gibbs samplers, which this work provides37. As a direct implication of this work, we provide a candidate algorithm for preparing a gapped ground state at a potentially very 1 1 ) and 𝛽 = Õ ( Δ𝑔𝑎 ) low depth: simply setting the Gaussian width to be 𝜎𝑡 ∼ Õ ( Δ𝑔𝑎 𝑝 𝑝 1 ensures the ground state to be approximately the common kernel of Õ ( Δ𝑔𝑎 𝑝 )-local Lindbladians38. From a complexity perspective, quantum Gibbs samplers provide a new dynamic angle to study the complexity of thermal states and even ground states. Indeed, existing complexity results for gapped ground states, especially the area law, have beaten the static properties to death (local gap, decay of correlation, etc.). Conceptually, these approaches neglect the instinctive experimental origin of ground states: cool the system in a fridge. It would be curious to bridge this thermodynamics process to the area-law literature (e.g., [10, 84, 106]). Practically, we hope noncommuting Gibbs samplers will inspire new tensor network algorithms or even new ansatz, which could lead to a better grasp of 2D gapped physics. 37The followup work [42] gives an even nicer noncommutative Gibbs sampler with exact detailed balance. 38This observation was later exploited in [61] for ground state preparation. 61 Acknowledgments We thank Simon Apers, Mario Berta, Garnet Chan, Alex Dalzell, Zhiyan Ding, Hsin-Yuan (Robert) Huang, Lin Lin, Yunchao Liu, Sam McArdle, Jonathan Moussa, Evgeny Mozgunov, Tobias Osborne, Wocjan Pawel, Patrick Rall, Mehdi Soleimanifar, Kristan Temme, Umesh Vazirani, and Tong Yu for helpful discussions. We also thank anonymous referees for their helpful feedback. AG thanks Chunhao Wang, Dávid Matolcsi, Cambyse Rouzé, and Daniel Stilck França for useful discussions. CFC is supported by the Eddlemen Fellowship and the AWS Center for Quantum Computing internship. AG acknowledges funding from the AWS Center for Quantum Computing. 2.5 Appendix:Operator Fourier Transform: properties and error bounds In this section, we study properties of the operator Fourier Transform. Given a Hamiltonian 𝑯, an operator 𝑨, and a complex-valued function 𝑓 : R → C, let 1 ∑︁ −i𝜔¯ 𝑡¯ ˆ e 𝑓 ( 𝑡¯) 𝑨( 𝑡¯) where 𝑨(𝑡) := ei𝑯𝑡 𝑨e−i𝑯𝑡 . (2.36) 𝑨 𝑓 ( 𝜔) ¯ := √ 𝑁 𝑡¯∈𝑆𝑡 0 Note the normalization √1 . The transformed operators 𝑨ˆ 𝑓 ( 𝜔) ¯ satisfy the desirable 𝑁 exact symmetry of a standard Fourier Transform as well as an operator version of Parseval’s identity. Proposition 2.5.1 (Symmetry and operator Parseval’s identity). For a set of matrices { 𝑨𝑎 } 𝑎∈𝐴 and a Hamiltonian 𝑯, consider their discrete operator Fourier Transform weighted by a complex-valued function 𝑓 : 𝑆𝑡0 → C as in (2.36). Then, the symmetry holds 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ † = ( 𝑨ˆ𝑎† ) 𝑓 ∗ (−𝜔), ¯ moreover ∑︁ ∑︁ ∑︁ ∑︁ ∑︁ | 𝑓 ( 𝑡¯)| 2 ei𝑯𝑡¯ 𝑨𝑎† 𝑨𝑎 e−i𝑯𝑡¯ ⪯ ∥ 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ † 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ = 𝑨𝑎† 𝑨𝑎 ∥∥ 𝑓 ∥ 22 · 𝑰 and 𝑎∈𝐴 𝜔∈𝑆 ¯ 𝜔0 𝑎∈𝐴 𝑡¯∈𝑆𝑡0 𝑎∈𝐴 ∑︁ ∑︁ ∑︁ ∑︁ ∑︁ (2.37) 𝑎∈𝐴 𝜔∈𝑆 ¯ 𝜔0 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ †= 𝑎∈𝐴 𝑡¯∈𝑆𝑡0 | 𝑓 ( 𝑡¯)| 2 ei𝑯𝑡¯ 𝑨𝑎 𝑨𝑎† e−i𝑯𝑡¯ ⪯ ∥ 𝑨𝑎 𝑨𝑎† ∥∥ 𝑓 ∥ 22 · 𝑰. 𝑎∈𝐴 (2.38) Í For our Lindbladian Gibbs samplers, the weights will be normalized 𝑡¯∈𝑆𝑡 | 𝑓 ( 𝑡¯)| 2 = 1, 0 which means that they can be implemented by amplitudes of a state. In the special Í case 𝑎∈𝐴 𝑨𝑎† 𝑨𝑎 = 𝑰 (i.e., these operators can be interpreted as a quantum channel), then the inequality (2.37) hold with equality, and as a consequence, the operators Í Í ˆ 𝑎 ¯ † 𝑨ˆ 𝑎 ( 𝜔) resolve the identity 𝑎∈𝐴 𝜔∈𝑆 ¯ 𝜔 𝑨 𝑓 ( 𝜔) 𝑓 ¯ = 𝑰. 0 62 Proof. First observe that by definition 1 ∑︁ i𝜔¯ 𝑡¯ ∗ 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ †=√ e 𝑓 ( 𝑡¯)( 𝑨𝑎 ( 𝑡¯)) † 𝑁 𝑡¯∈𝑆𝑡 0 ∑︁ 1 =√ ei𝜔¯ 𝑡¯ 𝑓 ∗ ( 𝑡¯) 𝑨𝑎† ( 𝑡¯) = ( 𝑨ˆ𝑎† ) 𝑓 ∗ ,−𝜔¯ . 𝑁 𝑡¯∈𝑆𝑡 (2.39) 0 Next, we prove (2.37) by direct computation as follows ∑︁ ∑︁ ∑︁ 1 ∑︁ ∑︁ ∑︁ ′ 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ † 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ = ei𝜔¯ 𝑡¯ 𝑓 ∗ ( 𝑡¯) 𝑨𝑎 ( 𝑡¯′) † e−i𝜔¯ 𝑡¯ 𝑓 ( 𝑡¯) 𝑨𝑎 ( 𝑡¯) 𝑁 ¯′ ¯ 𝑎∈𝐴 𝜔∈𝑆 ¯ 𝜔∈𝑆 ¯ 𝑎∈𝐴 𝜔0 𝑡 ∈𝑆 𝑡0 𝜔0 = ∑︁ ∑︁ 𝑡 ∈𝑆𝑡0 | 𝑓 ( 𝑡¯)| 2 𝑨𝑎 ( 𝑡¯) † 𝑨𝑎 ( 𝑡¯) 𝑎∈𝐴 𝑡¯∈𝑆𝑡 = ∑︁ ∑︁0 | 𝑓 ( 𝑡¯)| 2 ei𝑯𝑡¯ 𝑨𝑎† 𝑨𝑎 e−i𝑯𝑡¯ 𝑎∈𝐴 𝑡¯∈𝑆𝑡0 ⪯∥ ∑︁ 𝑨𝑎† 𝑨𝑎 ∥ =𝑰·∥ | 𝑓 ( 𝑡¯)| 2 ei𝑯𝑡¯ 𝑰e−i𝑯𝑡¯ 𝑡¯∈𝑆𝑡0 𝑎∈𝐴 ∑︁ ∑︁ ∑︁ 𝑨𝑎† 𝑨𝑎 ∥ 𝑎∈𝐴 | 𝑓 ( 𝑡¯)| 2 . 𝑡¯∈𝑆𝑡0 The second equality uses the Fourier representation of the discrete delta function ∑︁ ¯ 𝑡¯−𝑡¯′ ) (2.40) e−i𝜔( = 𝑁𝛿𝑡¯,𝑡¯′ . 𝜔∈𝑆 ¯ 𝜔0 The proof of (2.38) is completely analogous. ■ We also include the analogous analysis in the continuum limit where the discretization parameter 𝑁 disappears. We will assume throughout that the weight function 𝑓 is square integrable, i.e., 𝑓 ∈ ℓ2 (R). In the continuous case, the operator Fourier Transform is a matrix-valued function, and to emphasize this, we change the notation to 𝑨ˆ 𝑓 ( 𝜔) ¯ → 𝑨ˆ 𝑓 (𝜔). We could directly copy the above proof; however, arguing about the Dirac delta function in the continuous case is tricky. We resolve this by relying on Parseval-Plancherel’s identity. For studying the operator Fourier Transform, it is useful to decompose the operator according to the Bohr frequencies ∑︁ ei𝑯𝑡 𝑨e−i𝑯𝑡 = ei𝜈𝑡 𝑨𝜈 , (2.41) 𝜈∈𝐵(𝑯) where 𝑨𝜈 := ∑︁ 𝐸 2 −𝐸 1 =𝜈 𝑷 𝐸2 𝑨𝑷 𝐸1 satisfies that ( 𝑨𝜈 ) † = ( 𝑨† )−𝜈 , 63 where 𝑷 𝐸 denotes the orthogonal projector onto the subspace spanned by energy 𝐸 eigenstates of 𝑯. If 𝑓 ∈ ℓ1 (R), using this decomposition, we can conveniently express the operator Fourier Transform as follows ∫ ∞ 1 𝑨ˆ 𝑓 (𝜔) = √ e−i𝜔𝑡 𝑓 (𝑡)ei𝑯𝑡 𝑨e−i𝑯𝑡 d𝑡 2𝜋 ∫−∞ ∞ ∑︁ 1 e−i𝜔𝑡 𝑓 (𝑡)ei𝑯𝑡 𝑨𝜈 e−i𝑯𝑡 d𝑡 =√ 2𝜋 −∞ 𝜈∈𝐵(𝑯) ∫ ∞ ∑︁ ∑︁ 1 = e−i(𝜔−𝜈)𝑡 𝑓 (𝑡) 𝑨𝜈 d𝑡 = 𝑓ˆ(𝜔 − 𝜈) 𝑨𝜈 , (2.42) √ 𝜈∈𝐵(𝑯) 2𝜋 −∞ 𝜈∈𝐵(𝑯) ∫ ∞ where 𝑓ˆ(𝜔) = √1 −∞ 𝑓 (𝑡)e−i𝜔𝑡 d𝑡 is the Fourier Transform of the weight function 2𝜋 𝑓 (𝑡). More generally, if 𝑓 ∈ ℓ2 (R), then we use (2.42) as the definition of the operator Fourier Transform because the Fourier Transform uniquely extends to a unitary map F : ℓ2 (R) → ℓ2 (R). Proposition 2.5.2 (Symmetry and operator Parseval’s identity). For a set of matrices { 𝑨𝑎 } 𝑎∈𝐴 and a Hamiltonian 𝑯, consider their continuous operator Fourier Transform weighted by a complex-valued function 𝑓 ∈ ℓ2 (R) ∑︁ 𝑨ˆ 𝑎𝑓 (𝜔) := 𝑓ˆ(𝜔 − 𝜈) 𝑨𝜈𝑎 , 𝜈∈𝐵(𝑯) then ( 𝑨ˆ 𝑎𝑓 (𝜔)) † = ( 𝑨ˆ𝑎† ) 𝑓 ∗ (−𝜔), moreover ∑︁ ∫ ∞ ∑︁ ∫ ∞ ∑︁ 2 i𝑯𝑡 𝑎† 𝑎 −i𝑯𝑡 𝑎 † 𝑎 2 ˆ ˆ | 𝑓 (𝑡)| e 𝑨 𝑨 e 𝑨 𝑓 (𝜔) 𝑨 𝑓 (𝜔)d𝜔 = d𝑡 ⪯ ∥ 𝑓 ∥ 2 ∥ 𝑨𝑎† 𝑨𝑎 ∥ · 𝑰, 𝑎∈𝐴 −∞ ∑︁ ∫ ∞ 𝑎∈𝐴 𝑎∈𝐴 𝑨ˆ 𝑎𝑓 (𝜔) 𝑨ˆ𝑎 𝑓 (𝜔) † d𝜔 = −∞ −∞ ∑︁ ∫ ∞ 𝑎∈𝐴 −∞ 𝑎∈𝐴 (2.43) ∑︁ | 𝑓 (𝑡)| 2 ei𝑯𝑡 𝑨𝑎 𝑨𝑎† e−i𝑯𝑡 d𝑡 ⪯ ∥ 𝑓 ∥ 22 ∥ 𝑨𝑎 𝑨𝑎† ∥ · 𝑰. 𝑎∈𝐴 (2.44) Similarly as before, if Í 𝑎∈𝐴 𝑨 𝑎† 𝑨𝑎 = 𝑰, then the inequality (2.43) hold with equality. Proof. First observe that by definition ∑︁ 𝑨ˆ 𝑎𝑓 (𝜔) † = ( 𝑓ˆ(𝜔 − 𝜈)) ∗ ( 𝑨𝜈𝑎 ) † 𝜈∈𝐵(𝑯) = ∑︁ c 𝑓 ∗ (𝜈 − 𝜔)( 𝑨𝑎† )−𝐸 𝜈∈𝐵(𝑯) = ∑︁ −𝜈∈𝐵(𝑯) c 𝑓 ∗ (−𝜔 − 𝜈)( 𝑨𝑎† ) 𝜈 = 𝑨ˆ𝑎† 𝑓 ∗ (−𝜔). and 64 Next, we prove (2.43) by direct computation as follows: ∑︁ ∑︁ ∫ ∞ ∑︁ ∫ ∞ ∑︁ 𝑎 † ˆ𝑎 ˆ ( 𝑨𝜈𝑎 ) † ( 𝑓ˆ(𝜔 − 𝜈)) ∗ 𝑨 𝑓 (𝜔) 𝑨 𝑓 (𝜔)d𝜔 = 𝑎∈𝐴 −∞ −∞ 𝜈∈𝐵(𝑯) 𝑎∈𝐴 𝑓ˆ(𝜔 − 𝜈′) 𝑨𝜈𝑎′ d𝜔 𝜈 ′ ∈𝐵(𝑯) (by definition) = ∑︁ ∑︁ ( 𝑨𝜈𝑎 ) † 𝑨𝜈𝑎′ = ( 𝑓ˆ(𝜔)) ∗ 𝑓ˆ(𝜔 − (𝜈′ − 𝜈))d𝜔 −∞ 𝑎∈𝐴 𝜈,𝜈 ′ ∈𝐵(𝑯) ∑︁ ∑︁ ∫ ∞ (shift 𝜔 → 𝜔 − 𝜈 and use |𝐵| ≤ ∞) ∫ ∞ 𝑎† 𝑎 ′′ ( 𝑨 𝑨 )𝜈 ( 𝑓ˆ(𝜔)) ∗ 𝑓ˆ(𝜔 − 𝜈′′)d𝜔 𝑎∈𝐴 𝜈 ′′ ∈𝐵(𝑯) −∞ ∑︁ ∑︁ ∫ ∞ (by the definition of 𝑨𝜈𝑎 ) = ( 𝑨𝑎† 𝑨𝑎 ) 𝜈′′ ′′ ( 𝑓 (𝑡)) ∗ 𝑓 (𝑡)ei𝜈 𝑡 d𝑡 −∞ 𝑎∈𝐴 𝜈 ′′ ∈𝐵(𝑯) (since F is unitary) = ∑︁ ∑︁ ∫ ∞ 𝑎∈𝐴 𝜈 ′′ ∈𝐵(𝑯) | 𝑓 (𝑡)| 2 ei𝑯𝑡 ( 𝑨𝑎† 𝑨𝑎 ) 𝜈′′ e−i𝑯𝑡 d𝑡 −∞ (as in (2.42)) = ∑︁ ∫ ∞ | 𝑓 (𝑡)| 2 ei𝑯𝑡 𝑨𝑎† 𝑨𝑎 e−i𝑯𝑡 d𝑡 −∞ 𝑎∈𝐴 (since |𝐵(𝑯)| ≤ ∞) ⪯∥ ∑︁ ∫ ∞ 𝑨𝑎† 𝑨𝑎 ∥ 𝑎∈𝐴 | 𝑓 (𝑡)| 2 ei𝑯𝑡 𝑰e−i𝑯𝑡 d𝑡 −∞ Í Í (since 𝑎∈𝐴 𝑨𝑎† 𝑨𝑎 ⪯ ∥ 𝑎∈𝐴 𝑨𝑎† 𝑨𝑎 ∥ 𝑰) ∫ ∞ ∑︁ 𝑎† 𝑎 | 𝑓 (𝑡)| 2 d𝑡. =𝑰·∥ 𝑨 𝑨 ∥ 𝑎∈𝐴 −∞ The proof of (2.44) is completely analogous.39 2.6 ■ Appendix:Nonasymptotic secular approximation In this section, we define the secular approximation of the Fourier Transformed operators 𝑨ˆ 𝑓 ( 𝜔) ¯ and analyze the resulting error. The secular approximation applies truncation to the Fourier-transformed operators in the frequency domain by suppressing Bohr frequencies 𝜈 ∈ 𝐵(𝑯) that deviate substantially from the frequency label 𝜔 via some filter function 𝑠 ∈ ℓ∞ (R). For example truncation at 39 Intuitively speaking the fourth line can be viewed as consequence of the Fourier representation ∫∞ of the Dirac delta distribution −∞ ei𝜔𝑡 d𝜔 = 2𝜋𝛿(𝑡), analogous to (2.40). Not introducing delta functions makes the proof completely general. 1 O(tQP E) <latexit 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⇠ <latexit 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⌫ ! ¯ ! 0 <latexit 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|¯ ! ⌫| Figure 2.8: Left: the weight as the step function. Right: An illustration for the Fourier Í Í i(𝜈− 𝜔)𝑇 ¯ −i(𝜈− 𝜔)𝑇 ¯ Transformed amplitudes 𝑨𝜔¯ = 𝜈∈𝑆 𝜔 fˆ(¯ 𝑨! 𝜈 𝑓⌫)( 𝜔¯ − 𝜈) = 𝜈∈𝑆 𝜔 𝑨𝜈 e ei(𝜈−−e f (t̄) 𝜔)𝑡 ¯ 0 −1 0 0 for Bohr frequency 𝜈 given the energy label 𝜔. ¯ The expression coincides with the phase estimation profile. It peaks near energy 𝜈 = 𝜔¯ with a width ∼ 𝑇 −1 and decays polynomially. The profile in absolute value oscillates (blue), but we also display the norm bound to guide the eye (black). The secular approximation truncates the profile ! ¯ + µ̄ at an energy 𝜇¯ far in the tail 𝜇¯ ≫ 𝑇 −1 . <latexit 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f!¯ E <latexit 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T /2 ⇠e ⌦(T 2 |¯ ! ⌫|2 ) ⌫ energy difference 𝜇 can be achieved by setting 𝑠(𝜔) := 1 (|𝜔| < 𝜇) and defining the following secular-approximated operators as follows: ⌫¯ µ̄0 ∑︁ 𝑺ˆ 𝑓 ,𝑠 (𝜔) := 𝑓ˆ(𝜔 − 𝜈)𝑠(𝜔 − 𝜈) 𝑨𝜈 . 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¯ ! ¯ + µ̄0 <latexit sha1_base64="HwiqrWG/70k+r2EGMAC/0SPJDFw=">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</latexit> 𝜈∈𝐵(𝑯) We will sometimes consider alternative filter functions 𝑠 : R → C, so we will treat general 𝑠 throughout our discussion. The key observation in our analysis is that due to the definition of the operator Fourier Transform (2.42) we have 𝑺ˆ 𝑓 ,𝑠 (𝜔) = 𝑨ˆ 𝑓𝑠 (𝜔), where 𝑓𝑠 = F −1 ( 𝑓ˆ · 𝑠) is the inverse Fourier Transform of the function 𝑓ˆ(𝜔)𝑠(𝜔). Furthermore, this also implies that 𝑨ˆ 𝑓 (𝜔) − 𝑺ˆ 𝑓 ,𝑠 (𝜔) = 𝑨ˆ 𝑓 − 𝑓𝑠 (𝜔), where 𝑓 − 𝑓𝑠 = 𝑓1−𝑠 is the inverse Fourier Transform of the function 𝑓ˆ(𝜔)(1 − 𝑠(𝜔)). The significance of this equation is that it enables us to bound the error induced by the secular approximation via bounding the tail ∥ 𝑓ˆ(𝜔)(1 − 𝑠(𝜔))∥ 2 . Proposition 2.6.1. If 𝑓 (𝑡) is real and 𝑠(𝑡) is real and even, then 𝑓𝑠 (𝑡) is real. <latexit sha1_base64="/DQ4XnesXJ9Z1PH/gWZB1p5YDGQ=">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</latexit> <latexit sha1_base64="qDkQ6YR3fvnY9V483RKwFrRcT0s=">AAAOUnichZfLcts2FIbppJfUjV27XXbDqcYzSaf2SJo06SYzsWn5fpFtSXZtyh6QgiSMwcuAIBoNw75Cn6bb9iW66at0VZCQTBhHbbgRcL7/4HKInxS9mJKE1+t/Lzx5+smnn33+7IvFL58vLX+1svp1L4lS5uOuH9GIXXkowZSEuMsJp/gqZhgFHsWX3r1T8EuBWUKisMMnMe4HaBSSIfERl6G7lZduQgIX32br7mmAR+gFv8vi8SS/bX5wvaiI2OutD7fNl/ndSq2+US8vGzYa00bNml7tu9WlVXcQ+WmAQ+5TlCQ3jXrM+xlinPgU54uLbprgGPn3aIQzFCTJJPByey1AfJyYrAjOYzcpH/7Uz0gYpxyHvpRINkypzSO72K89IAz7nE5kA/mMyKltf4wY8rmsytxZHq/LC2RgzdanLKsuC36fvOUsxT/YiNIylrz1aIr72VhOzBgeykxjMIbuMc8X1/RgiLhHvGKWNeR5DItFl+EQ/+JHQYDCQeYK7OeZWyzE1cPOpowWK/YRzTZzk25pdAtQRB2NOybfmlHPg9DZ1lK3AW1ptAUGblUDA+jsaKk7gO5qdBfQPY3uAbqv0X1ADzR6AOihRg8BPdLoEaDHGj0G9ESjJ6BUJ1WpAHROtdRTQNsabYOB29XAADpnWuoZSD2rUgF0zrXUc5B6XqUC6FxoqReAdjTaAbSr0S6gPY32AL3U6CWgVxq9AvRnW8OyY/JrDV8/pt8Xro4TKRCIyQahUVg8LR4NIDZL20NvC6QAAmBLAeB34SngAeAoAEwuthUA/hYtBYB7xY4CwLhiVwHgWbGnALCr2FcAOFUcKABMKg4VAP4URwoAa4pjBYArxYkCwHPiVAFgN9FWAJhJnCkArCLOFQBGEBcKAA+IjgLg+IuuAuDki54C4NCLSwXAeRdXCoCjLsqjLgk85eJakWsAEjIKpgfVdlXHkHiVBjEpEnNVgo2j2TBFMzffj8UdpnjIjbyixIyMxhwYywvT6ZyubBlp4xKOEZ8HvVkiJMHDmAGE5VuihE5xENce09YMAk+pv2AzrDqGZIBGo1z9YJYbO6XFGaA9zMzisAKwOUD+WeSEkPym0c9UWcuQe49ZuF7faP6I338satcatqo8FH00bG4vjFhQLsYtFmtntUZuu3PWHSths69SslozvyvEc4aLdVmjkDVNGdV09P+Ew2p9lax86rPySQhuCEVeovQeGY1UvdSmHvrmWh4yqgqrDK1qc5av5fTMnDkFxPnDurG5S5KGhFecmJyR/4TyTcdZRTsM4Diik0rQLnpghElcKYqOKejogs4cAWLyyfJ+qoliueTAlrFfi6BLSUB4YqaQQV7K5K9BLtJYm67smfeZhIPiGyuabd2Tjy95GOVHVMP8ZIKNXnOj8Xrj1dmr2rut6efUM+tb6zvrhdWw3ljvrD2rbXUt3/rN+t36w/pz6a+lf5YXlp8q6ZOFac431qNr+fm/uFdz9w==</latexit> ⇠ 2 ⇠ e ⌦(tphy |¯! |E 66 Proof. If 𝑓 is real, then 𝑓ˆ has an even real and odd imaginary part, which remains true for 𝑓ˆ · 𝑠, and therefore F −1 ( 𝑓ˆ · 𝑠) is real as well.40 ■ Analogously, we define the secular approximation for the discrete Fourier Transform as   ∑︁ 𝑺ˆ 𝑓 ,𝑠 ( 𝜔) ¯ := F̄ 𝑓 ( 𝑡¯) · ei𝜈𝑡¯ ( 𝜔) ¯ · 𝑠( 𝜔¯ − 𝜈) 𝑨𝜈 , 𝜈∈𝐵(𝑯) where F̄ denotes the discrete Fourier  Transform.  In case 𝜈 is an integer multiple of ¯ (i𝜈 𝑡 ) the base frequency 𝜔0 of F̄ , then F̄ 𝑓 ( 𝑡¯) · e ( 𝜔) ¯ above simplifies to 𝑓ˆ( 𝜔¯ − 𝜈) = F̄ ( 𝑓 ) ( 𝜔¯ − 𝜈). Indeed, the discrete Fourier Transform 𝑓ˆ is only defined at points 𝜔¯ ∈ 𝑆𝜔0 , and thus translation by some value 𝜈 which is not an integer multiple of 𝜔0 can cause troubles. The simple analysis working nicely in the continuous case can be directly translated to the discrete case if all the Bohr frequencies are multiples of the base frequency 𝜔0 of F̄ , i.e., 𝐵 ⊂ 𝜔0 Z. This is the reason why we introduce a slightly rounded Hamiltonian in the analysis of the secular approximation. Proposition 2.6.2 (Truncation via modifying weight function). Let 𝑓 : 𝑆𝑡0 → C and 𝑠 : 𝑆𝜔0 → C and suppose that 𝐵(𝑯) ⊂ 𝜔0 Z, then the secular-approximated operator can be represented as plain operator Fourier Transform corresponding to a perturbed weight function 𝑺ˆ 𝑓 ,𝑠 ( 𝜔) ¯ = 𝑨ˆ 𝑓𝑠 ( 𝜔), ¯ where 𝑓𝑠 ( 𝜔) ¯ is the inverse discrete Fourier Transform of the function 𝑓ˆ( 𝜔)𝑠( ¯ 𝜔) ¯ and Í ¯ 1 −i 𝜔 ¯ 𝑡 𝑓ˆ( 𝜔) ¯ := √ 𝑓 ( 𝑡¯). 𝑡¯∈𝑆𝑡 e 𝑁 0 40 If we work with the discrete Fourier Transform, and 𝑁 is even, then −𝜔 0 · 𝑁/2 (or −𝑡 0 · 𝑁/2 if we work in the time domain) should be treated as its own inverse due to reasons of parity and modular arithmetic. In particular an even function can take arbitrary value on −𝜔0 · 𝑁/2, but an odd function must be 0, similarly to how such functions must behave on 0. 67 Proof. 𝑺ˆ 𝑓 ,𝑠 ( 𝜔) ¯ = ∑︁ 𝑓ˆ( 𝜔¯ − 𝜈) ¯ · 𝑠( 𝜔¯ − 𝜈) ¯ 𝑨𝜈¯ 𝜈∈𝐵(𝑯) ¯ = ∑︁ 𝑓ˆ𝑠 ( 𝜔¯ − 𝜈) ¯ 𝑨𝜈¯ 𝜈∈𝐵(𝑯) ¯ 1 ∑︁ −i(𝜔−𝜈) e ¯ 𝑡¯ 𝑓𝑠 ( 𝑡¯) 𝑨𝜈¯ √ 𝑁 𝑡¯∈𝑆𝑡 𝜈∈𝐵(𝑯) ¯ 0 1 ∑︁ −i𝜔¯ 𝑡¯ =√ e 𝑓𝑠 ( 𝑡¯) 𝑨( 𝑡¯). 𝑁 𝑡¯∈𝑆𝑡 = ∑︁ 0 The last equality recombines the Bohr frequencies gously to (2.42). Í ei𝜈¯𝑡¯ 𝑨𝜈¯ = 𝑨( 𝑡¯) analo𝜈∈𝐵(𝑯) ¯ ■ To reiterate, the condition 𝐵 ⊆ 𝜔0 Z need not hold for the original Hamiltonian 𝑯. Proceeding with the discretized Hamiltonian 𝑯¯ introduces a small additive error.41 We present error bounds for both the Lindbladian and our discriminant proxy. Lemma 2.6.1 (Perturbation bounds). Let 𝑓 , 𝑓 ′ ∈ C𝑆𝑡0 , and 𝛾 ∈ C𝑆 𝜔0 such that Í ∥ 𝑓 ∥ 2 , ∥ 𝑓 ′ ∥ 2 , ∥𝛾∥ ∞ ≤ 1. If ∥ 𝑎∈𝐴 𝑨𝑎† 𝑨𝑎 ∥ ≤ 1, then for every 𝑇 > 0 and 𝑓𝑇 (𝑡) := 𝑓 (𝑡) 1 (|𝑡| ≤ 𝑇) we have that ∑︁ √︁ √︁ ∥ 𝛾( 𝜔)| ¯ 𝜔, ¯ 𝑎⟩⟨0̄| ⊗ 𝑨ˆ 𝑎( 𝑓 ,𝑯) ( 𝜔) ¯ 𝜔, ¯ 𝑎⟩⟨0̄| ⊗ 𝑨ˆ 𝑎( 𝑓 ′ ,𝑯′ ) ( 𝜔)∥ ¯ − 𝛾( 𝜔)| ¯ 𝜔∈𝑆 ¯ 𝜔0 ,𝑎∈𝐴 ≤ ∥ 𝑓 − 𝑓𝑇 ∥ 2 + ∥ 𝑓𝑇 − 𝑓 ′ ∥ 2 + 2𝑇 ∥𝑯 − 𝑯′ ∥. Proof. This directly follows from the (not necessarily unitary) block-encoding construction of Figure 2.5-Figure 2.4 and triangle inequalities. Indeed, for any function 𝑔 ∈ C𝑆𝑡0 let 𝑷𝒓𝒆 𝒑 𝑔 := |𝑔⟩⟨0̄| and let 𝑩 (𝑔,𝑯) denote the block-encoding given Í by Figure 2.4 when setting 𝑷𝒓𝒆 𝒑 ← 𝑷𝒓𝒆 𝒑 𝑔 , 𝑽 𝑗 𝑝 ← 𝑎 |𝑎⟩ ⊗ 𝑨𝑎 , and using the Hamiltonian 𝑯. Then, we have that ∥ 𝑩 (𝑔,𝑯) − 𝑩 ( 𝑓𝑇 ,𝑯′ ) ∥ ≤ ∥ 𝑩 (𝑔,𝑯) − 𝑩 ( 𝑓𝑇 ,𝑯) ∥ + ∥ 𝑩 ( 𝑓𝑇 ,𝑯) − 𝑩 ( 𝑓𝑇 ,𝑯′ ) ∥ ≤ ∥𝑔 − 𝑓𝑇 ∥ 2 ∥𝑽 𝑗 𝑝 ∥ + 2𝑇 ∥𝑯 − 𝑯′ ∥∥𝑽 𝑗 𝑝 ∥ ≤ ∥𝑔 − 𝑓𝑇 ∥ 2 + 2𝑇 ∥𝑯 − 𝑯′ ∥. 41This differs from the unphysical rounding assumption [180] where the Hamiltonian needs to have “large” gaps in the spectrum. Here, 𝜔0 is not related to the resolution of energy estimates but rather the discretization of the register. 68 Using the above inequality twice, we obtain the desired result   ∑︁ √︁ ∥ 𝛾( 𝜔)| ¯ 𝜔, ¯ 𝑎⟩⟨0̄| ⊗ 𝑨ˆ 𝑎( 𝑓 ,𝑯) ( 𝜔) ¯ − 𝑨ˆ 𝑎( 𝑓 ′ ,𝑯′ ) ( 𝜔) ¯ ∥ 𝜔∈𝑆 ¯ 𝜔0 ,𝑎∈𝐴 = ∥ 𝑩 ( 𝑓 ,𝑯) − 𝑩 ( 𝑓 ′ ,𝑯′ ) ∥ ≤ ∥ 𝑩 ( 𝑓 ,𝑯) − 𝑩 ( 𝑓𝑇 ,𝑯′ ) ∥ + ∥ 𝑩 ( 𝑓𝑇 ,𝑯′ ) − 𝑩 ( 𝑓 ′ ,𝑯′ ) ∥ ≤ ∥ 𝑓 − 𝑓𝑇 ∥ 2 + ∥ 𝑓𝑇 − 𝑓 ′ ∥ 2 + 2𝑇 ∥𝑯 − 𝑯′ ∥. ■ Corollary 2.6.1 (Perturbation bounds on Lindbladians and discriminant proxies). Consider   ∑︁ 1 𝑎 𝑎 † 𝑎 † 𝑎 L̄ ( 𝑓 ,𝑯) := 𝛾( 𝜔) ¯ 𝑨ˆ ( 𝑓 ,𝑯) ( 𝜔) ¯ [·] 𝑨ˆ ( 𝑓 ,𝑯) ( 𝜔) ¯ − { 𝑨ˆ ( 𝑓 ,𝑯) ( 𝜔) ¯ 𝑨ˆ ( 𝑓 ,𝑯) ( 𝜔), ¯ ·} , and 2 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 ∑︁ √︁ 𝛾( 𝜔)𝛾(− ¯ 𝜔) ¯ 𝑨ˆ 𝑎( 𝑓 ,𝑯) ( 𝜔) ¯ ⊗ 𝑨ˆ 𝑎( 𝑓 ,𝑯) ( 𝜔) ¯ ∗ D̄ ( 𝑓 ,𝑯) := 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 −  𝛾( 𝜔) ¯  ˆ𝑎 𝑨 ( 𝑓 ,𝑯) ( 𝜔) ¯ † 𝑨ˆ 𝑎( 𝑓 ,𝑯) ( 𝜔) ¯ ⊗ 𝑰 + 𝑰 ⊗ 𝑨ˆ 𝑎( 𝑓 ,𝑯) ( 𝜔) ¯ ∗† 𝑨ˆ 𝑎( 𝑓 ,𝑯) ( 𝜔) ¯ ∗ . 2 Assuming the conditions and the notation of Lemma 2.6.1 hold, we have ∥ L̄ ( 𝑓 ,𝑯) − L̄ ( 𝑓 ′ ,𝑯′ ) ∥ 1−1 ≤ 4(∥ 𝑓 − 𝑓𝑇 ∥ 2 + ∥ 𝑓𝑇 − 𝑓 ′ ∥ 2 + 2𝑇 ∥𝑯 − 𝑯′ ∥). Further assuming the symmetry and normalization conditions (2.10),(2.11) are satisfied by 𝑓 , 𝑓 ′ and { 𝑨𝑎 : 𝑎 ∈ 𝐴}, we have ∥ D̄ ( 𝑓 ,𝑯) − D̄ ( 𝑓 ′ ,𝑯′ ) ∥ ≤ 4(∥ 𝑓 − 𝑓𝑇 ∥ 2 + ∥ 𝑓𝑇 − 𝑓 ′ ∥ 2 + 2𝑇 ∥𝑯 − 𝑯′ ∥). Proof. The superoperator L ( 𝑓 ,𝑯) acts as follows: L̄ ( 𝑓 ,𝑯) [𝝆]   ∑︁ √︁ © ∑︁ √︁  ª © ª 𝑎 𝑎 † 𝛾( 𝜔)| ¯ 𝜔, ¯ 𝑎⟩⟨0̄| 𝑨ˆ ( 𝜔) ¯ ® | 0̄⟩⟨0̄| ⊗ 𝝆 ­ 𝛾( 𝜔)| ¯ 0̄⟩⟨𝜔, ¯ 𝑎| 𝑨ˆ ( 𝜔) ¯ ® = Tr𝑎,𝜔¯ ­  𝑎∈𝐴,𝜔∈𝑆  ¯ 𝜔0 ¯ 𝜔0 ¬ «𝑎∈𝐴,𝜔∈𝑆 ¬ «     ∑︁ √︁  © ∑︁ √︁   1  ª © ª 𝑎 † 𝑎 ˆ ˆ − Tr0̄ ­ 𝛾( 𝜔)| ¯ 0̄⟩⟨𝜔, ¯ 𝑎| 𝑨 ( 𝜔) ¯ ®­ 𝛾( 𝜔)| ¯ 𝜔, ¯ 𝑎⟩⟨0̄| 𝑨 ( 𝜔) ¯ ®, | 0̄⟩⟨0̄| ⊗ 𝝆 .  2   𝑎∈𝐴,𝜔∈𝑆  ¯ 𝜔0 ¯ 𝜔0 ¬«𝑎∈𝐴,𝜔∈𝑆 ¬ «  The conclusion about ∥ L̄ ( 𝑓 ,𝑯) − L̄ ( 𝑓 ′ ,𝑯′ ) ∥ 1−1 follows from Lemma 2.6.1 using the triangle and Hölder inequalities and that taking partial trace contracts trace-distance. 69 The proof of Proposition 2.3.5 shows that if 𝑩 is a (nonunitary) block-encoding of ′ L̄ ( 𝑓 ,𝑯) , then 𝑩 † ·( 𝑹−𝑰)·𝑩′ is a block-encoding of D̄ ( 𝑓 ,𝑯) . Since ∥ 𝑓 ∥ 2 , ∥ 𝑓 ′ ∥ 2 , ∥𝛾∥ ∞ ≤ 1 without loss of generality we can assume ∥ 𝑩′ ∥ ≤ 1, which together with ∥ 𝑹 − 𝑰∥ ≤ 2 implies the bound on ∥ D̄ ( 𝑓 ,𝑯) − D̄ ( 𝑓 ′ ,𝑯′ ) ∥ via a triangle inequality. ■ Note that under the conditions of Theorem 2.8.1, the same bounds also hold in the continuous case, as can be shown by a limit argument using the results of Theorem 2.8.1. Corollary 2.6.2 (Perturbation bounds on continuous Lindbladians and discriminant proxies). Consider   ∑︁ ∫ ∞ 1 𝑎 𝑎 † 𝑎 † 𝑎 L ( 𝑓 ,𝑯) := 𝛾(𝜔) 𝑨ˆ ( 𝑓 ,𝑯) (𝜔) [·] 𝑨ˆ ( 𝑓 ,𝑯) (𝜔) − { 𝑨ˆ ( 𝑓 ,𝑯) (𝜔) 𝑨ˆ ( 𝑓 ,𝑯) (𝜔), ·} d𝜔, 2 −∞ 𝑎∈𝐴 ∑︁ ∫ ∞√︁ 𝛾(𝜔)𝛾(−𝜔) 𝑨ˆ 𝑎( 𝑓 ,𝑯) (𝜔) ⊗ 𝑨ˆ 𝑎( 𝑓 ,𝑯) (𝜔) ∗ D ( 𝑓 ,𝑯) := 𝑎∈𝐴 − −∞  𝛾(𝜔)  ˆ 𝑎 𝑨 ( 𝑓 ,𝑯) (𝜔) † 𝑨ˆ 𝑎( 𝑓 ,𝑯) (𝜔) ⊗ 𝑰 + 𝑰 ⊗ 𝑨ˆ 𝑎( 𝑓 ,𝑯) (𝜔) ∗† 𝑨ˆ 𝑎( 𝑓 ,𝑯) (𝜔) ∗ d𝜔. 2 If 𝛾 ∈ ℓ∞ (R), 𝑓 ∈ ℓ2 (R), and 𝛾, 𝑓 are continuous almost everywhere (i.e., the set of points of discontinuity has measure zero) while 𝑓 is bounded on every finite interval, then assuming the conditions and the notation of Lemma 2.6.1 hold, we have L ( 𝑓 ,𝑯) − L ( 𝑓 ′ ,𝑯′ ) 1−1 ≤ 4(∥ 𝑓 − 𝑓𝑇 ∥ 2 + ∥ 𝑓𝑇 − 𝑓 ′ ∥ 2 + 2𝑇 ∥𝑯 − 𝑯′ ∥). Further assuming the symmetry and normalization conditions (2.10),(2.11) are satisfied by 𝑓 , 𝑓 ′ and { 𝑨𝑎 : 𝑎 ∈ 𝐴}, we have ∥ D ( 𝑓 ,𝑯) − D ( 𝑓 ′ ,𝑯′ ) ∥ ≤ 4(∥ 𝑓 − 𝑓𝑇 ∥ 2 + ∥ 𝑓𝑇 − 𝑓 ′ ∥ 2 + 2𝑇 ∥𝑯 − 𝑯′ ∥). Proof. The objects L ( 𝑓 ,𝑯) , L ( 𝑓 ′ ,𝑯′ ) , D ( 𝑓 ,𝑯) , D ( 𝑓 ,𝑯) can be obtained as limits of their respective discretizations L̄ ( 𝑓 ,𝑯) , L̄ ( 𝑓 ′ ,𝑯′ ) , D̄ ( 𝑓 ,𝑯) , D̄ ( 𝑓 ,𝑯) as per Theorem 2.8.1, for which the discretized versions of these bounds hold due to Corollary 2.6.2. As shown in the proof of Theorem 2.8.1, if a function 𝑔 ∈ ℓ2 (R) is continuous almost everywhere while also bounded on every finite interval, then ∥𝑔∥ 22 = √︁ Í √ lim𝐾→∞ lim𝑁→∞ 𝑡¯∈𝑆 ⌈ 𝑁 ⌋ | 𝑔¯ 𝐾 ( 𝑡¯)| 2 , where 𝑡0 = 2𝜋/𝑁 and 𝑔¯ 𝐾 (𝑡) = 𝑡0 𝑔(𝑡) 1 (|𝑡| ≤ 𝑡0 𝐾), therefore the RHS of the discretized bounds also converge to their continuous counterpart implying the validity of the continuous versions of these bounds. ■ and 70 Lemma 2.6.2 (Secular approximation). Let ∥𝛾∥ ∞ ≤ 1, and consider the Lindbladian L 𝛽 (2.21) and discriminant D 𝛽 (2.26) with 𝑨ˆ 𝑎 ( 𝜔) ¯ being the operator Fourier 𝑎 Transforms of 𝑨 with 𝑯 and their secular approximations   ∑︁ 1 𝑎 𝑎 † 𝑎 † 𝑎 L 𝑠𝑒𝑐 [·] = 𝛾( 𝜔) ¯ 𝑺ˆ ( 𝜔) ¯ [·] 𝑺ˆ ( 𝜔) ¯ − { 𝑺ˆ ( 𝜔) ¯ 𝑺ˆ ( 𝜔), ¯ ·} , 2 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔 ∑︁ 0 √︁ 𝛾( 𝜔)𝛾(− ¯ 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ D 𝑠𝑒𝑐 = 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 −  𝛾( 𝜔) ¯  ˆ𝑎 𝑺 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑰 + 𝑰 ⊗ 𝑺ˆ 𝑎 ( 𝜔) ¯ †∗ 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ , 2 with the operators ∑︁ ¯ := 𝑺ˆ 𝑎 ( 𝜔) 𝑓ˆ𝑠 ( 𝜔¯ − 𝜈) ¯ 𝑨𝜈𝑎¯ where 𝑓ˆ𝑠 ( 𝜔) ¯ := 𝑓ˆ( 𝜔) ¯ · 1 (| 𝜔| ¯ < 𝑚𝜔0 ) ¯ 𝜈∈𝐵( ¯ 𝑯) defined by the discretized Hamiltonian 𝑯¯ and cut-off frequency 𝑚𝜔0 . If ∥ 𝑓 ∥ 2 ≤ 1 Í and ∥ 𝑎∈𝐴 𝑨𝑎† 𝑨𝑎 ∥ ≤ 1, then for every 𝑇 > 0 and 𝑓𝑇 (𝑡) := 𝑓 (𝑡) 1 (|𝑡| ≤ 𝑇), we have L 𝛽 − L 𝑠𝑒𝑐 1−1 ≤ 4∥ 𝑓ˆ − 𝑓ˆ𝑠 ∥ 2 + 8∥ 𝑓 − 𝑓𝑇 ∥ 2 + 4𝑇𝜔0 . Moreover, assuming the symmetry and normalization conditions (2.10),(2.11), we have D 𝑠𝑒𝑐 = D†𝑠𝑒𝑐 and ∥ D 𝛽 − D 𝑠𝑒𝑐 ∥ ≤ 4∥ 𝑓ˆ − 𝑓ˆ𝑠 ∥ 2 + 8∥ 𝑓 − 𝑓𝑇 ∥ 2 + 4𝑇𝜔0 . The truncation introduces an error scaling with the tail in the frequency domain, while the last two error terms arise from discretizing the Hamiltonian spectrum 𝑯 → 𝑯¯ for discrete Fourier Transforms; this is more of a technical artifact and merely introduces a minor error shrinking with finer Fourier frequency resolution 𝜔0 . Proof. Since the secular approximation amounts to changing the real function and discretizing the Hamiltonian, i.e., L 𝑠𝑒𝑐 = L ( 𝑓𝑠 , 𝑯) ¯ we can apply Corollary 2.6.1. The ¯ ≤ 𝜔0 /2, and ∥ 𝑓 − 𝑓𝑠 ∥ = final bound follows using the observation that ∥𝑯 − 𝑯∥ ∥ 𝑓ˆ − 𝑓ˆ𝑠 ∥ since the discrete Fourier Transformation is unitary. Finally, since D 𝑠𝑒𝑐 = D ( 𝑓𝑠 , 𝑯) ¯ we have that D 𝑠𝑒𝑐 is self-adjoint due to Proposition 2.6.1 and Corollary 2.3.2. ■ Taking the continuum limit further simplifies the expression. 71 Lemma 2.6.3 (Secular approximation in the continuum). Let ∥𝛾∥ ∞ ≤ 1, and consider the Lindbladian L 𝛽 (2.21) and discriminant D 𝛽 (2.26) with 𝑨ˆ 𝑎 (𝜔) being the operator Fourier Transforms of 𝑨𝑎 with 𝑯 and their secular approximations   ∑︁ ∫ ∞ 1 𝑎 𝑎 † 𝑎 † 𝑎 L 𝑠𝑒𝑐 [·] = 𝛾(𝜔) 𝑺ˆ (𝜔) [·] 𝑺ˆ (𝜔) − { 𝑺ˆ (𝜔) 𝑺ˆ (𝜔), ·} d𝜔, 2 −∞ 𝑎∈𝐴 ∫ ∑︁ ∞ √︁ D 𝑠𝑒𝑐 = 𝛾(𝜔)𝛾(−𝜔) 𝑺ˆ 𝑎 (𝜔) ⊗ 𝑺ˆ 𝑎 (𝜔) ∗ 𝑎∈𝐴 − −∞  𝛾(𝜔)  ˆ 𝑎 𝑺 (𝜔) † 𝑺ˆ 𝑎 (𝜔) ⊗ 𝑰 + 𝑰 ⊗ 𝑺ˆ 𝑎 (𝜔) †∗ 𝑺ˆ 𝑎 (𝜔) ∗ d𝜔, 2 with the operators ∑︁ 𝑺ˆ 𝑎 (𝜔) := 𝑓ˆ𝑠 (𝜔 − 𝜈) 𝑨𝜈𝑎 where 𝑓ˆ𝑠 (𝜔) := 𝑓ˆ(𝜔) · 1 (|𝜔| < 𝜇). 𝜈∈𝐵(𝑯) If ∥ 𝑓 ∥ 2 ≤ 1 and ∥ Í 𝑎∈𝐴 𝑨 𝑎† 𝑨𝑎 ∥ ≤ 1, then, L 𝛽 − L 𝑠𝑒𝑐 1−1 ≤ 4∥ 𝑓ˆ − 𝑓ˆ𝑠 ∥ 2 Moreover, assuming the symmetry and normalization conditions (2.10),(2.11), we have D 𝑠𝑒𝑐 = D†𝑠𝑒𝑐 and ∥ D 𝛽 − D 𝑠𝑒𝑐 ∥ ≤ 4∥ 𝑓ˆ − 𝑓ˆ𝑠 ∥ 2 . Let us quickly note that the above argument controls the implementation error for truncating the Hamiltonian simulation. Indeed, in practice, we will only implement Hamiltonian simulation up to time 𝑇, and this is perfectly accounted for by Lemma 2.6.1,Corollary 2.6.1 by setting 𝑓 ′ = 𝑓𝑇 and 𝑯′ = 𝑯. Uniform weights Consider the simplest Fourier Transform with uniform weights ∑︁ 1 e−i𝜔¯ 𝑡¯ 𝑨( 𝑡¯). 𝑨ˆ 𝑓 ( 𝜔) ¯ := √︁ 2𝑁𝑇/𝑡0 −𝑇 ≤𝑡¯<𝑇 Proposition 2.6.3 (Preparing uniform weights). Suppose that 𝑇/𝑡 0 = 2 𝑘 . Then, the state ∑︁ 𝑡¯ 𝑓 ( 𝑡¯)| 𝑡¯⟩ for    1 if − 𝑇 ≤ 𝑡¯ < 𝑇  𝑓 ( 𝑡¯) = √︁ · 2𝑇/𝑡 0   0 else.  1 can be prepared using 𝑘 + 1 Hadamard gates and 𝑛 − 𝑘 − 1 CNOT gates. 72 Proof. Prepare with the GHZ state on the first 𝑛 − 𝑘 qubits using 1 Hadamard gate and 𝑛 − 𝑘 − 1 CNOT gates  1  √ 1𝑛−𝑘 + 0𝑛−𝑘 0 𝑘 2 and then apply Hadamard gates on the last 𝑘 qubits. ■ Í Since the weights are real 𝑓 (𝑡) = 𝑓 ∗ (𝑡) and normalized 𝑡¯∈𝑆𝑡 | 𝑓 ( 𝑡¯)| 2 = 1, the 0 transformed operator satisfies the properties listed in Proposition 2.5.1. Gaussian ansatz Instead of the plain Fourier Transform, consider the Gaussian-weighted Fourier Transform 1 ∑︁ −i𝜔¯ 𝑡¯ 𝑨ˆ 𝑓 ( 𝜔) ¯ := √ e 𝑓 ( 𝑡¯) 𝑨( 𝑡¯) 𝑁 𝑡¯∈𝑆𝑡 for Í 0 2 1 𝑓 ( 𝑡¯) := √︂ ∑︁ 2 − 𝑡¯ 2 2𝜎𝑡 𝑡¯∈𝑆𝑡0 e − 𝑡¯ 2 e 4𝜎𝑡 | 𝑡¯⟩. 𝑡¯∈𝑆𝑡0 ∫∞ Again, since the weight is real 𝑓 (𝑡) = 𝑓 ∗ (𝑡) and normalized −∞ | 𝑓 (𝑡)| 2 d𝑡 = 1, the transformed operator satisfies the symmetry properties listed in Proposition 2.5.1. To implement the above operator, we just need to prepare the initial state approximately. Proposition 2.6.4 (Preparing a truncated Gaussian state [127]). Suppose (𝑁𝑡 0 ) 2 /16𝜎𝑡2 ≥ log(1/𝜖). Then, the state 2 1 √︂ Í ∑︁ 2 𝑡¯∈𝑆𝑡0 e − 𝑡¯ 2 e 4𝜎𝑡 | 𝑡¯⟩ − 𝑡¯ 2 𝑡¯∈𝑆𝑡 2𝜎𝑡 0 can be prepared using O (𝑛 log(1/𝜖) 5/4 ) gates up to error 𝜖. The main advantage of using a Gaussian weight is that its Fourier Transform remains a Gaussian, which has a rapidly decaying tail. Indeed, for the continuous Gaussian, we can evaluate the Gaussian integral by completing the square 1 √︃ √ 𝜎𝑡 2𝜋 ∫ ∞ 2 −i𝜔𝑡 e − 𝑡 2 e 4𝜎𝑡 √︃ √ 2 2 d𝑡 = 𝜎𝑡 2𝜋e−𝜔 𝜎𝑡 . −∞ The uncertainty in energy is inversely proportional to the uncertainty in time 𝜎𝑡−1 , as a manifestation of the energy-time uncertainty principle. 73 Discretizing continuous functions via periodic summation It is not obvious how to carefully derive bounds on the discretization errors that appear in the Riemann sums of the discrete Fourier Transform, as the Fourier phases are highly oscillatory. Nevertheless, the discrete Fourier Transform remains Gaussian, up to a well-controlled error. Proposition 2.6.5 (DFT of Gaussian). There is a choice of parameter (𝑁𝑡 0 ) 2 /𝜎𝑡2 = Ω(log(1/𝜖)) and (𝑁𝜔0 ) 2 𝜎𝑡2 = Ω(log(1/𝜖)) such that the discrete Fourier Transform for 𝑓 ( 𝑡¯) = √︂ Í 2 − 𝑡¯ 2 1 2 e 4𝜎𝑡 is approximately 2𝜎𝑡 2 2 2 2 e−𝜔¯ 𝜎𝑡 −𝜔¯ 2 𝜎𝑡2 𝜔∈𝑆 ¯ 𝜔0 e − 𝑡¯ 2 𝑡¯∈𝑆𝑡0 e 1 √︃Í 2 2 2 2 up to error O ( 𝑁𝑡10 𝜎𝑡 e−𝑁 𝑣 0 𝜎𝑡 /2 + 𝑁𝑡01/𝜎𝑡 e−𝑁 𝑡0 /16𝜎𝑡 ) in 2-norm. To relate the continuous Fourier Transform to the discrete one, we apply the discretization to a continuous-variable function after periodic summation. This is related to the Poisson Summation Formula [190, Chapter II §13], but pushes the idea one step further to the realm of discrete Fourier Transform. Similar ideas are used, e.g., in lattice cryptography (c.f., [151]), but we include a self-contained treatment for completeness. To state the following general result, we introduce the notation ℓ1 (R) for (Lebesque) integrable R → C functions. ∫∞ Fact 2.6.1. Consider the Fourier Transform 𝑓ˆ(𝜔) := √1 −∞ e−i𝜔𝑡 𝑓 (𝑡)d𝑡 of a 2𝜋 function 𝑓 ∈ ℓ1 (R). Suppose that a “wrapped around” version of 𝑓 can be Í defined such that 𝑝(𝑡) = ∞ 𝑛=−∞ 𝑓 (𝑡 + 𝑛𝑁𝑡 0 ) for almost every 𝑡 ∈ R (i.e., the set of points where the equality does not hold has Lebesgue measure 0), 𝑝(𝑡) is continuous at every 𝑡¯ ∈ 𝑆𝑡0 and Riemann integrable on the interval [− 𝑁2 𝑡0 , 𝑁2 𝑡 0 ]. If the sequence 𝑘 · 𝑓ˆ(𝑘𝜔0 ) : 𝑘 ∈ Z is bounded in absolute value, then the limit Í𝐵 𝑝( ˆ 𝜔) ¯ := lim𝐵→∞ ℓ=−𝐵 𝑓ˆ( 𝜔¯ + ℓ𝑁𝜔0 ) exists for every 𝜔¯ ∈ 𝑆𝜔0 and 𝑝( ˆ 𝜔) ¯ is the discrete Fourier Transform of 𝑝( 𝑡¯), i.e., 𝑡 0 ∑︁ −i𝜔¯ 𝑡¯ 𝑝( ˆ 𝜔) ¯ =√ e 𝑝( 𝑡¯). 2𝜋 𝑡¯∈𝑆𝑡 0 74 Proof. For every 𝜔 ∈ 𝜔0 · Z we have that ∫ ∞ ˆ𝑓 (𝜔) = √1 e−i𝜔𝑡 𝑓 (𝑡)d𝑡 (since 𝑓 ∈ ℓ1 (R)) 2𝜋 −∞ ∞ ∫ 𝑁 𝑡0 2 1 ∑︁ =√ e−i𝜔(𝑡+𝑛𝑁𝑡0 ) 𝑓 (𝑡 + 𝑛𝑁𝑡0 )d𝑡 𝑁 2𝜋 𝑛=−∞ − 2 𝑡0 (by Fubini’s theorem since 𝑓 ∈ ℓ1 (R)) 𝑁 ∫ ∞ 2 𝑡0 1 ∑︁ =√ e−i𝜔𝑡 𝑓 (𝑡 + 𝑛𝑁𝑡0 )d𝑡 (since 𝜔 ∈ 𝜔0 · Z and 𝜔0 𝑡0 = 2𝜋 𝑁) 𝑁 2𝜋 𝑛=−∞ − 2 𝑡0 ∫ 𝑁 𝑡0 ∑︁ ∞ 2 1 =√ e−i𝜔𝑡 𝑓 (𝑡 + 𝑛𝑁𝑡0 )d𝑡 𝑁 2𝜋 − 2 𝑡0 𝑛=−∞ (by Fubini’s theorem since 𝑓 ∈ ℓ1 (R)) ∫ 𝑁 𝑡0 2 Í 1 e−i𝜔𝑡 𝑝(𝑡)d𝑡. (since 𝑝(𝑡) = ∞ =√ 𝑛=−∞ 𝑓 (𝑡 + 𝑛𝑁𝑡 0 ) a.s.) 2𝜋 − 𝑁2 𝑡0 √ The above equality means that 𝑐 𝑘 := 𝑁𝑡2𝜋0 · 𝑓ˆ(𝑘 · 𝜔0 ) is the Fourier series of 𝑝(𝑡). Since 𝑝 is Riemann integrable on the interval [− 𝑁2 𝑡0 , 𝑁2 𝑡0 ], continuous at every 𝑡¯ ∈ 𝑆𝑡0 point and 𝑘 · 𝑐 𝑘 is bounded by assumption, we have for all 𝑡¯ ∈ 𝑆𝑡0 that 𝑝( 𝑡¯) = lim 𝐵 ∑︁ 𝑐 𝑘 ei𝑡¯𝑘𝜔0 (due to [105, Theorem 15.3]) 𝐵→∞ = lim 𝑘=−𝐵 𝐵𝑁 ∑︁ 𝑐 𝑘 ei𝑡¯𝑘𝜔0 𝐵→∞ = lim 𝑘=−𝐵𝑁 𝐵𝑁+⌊(𝑁−1)/2⌋ ∑︁ i𝑡¯𝑘𝜔0 𝑐𝑘 e 𝐵→∞   (since |𝑐 𝑘 | = O 1𝑘 ) 𝑘=−𝐵𝑁−⌈(𝑁−1)/2⌉ = lim 𝐵 ∑︁ ⌊(𝑁−1)/2⌋ ∑︁ 𝑐 ℓ𝑁+ 𝑗 ei𝑡¯(ℓ𝑁+ 𝑗)𝜔0 (set 𝑘 = ℓ𝑁 + 𝑗) 𝐵→∞ ℓ=−𝐵 𝑗=−⌈(𝑁−1)/2⌉ = lim 𝐵 ∑︁ ∑︁ √2𝜋 𝐵→∞ ℓ=−𝐵 𝜔∈𝑆 ¯ 𝜔0 𝑁𝑡 0 𝑓ˆ(ℓ𝑁𝜔0 + 𝜔)e ¯ i𝑡¯𝜔¯ . (set 𝜔¯ = 𝑗𝜔0 ; use 𝑡¯ ∈ 𝑡0 · Z and 𝜔0 𝑡0 = 2𝜋 𝑁) ! ¯ + µ̄ <latexit 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⌫ ! ¯ ! 0 <latexit 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|¯ ! ⌫| 75 fˆ(¯ ! <latexit 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f (t̄) f!¯ E <latexit 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t0 2 ⇠ e ⌦(T |¯! ⌫| ) <latexit 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⌫ ! ¯ ! 0 <latexit 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Figure 2.9: Left: the weight function as (approximately) the Gaussian distribution but truncated at 𝑇.⌫¯ µ̄Right: An illustration for the Fourier Transformed amplitudes, 0 which is also (approximately) Gaussian. It peaks near energy 𝜈 = 𝜔¯ with a width ∼ 𝑇 −1 and decays exponentially. The secular approximation truncates the profile at an energy 𝜇¯ in the tail 𝜇¯ ∼ 𝑇 −1 . <latexit sha1_base64="8nMmsIZisDv1p/hMjDrYYGEU6OM=">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</latexit> <latexit sha1_base64="8nMmsIZisDv1p/hMjDrYYGEU6OM=">AAAOLnichZfZbuM2FIY1022a1u2kveyNUMNAUbSBHUyXmwEmUfbVSbxNIiMgZdomTC2gKHYMQ32H3rYv0acp0Iuit32MUqIdMTxuRzcmz/cfLkf8JQsnjKai2fzzydN33n3v/Q+efbjx0ce1Tz59vvlZL40zHpBuELOYDzBKCaMR6QoqGBkknKAQM9LHM6/gfUl4SuOoI+YJGYZoEtExDZBQob6P45BM0P3zenOrWV4ubLSWjbqzvNr3m7VNfxQHWUgiETCUpnetZiKGC8QFDRjJNzb8LCUJCmZoQhYoTNN5iHO3ESIxTW1WBNexu0yMfxwuaJRkgkSBkig2zpgrYrfYizuinASCzVUDBZyqqd1gijgKhNrx2lkerwuHKtBwzSnLiqpiztKXgmfkGxcxVsbSl5hlZLiYqok5J2OVaQ3G0YyIfKNhBiMkMMXFLA2EMSdyw+ckIj8FcRiiaLTwJQnyhV8sxDfD3o6KFisOEFvs5DbdNeguoIh5BvdsvruiGEPo7Rmpe4DuG3QfDLxfDQygd2CkHgB6aNBDQI8MegTosUGPAT0x6AmgpwY9BfTMoGeAnhv0HNALg16AUl1UpQLQuzRSLwFtG7QNBm5XAwPoXRmpVyD1qkoF0Ls2Uq9B6nWVCqB3Y6TeANoxaAfQrkG7gPYM2gO0b9A+oAODDgB97RpYdWx+a+Dbx/TrwtVJqgQScdWgLI6Kp8WjAeROaXvobYk0QADsagD8LrEGGABPA2ByuacB8Lfc1wC4Vx5oAIwrDzUAnpVHGgC7ymMNgFPliQbApPJUA+BPeaYBsKY81wC4Ul5oADwnLzUAdpNtDYCZ5JUGwCryWgNgBHmjAfCA7GgAjr/sagBOvuxpAA697GsAzrscaACOuiyPuiLwlMtbTW4BSOkkXB5U19cdS4IrDeJKJNeqJJ/Gq2GKZm6/H4s7zMhYWHlFiTmdTAUwFo6y5Zy+allp0xJOkVgH8SoRkvBhzBDC8i1RQq84iI3HdH8Fgaf0X68V1h1LMkKTSa5/CM+tnbLiDLAe4XZxeAH4GqD+CApKaX7XGi50WcuQPyM8+ra5tf0defO2qFtvubryUPTWsL29KOZhuRi/WKy7qLdy11+z7kQLt4c6ZVHfzu8L8ZrhElPWKmTbtowZOvZ/wnG1vkpWPvV5+SQEN4QhnGo9ppOJrpfe1EPfXstDRlVhnWFUbc3yjZyenbOmgCR/WDexd0mziIqKU5tz+p9QvekEr2iHA5zEbF4J2kUPjDBPKkXRsQUdU9BZI0BcPVneLDVxopYcuir2cxH0GQ2pSO0UOspLmfq1yE2WGNOVPfs+02hUfD/Fq61j9fhSh1F9RLXsTybY6G1vtb7fenH1ov5qd/k59cz5wvnS+cppOT84r5wjp+10ncCZOb84vzq/1X6v/VH7q/a3lj59ssz53Hl01f75F1pMaSo=</latexit> ! ¯ <latexit sha1_base64="JN1tWmp3eokyvRsD817HF0U4ZzM=">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</latexit> <latexit sha1_base64="dtarvXYF+rcQgO8BaY6yywvHBQU=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPBi94q2A9IQtlst+3SzSbsToQS+jO8eFDEq7/Gm//GbZuDtj4YeLw3w8y8KJXCoOt+O6W19Y3NrfJ2ZWd3b/+genjUNkmmGW+xRCa6G1HDpVC8hQIl76aa0ziSvBONb2d+54lrIxL1iJOUhzEdKjEQjKKV/CCiOg9UNu25vWrNrbtzkFXiFaQGBZq96lfQT1gWc4VMUmN8z00xzKlGwSSfVoLM8JSyMR1y31JFY27CfH7ylJxZpU8GibalkMzV3xM5jY2ZxJHtjCmOzLI3E//z/AwHN2EuVJohV2yxaJBJggmZ/U/6QnOGcmIJZVrYWwkbUU0Z2pQqNgRv+eVV0r6oe1f1y4fLWuO+iKMMJ3AK5+DBNTTgDprQAgYJPMMrvDnovDjvzseiteQUM8fwB87nD1FGkU8=</latexit> 0 ! ¯ ! ¯ + µ̄0 <latexit sha1_base64="HwiqrWG/70k+r2EGMAC/0SPJDFw=">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</latexit> Finally, for all 𝜔¯ ′ ∈ 𝑆𝜔0 we have 𝐵 ∑︁ ∑︁ ∑︁ 1 𝑡0 ∑︁ −i𝑡¯𝜔¯ ′ ¯𝜔¯ ′ −i 𝑡 e 𝑝( 𝑡¯) = e lim 𝑓ˆ( 𝜔¯ + ℓ𝑁𝜔0 )ei𝑡¯𝜔¯ √ 𝐵→∞ 𝑁 2𝜋 𝑡¯∈𝑆𝑡 ℓ=−𝐵 𝜔∈𝑆 ¯ 𝜔 𝑡¯∈𝑆𝑡 0 0 0 𝐵 ∑︁ ∑︁ 1 ∑︁ ′ 𝑓ˆ( 𝜔¯ + ℓ𝑁𝜔0 ) e−i𝑡¯𝜔¯ ei𝑡¯𝜔¯ 𝐵→∞ 𝑁 ¯ ℓ=−𝐵 𝜔∈𝑆 ¯ = lim 𝑡 ∈𝑆𝑡0 𝜔0 = lim 𝐵→∞ 𝐵 ∑︁ 𝑓ˆ( 𝜔¯ ′ + ℓ𝑁𝜔0 ) ℓ=−𝐵   Í ¯ 𝜔¯ ′ ) = 𝑁 · 𝛿 since 𝑡¯∈𝑆𝑡 ei𝑡¯(𝜔− 𝜔, ¯ 𝜔¯ ′ 0 = 𝑝( ˆ 𝜔¯ ′). ■ Using the above, we prove Proposition 2.6.5. Proof of Proposition 2.6.5. Apply Fact 2.6.1 for the Gaussian 𝑓 (𝑡) = √︁ ⇠ 2 2 1 ⇠ e ⌦(tphy |¯! E| ) <latexit sha1_base64="qJVUOHSM9hpCm3uSHow9M47lD+4=">AAAOKnichZfLbuM2FIY109s0rdtJu+xGqBGgKNrADqaXzQCTKPerk9iOJ5ERkDJtE6EuoCh2AkN9g27bl+jTdDfotg9SUrQjhsftaGPyfP/h5Yi/ZOGM0Vy0Wm+fPH3v/Q8+/OjZxyuffNr47PPnq1/087TgEelFKUv5AKOcMJqQnqCCkUHGCYoxI1f4LtD8ShKe0zTpivuMDGM0SeiYRkio0GWYFLfPm631VnX5sNGeN5re/OrcrjZWw1EaFTFJRMRQnt+0W5kYzhAXNGKkXFkJi5xkKLpDEzJDcZ7fx7j012IkprnLdHAZuynE+OfhjCZZIUgSKYli44L5IvX1PvwR5SQS7F41UMSpmtqPpoijSKjdLp3l8bpwrAJrvj1lVU1VyLv8peAF+c5HjFWx/CVmBRnOpmpizslYZTqDcXRHRLmyZgcTJDDFepY1hDEnciXkJCG/RGkco2Q0CyWJylmoFxLa4WBTRfWKI8Rmm6VLtyy6BShigcUDl28tKMYQBttW6jagOxbdAQPv1AMDGOxaqbuA7ll0D9B9i+4DemDRA0APLXoI6JFFjwA9tugxoCcWPQH01KKnoFSndakADM6s1DNAOxbtgIE79cAABudW6jlIPa9TAQwurNQLkHpRpwIYXFqpl4B2LdoFtGfRHqB9i/YBvbLoFaADiw4Afe1bWHVcfm3h68f0W+3qLFcCibhqUJYm+mnxaAC5WdkeelsiAxAAWwYAv0tsAAYgMACYXG4bAPwtdwwA7pW7BgDjyj0DgGflvgHArvLAAOBUeWgAMKk8MgD4Ux4bAKwpTwwArpSnBgDPyTMDgN1kxwBgJnluALCKvDAAGEFeGgA8ILsGgOMvewaAky/7BoBDL68MAOddDgwAR11WR10ReMrltSHXAOR0Es8Pqh+ajiPBtQZxJZJLVZJP08Uwulm670d9hxkZCydPl5jTyVQAY+GkmM+p/te4000rOEViGcSLREjihzFjCKu3RAUDfRDXHtOdBQSewmlMJosChabjSEZoMinND+Gls1OmzwDrE+4Wh2vAlwD1J1BQSsub9nBmylqFwjvCk+9b6xs/kDfvivrNtm8qD0XvDLvbS1IeV4sJ9WL9WbNd+uGSdWdGuDE0KbPmRnmrxUuGy2xZW8s2XBmzdOz/hON6fbWseurz6kkIbghDODd6TCcTUy+zqYe+u5aHjLrCJsOq2pLlWzl9N2dJAUn5sG7i7pIWCRU1py7n9D+hetMJXtMuBzhL2X0t6OgeGOE+qxW64wq6tqC7RIC4erK8mWvSTC059lXsVx0MGY2pyN0UOiormfp1yGWRWdNVPfc+02Skv53Sxdaxenypw6g+otruJxNs9DfW2z+uvzh/0Xy1Nf+ceuZ95X3tfeO1vZ+8V96+1/F6XuRNvN+8370/Gn82/mq8bfxtpE+fzHO+9B5djX/+BXkWZ3o=</latexit> 𝑔𝜎 (𝑡) = 2 − 𝑡 2 √ 1√ e 4𝜎𝑡 . The problem reduces to implementing the periodic sum 𝑝( 𝑡¯) (Fig𝜎𝑡 2𝜋 ure 2.9) approximately. Up to a small error from the Gaussian tail, we may only keep the centered Gaussian (𝑛 = 0) as long as the Gaussian is largely confined in the window for both the time domain (𝑁𝑡 0 ) 2 ≫ 𝜎𝑡2 and frequency domain (𝑁𝜔0 ) 2 ≫ 𝜎𝑡2 . ■ Tail bounds We evaluate the tail bounds that appear in the analysis of the secular approximation. First, we consider the case of the uniform weights whose Fourier Transform has a |E 76 heavy tail impacting the accuracy. √ ≤𝑡 <𝑇) ei𝜈𝑡¯. Then, Proposition 2.6.6 (Tail bound for uniform weights). Let 𝑓 ( 𝑡¯) := 1 (−𝑇 ¯ 2𝑇/𝑡0 its discrete Fourier Transform is 1 e 𝑓ˆ( 𝜔) ¯ = √︁ 2𝑇 𝑁/𝑡0 i(𝜈−𝜔)𝑇 ¯ − e−i(𝜈−𝜔)𝑇 ¯ ¯ 0 −1 ei(𝜈−𝜔)𝑡 with a tail bound ∑︁ 2 𝑓ˆ( 𝜔) ¯ ≤ | 𝜔|>𝑚𝜔 ¯ 0 𝜋 . 2𝑚𝜔0𝑇 ¯ 𝑡¯ Proof. The Fourier Transform gives a geometric series with ratio ei(𝜈−𝜔) ¯ − e−i(𝜈−𝜔)𝑇 ¯ ∑︁ 1 ei(𝜈−𝜔)𝑇 1 1 ∑︁ −i𝜔¯ 𝑡¯ ¯ 𝑡¯ ei(𝜈−𝜔) = √︁ e 𝑓 ( 𝑡¯) = √︁ . √ ¯ 0 −1 ei(𝜈−𝜔)𝑡 𝑁 𝑡¯∈𝑆𝑡 2𝑇 𝑁/𝑡0 −𝑇 ≤𝑡¯<𝑇 2𝑇 𝑁/𝑡 0 0 This function scales inversely with 𝜔¯ ∑︁ ∑︁ 1 4 2 𝑓ˆ( 𝜔) ¯ ≤ 2𝑁𝑇/𝑡0 ¯ 0 −12 ei𝜔𝑡 | 𝜔|>𝑚𝜔 ¯ 0 | 𝜔|>𝑚𝜔 ¯ 0 ≤ 𝑡0 2𝑁𝑇 𝜋2 ( 𝜔𝑡 ¯ 0)2 | 𝜔|>𝑚𝜔 ¯ ∑︁ (since ei𝑥 − 1 ≥ 𝜋2 |𝑥| for 𝑥 ∈ [−𝜋, 𝜋]) 0 ∞ ∑︁ 1 ≤ 𝑁𝑇𝑡0 𝜔20 𝑛=𝑚+1 𝑛2 𝜋2 ∞ ∑︁ 1 𝜋 2𝑇𝜔0 𝑛=𝑚+1 𝑛(𝑛 − 1) 𝜋 . = 2𝑚𝜔0𝑇 ≤ (since 𝜔0 𝑡0 = 2𝜋 𝑁) ■ In retrospect, it is very important that we consider the 2-norm of the tail here; the 1-norm would be divergent. Now, we consider Gaussians, which have a rapidly decaying tail, greatly improving the accuracy. Proposition 2.6.7 (Tail bound for Gaussian weights). For the function 𝑓 ( 𝑡¯) = 2 2 − 𝑡¯ 2 − 𝑡¯ 2 Í 2𝜎𝑡 −1/2 4𝜎𝑡 ) e , the Fourier-transformed tail satisfies ( 𝑡¯∈𝑆𝑡 e 0 ! v u ∑︁ u 2 2 2 2 2 2 2 2 1 1 1 2 u t 𝑓ˆ( 𝜔) ¯ ≤O √ e−𝑁 𝜔0 𝜎𝑡 /2 + √︁ e−𝑁 𝑡0 /16𝜎𝑡 + √ e− 𝜇¯ 𝜎𝑡 . 𝜇𝜎 ¯ 𝑡 𝑁𝜔0 𝜎𝑡 𝑁𝑡 0 /𝜎𝑡 ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 | 𝜔¯ | ≥ 𝜇¯ 77 Proof. By Proposition 2.6.5, the discrete Fourier Transform 𝑓ˆ( 𝜔) ¯ is approximately 2 2 2 2 2 2 2 2 1 e−2𝜔¯ 𝜎𝑡 up to error O ( √𝑁𝜔1 𝜎 e−𝑁 𝑣 0 𝜎𝑡 /2 + √ 1 e−𝑁 𝑡0 /16𝜎𝑡 ). We Í − 𝜔¯ 2 𝜎 2 𝜔¯ ∈𝑆 𝜔0 e 𝑁𝑡 0 /𝜎𝑡 0 𝑡 𝑡 then control the tail bound 1 Í ∑︁ −𝜔¯ 2 𝜎𝑡2 𝜔∈𝑆 ¯ 𝜔 e 0 2.7 2 2 e−2𝜔¯ 𝜎𝑡 = O ( | 𝜔|≥ ¯ 𝜇¯ 1 −2 𝜇¯ 2 𝜎𝑡2 e ). 𝜇𝜎 ¯ 𝑡 ■ Appendix:Proving approximate detailed balance In this section, we prove approximate detailed balance (or discriminant proxy) for the constructed discriminant. It amounts to controlling the error arising from Boltzmann factors due to the finite resolution of the operator Fourier Transform. A simpler but weaker bound We begin with a simpler but weaker bound. This will be enough for the Gaussiandamped discriminant due to its rapidly decaying tail. We can bootstrap this weaker bound using a more careful truncation scheme for the special case of uniform weight (which has a heavy tail) as shown in section 2.7. To prove our error bound, we introduce two useful technical lemmas. Lemma 2.7.1 (Norm bound on block-band matrices). Let 𝑉𝑖 ⊆ H and 𝑊𝑖 ⊆ H ′ be É systems of mutually orthogonal subspaces of H and H ′ respectively. If 𝑴 = 𝑖 𝑩𝑖 where 𝑩𝑖 : 𝑉𝑖 → 𝑊𝑖 , then ∥ 𝑴 ∥ = max𝑖 ∥ 𝑩𝑖 ∥. Proof. We can get a singular value decomposition of 𝑴 by taking singular value decompositions of each 𝑩𝑖 and then merging them. Since ∥ 𝑴 ∥ is the largest singular value, we get the claimed equality. ■ Lemma 2.7.2 (Norm bounds on sums of tensor products of matrices). Let |𝐼 | ≤ ∞ ′ ′ and 𝑨𝑖 ∈ C𝑛×𝑚 , 𝑩𝑖 ∈ C𝑛 ×𝑚 for each 𝑖 ∈ 𝐼, then √︄ ∑︁ ∑︁ ∑︁ ∑︁ † ∗ 𝑨𝑖 𝑨𝑖† ∥∥ 𝑩𝑖† 𝑩𝑖 ∥. ∥ 𝑨𝑖 [·] 𝑩𝑖 ∥ 2−2 = ∥ 𝑨𝑖 ⊗ 𝑩 ∥ ≤ ∥ 𝑖∈𝐼 Proof. Define the maps ∑︁ 𝑽 := 𝑨𝑖 ⊗ 𝑰 ⊗ ⟨𝑖| 𝑖∈𝐼 ∥ ∑︁ 𝑖∈𝐼 𝑖∈𝐼 and 𝑼 := ∑︁ 𝑖∈𝐼 𝑰 ⊗ 𝑩∗ ⊗ |𝑖⟩, then 𝑖∈𝐼 𝑨𝑖 ⊗ 𝑩∗ ∥ = ∥𝑽𝑼∥ ≤ ∥𝑽 ∥∥𝑼∥ ≤ ∥ ∑︁ 𝑖∈𝐼 Take complex conjugate to conclude the proof. 𝑖∈𝐼 𝑨𝑖 𝑨𝑖† ∥ 1/2 ∥ ∑︁ 𝑩∗† 𝑩∗ ∥ 1/2 . 𝑖∈𝐼 ■ 78 Lemma 2.7.3 (Secular approximation gives discriminant proxy). Consider the following discriminant proxy and a closely related Lindbladian ∑︁ ∑︁ √︁ D 𝑠𝑒𝑐 := 𝛾( 𝜔)𝛾(− ¯ 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ 𝑎∈𝐴 𝜔∈𝑆 ¯ 𝜔0  𝛾( 𝜔) ¯  ˆ𝑎 𝑺 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑰 + 𝑰 ⊗ 𝑺ˆ 𝑎 ( 𝜔) ¯ †∗ 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ , 2 ∑︁ ∑︁ 𝛾( 𝜔) ¯ ˆ𝑎 L 𝑠𝑒𝑐 = 𝛾( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ [·] 𝑺ˆ 𝑎 ( 𝜔) ¯ †− { 𝑺 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔), ¯ ·}, 2 𝑎∈𝐴 𝜔∈𝑆 ¯ − 𝜔0 such that the nonnegative weights satisfy42 𝛾( 𝜔)/𝛾(− ¯ 𝜔) ¯ = e−𝛽𝜔¯ or 𝛾( 𝜔) ¯ = 𝛾(−𝜔) ¯ = 0 for each 𝜔¯ ∈ 𝑆𝜔0 . (2.46) Suppose the operators satisfy that ⟨𝜓𝑖 | 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝜓𝑗 = 0 whenever (𝐸𝑖 − 𝐸 𝑗 ) − 𝜔¯ > 𝜇¯ (2.47) Í for the eigenvalue decomposition of 𝑯 = 𝑗 𝐸 𝑗 |𝜓 𝑗 ⟩⟨𝜓 𝑗 |. Then, for any 𝛽, 𝜇¯ > 0 such that 𝛽 𝜇¯ ≤ 1 and the Gibbs state 𝝆 = e−𝛽𝑯 /Tr[e−𝛽𝑯 ], ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 )∥ √︄ ∑︁ ≤ 𝛽 𝜇¯ ∥ 𝛾( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)∥ ¯ 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 √︄ √︄ ∑︁ ∑︁ © ª 𝑎 † 𝑎 ˆ ˆ 𝛾( 𝜔) ¯ 𝑺 ( 𝜔) ¯ 𝑺 ( 𝜔)∥ ¯ + 125 ∥ 𝛾(−𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † ∥ ®, ­7 ∥ 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 « ¬ where D( 𝝆, L 𝑠𝑒𝑐 ) is the vectorization of D ( 𝝆, L 𝑠𝑒𝑐 ). Finally, if there is a permuta√︁ √︁ ′ tion 𝑷 : 𝑎 → 𝑎′ such that42 𝛾(−𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † = 𝛾(−𝜔) ¯ 𝑺ˆ 𝑎 (−𝜔) ¯ for each 𝑎 and 𝜔, ¯ † then we have that D𝑠𝑒𝑐 = D𝑠𝑒𝑐 and that ∑︁ ∑︁ ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 ) † ∥ = ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 )∥ ≤ 132𝛽 𝜇∥ ¯ 𝛾( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)∥. ¯ 𝑎∈𝐴 𝜔∈𝑆 ¯ 𝜔0 42 Potentially allowing zero values of 𝛾( 𝜔) ¯ = 𝛾(−𝜔) ¯ = 0 might be needed for dealing with the case when 𝑁 is even and therefore the smallest label in 𝑆 𝜔0 would be its own inverse (due to parity and the modulo arithmetic of 𝑆 𝜔0 ). Also note that if 𝑺ˆ 𝑎 ( 𝜔) ¯ = 𝑺 𝑎− 𝜔¯ = 0 for all 𝑎 ∈ 𝐴, then we can assume without loss of generality that the corresponding weight is 𝛾( 𝜔) ¯ = 𝛾(−𝜔) ¯ = 0. 79 This also quickly leads to approximate detailed balance for Lindbladians (Lemma 2.2.2) by a triangle inequality D ( 𝝆, L 𝑠𝑒𝑐 ) − D ( 𝝆, L 𝑠𝑒𝑐 ) † 2−2 ≤ ∥D ( 𝝆, L 𝑠𝑒𝑐 ) − D𝑠𝑒𝑐 ∥ 2−2 + ∥D𝑠𝑒𝑐 − D ( 𝝆, L 𝑠𝑒𝑐 ) † ∥ 2−2 = ∥ D( 𝝆, L 𝑠𝑒𝑐 ) − D 𝑠𝑒𝑐 ∥ + ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 ) † ∥. Í Note that the term ∥ 𝑎,𝜔¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)∥ ¯ can be thought of as the “strength” of the interaction, and in our case can be simply bounded by 1 due to Proposition 2.5.1 as follows: † ∑︁ ∑︁  ∑︁ ∥ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)∥ ¯ =∥ 𝑨𝑎𝑓𝑠𝑒𝑐 ( 𝜔) ¯ 𝑨𝑎𝑓𝑠𝑒𝑐 ( 𝜔)∥ ¯ =∥ 𝑨𝑎† 𝑨𝑎 ∥ · ∥| 𝑓𝑠𝑒𝑐 ⟩∥ 2 ≤ 1. 𝑎,𝜔¯ 𝑎,𝜔¯ 𝑎∈𝐴 Proof of Lemma 2.7.3. Our proof adapts from the strategy of [38] for dealing with approximate detailed balance. First, let us define projectors that partition the spectrum per truncation frequency 𝜇¯ as follows: ∑︁ ∑︁ 𝑷𝑖 := |𝜓 𝑗 ⟩⟨𝜓 𝑗 | for each 𝑖 ∈ Z such that 𝑷𝑖 = 𝑰 and 𝑷𝑖 𝑯 = 𝑯𝑷𝑖 . 𝑗: 𝐸𝑗 1 1 𝜇¯ ∈[(𝑖− 2 ),(𝑖+ 2 )) 𝑖 In other words, these projectors provide a resolution of the identity, and moreover, they commute with 𝑯. We proceed by decomposing the difference of the operators D 𝑠𝑒𝑐 − D′ as follows: 𝛿A:= ′ D 𝑠𝑒𝑐 − D = z ∑︁ √︁ ˆ𝑎 ∗ ˆ𝑎 }| 𝛾( 𝜔)𝛾(− ¯ 𝜔) ¯ 𝑺 ( 𝜔) ¯ ⊗ 𝑺 ( 𝜔) ¯ − 𝛾( 𝜔) ¯ 𝝆 −1/4 ˆ 𝑎 𝑺 ( 𝜔) ¯ 𝝆 1/4 ⊗𝝆 ∗−1/4 ˆ 𝑎 𝑺 ( 𝜔) ¯ 𝝆 𝑎,𝜔¯ + ∑︁ 𝛾( 𝜔) ¯  𝑎,𝜔¯ 2 | + {z 1 2 𝛿R⊗𝑰:= ∑︁ 𝛾( 𝜔) ¯  𝑎,𝜔¯ | 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑰 − 𝝆 −1/4 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝝆 1/4 ⊗ 𝑰 2  }  𝑎 †∗ ˆ 𝑎 ∗ ∗−1/4 ˆ 𝑎 ∗† ˆ 𝑎 ∗ ∗1/4 ˆ ¯ −𝑰⊗𝝆 𝑺 ( 𝜔) ¯ 𝑺 ( 𝜔) ¯ 𝝆 𝑰 ⊗ 𝑺 ( 𝜔) ¯ 𝑺 ( 𝜔) . {z ∗ 1 2 𝑰⊗𝛿R = } (2.48) { ∗ ∗1/4 80 By the triangle inequality we have that ∥ D 𝑠𝑒𝑐 − D′ ∥ ≤ ∥𝛿A∥ + 12 ∥𝛿R ⊗ 𝑰∥ + ∗ 1 2 ∥ 𝑰 ⊗ 𝛿R ∥ = ∥𝛿A∥ + ∥𝛿R∥. Now, we bound the above two terms individually, starting from the term ∥𝛿R∥. Our proof crucially relies on the fact that the frequency label of 𝑺ˆ 𝑎 ( 𝜔) ¯ closely approximates the true Bohr frequency, up to the truncation frequency 𝜇¯ as expressed by (2.47). This implies that 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ roughly preserves energy that ⟨𝜓𝑖 | 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝜓 𝑗 = 0 whenever 𝐸𝑖 − 𝐸 𝑗 > 2 𝜇. ¯ Indeed, ⟨𝜓𝑖 | 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝜓𝑗 = ∑︁ ⟨𝜓𝑖 | 𝑺ˆ 𝑎 ( 𝜔) ¯ † |𝜓 𝑘 ⟩⟨𝜓 𝑘 | 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝜓𝑗 𝑘 ∑︁ = ⟨𝜓𝑖 | 𝑺ˆ 𝑎 ( 𝜔) ¯ † |𝜓 𝑘 ⟩⟨𝜓 𝑘 | 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝜓𝑗 , |(𝐸 −𝐸 )−𝜔|≤ ¯ 𝜇¯ & 𝑘 : |(𝐸 𝑘 −𝐸 𝑖 )−𝜔|≤ ¯ 𝜇¯ 𝑘 𝑗 (due to (2.47)) since 𝐸𝑖 − 𝐸 𝑗 = (𝐸 𝑘 − 𝐸 𝑗 ) − 𝜔¯ − ((𝐸 𝑘 − 𝐸𝑖 ) − 𝜔) ¯ ≤ (𝐸 𝑘 − 𝐸 𝑗 ) − 𝜔¯ +|(𝐸 𝑘 − 𝐸𝑖 ) − 𝜔|, ¯ meaning that the above summands can only be nonzero when 𝐸𝑖 − 𝐸 𝑗 ≤ 2 𝜇. ¯ This observation enables us to introduce the following decomposition 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ = ∑︁ 𝑷𝑖 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑗= 𝑖, 𝑗 2 ∑︁ ∑︁ 𝑷𝑖 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑖+ℓ .(2.49) ℓ=−2 𝑖 Let us define 𝛿𝑯 𝑗 := 4𝛽 (𝑷 𝑗 𝑯𝑷 𝑗 − 𝑗 𝜇𝑷 ¯ 𝑗 ) and 𝛿𝑯 := commute and 𝑷 𝑗 = 𝑷2𝑗 we have 1 1 1 Í 𝑗 𝛿𝑯 𝑗 . Since 𝑯 and 𝑷 𝑗 𝛽 𝜇¯ 𝝆 ∓ 4 𝑷 𝑗 = 𝑷 𝑗 𝝆 ∓ 4 = (Tr(e−𝛽𝑯 )) ± 4 · e± 4 𝑗 𝑷 𝑗 e±𝛿𝑯 . (2.50) We use (2.49)-(2.50) to exploit the “approximate energy preservation” of the operator 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ as follows: 𝝆 −1/4 ˆ 𝑎 † ˆ𝑎 𝑺 ( 𝜔) ¯ 𝑺 ( 𝜔) ¯ 𝝆 1/4 = = 2 ∑︁ ∑︁ ℓ=−2 𝑖 2 ∑︁ ∑︁ ℓ=−2 𝑖 𝝆 −1/4 𝑷𝑖+ℓ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑖 𝝆 1/4 𝛽 𝜇¯ e 4 ℓ 𝑷𝑖+ℓ e𝛿𝑯 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)e ¯ −𝛿𝑯 𝑷𝑖 . (2.51) 81 Let us define 𝑺′ := ∥𝛿R∥: ∥𝛿R∥ = ∑︁ Í ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔), ¯ then we get the following bound on ¯ 𝑺 𝑎,𝜔¯ 𝛾( 𝜔)   𝛾( 𝜔) ¯ 𝝆 −1/4 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝝆 1/4 − 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ (by (2.48)) 𝑎,𝜔¯ = ∑︁ 𝛾( 𝜔) ¯ 𝑎,𝜔¯ 2 ∑︁ ∑︁ e 𝛽 𝜇¯ 4 ℓ ! 𝑷𝑖+ℓ e𝛿𝑯 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)e ¯ −𝛿𝑯 𝑷𝑖 − 𝑷𝑖+ℓ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑖 ℓ=−2 𝑖 (by (2.49)-(2.51)) 2 ∑︁  𝛽 𝜇¯  ∑︁ ∑︁ ℓ 𝛿𝑯 ˆ 𝑎 † ˆ𝑎 −𝛿𝑯 𝑎 † ˆ𝑎 ˆ 4 𝑷𝑖+ℓ 𝛾( 𝜔) ¯ e e 𝑺 ( 𝜔) ≤ ¯ 𝑺 ( 𝜔)e ¯ − 𝑺 ( 𝜔) ¯ 𝑺 ( 𝜔) ¯ 𝑷𝑖 ℓ=−2 𝑎,𝜔¯ 𝑖 (by triangle inequality) 2  𝛽 𝜇¯  ∑︁ ∑︁ ℓ 𝛿𝑯 ˆ 𝑎 † ˆ𝑎 −𝛿𝑯 𝑎 † ˆ𝑎 ˆ 4 = max 𝑷𝑖+ℓ 𝛾( 𝜔) ¯ e e 𝑺 ( 𝜔) ¯ 𝑺 ( 𝜔)e ¯ − 𝑺 ( 𝜔) ¯ 𝑺 ( 𝜔) ¯ 𝑷𝑖 . 𝑖 𝑎,𝜔¯ ℓ=−2 (by Lemma 2.7.1) We may now drop the project 𝑷𝑖 and 𝑷𝑖+ℓ and simplify via elementary bounds. ′ (𝑐𝑜𝑛𝑡 𝑑) ≤ ≤ ≤ 2 ∑︁ 𝛽 𝜇¯ e (𝛿𝑯− 4 ℓ𝑰) 𝑺′e−𝛿𝑯 − 𝑺′ ℓ=−2 2 ∑︁ ℓ=−2 2 ∑︁ 𝛽 𝜇¯ e (𝛿𝑯− 4 ℓ𝑰) 𝑺′e−𝛿𝑯 − 𝑺′e−𝛿𝑯 + 𝑺′e−𝛿𝑯 − 𝑺′ 𝛽 𝜇¯ ∥e (𝛿𝑯− 4 ℓ𝑰) − 𝑰∥∥𝑺′ ∥∥e−𝛿𝑯 ∥ + ∥𝑺′ ∥ e−𝛿𝑯 − 𝑰 ℓ=−2 ′ ≤ ∥𝑺 ∥ 2 ∑︁ ℓ=−2 ∥2𝛿𝑯 − 𝛽 𝜇¯ ℓ𝑰∥ (1 + ∥2𝛿𝑯∥) + ∥2𝛿𝑯∥ 2 (since |e𝑥 − 1| ≤ 2|𝑥| for |𝑥| ≤ 54 )    2 ∑︁ 1 |ℓ| 𝛽 𝜇¯ 𝛽 𝜇¯ ′ ≤ ∥𝑺 ∥ 𝛽 𝜇¯ + 1+ + 4 2 4 4 ℓ=−2 (since ∥𝛿𝑯∥ ≤ 𝛽8𝜇¯ and 𝛽 𝜇¯ ≤ 1) ≤ 7𝛽 𝜇∥𝑺 ¯ ′ ∥. Next, we bound 𝛿A in a similar fashion. The expression will be more cumbersome 82 Í because of the double Hilbert spaces. Decompose 𝛿A = 𝑎∈𝐴 𝛿A 𝑎 , where ∑︁ √︁ ∑︁ 𝛿A 𝑎 := 𝛾( 𝜔)𝛾(− ¯ 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗− 𝛾( 𝜔) ¯ 𝝆 −1/4 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝝆 1/4 ⊗ 𝝆 ∗−1/4 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ 𝝆 ∗1/4 | {z } 𝜔∈𝑆 ¯ 𝜔 𝜔∈𝑆 ¯ 𝜔 0 (by (2.46)) ∑︁ = 0 𝛾( ˚ 𝜔):= ¯ 𝛽 ¯ 𝝆 1/4 ⊗ 𝝆 ∗−1/4 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ 𝝆 ∗1/4 . 𝛾( ˚ 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ ⊗ 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ − 𝛾( ˚ 𝜔)e ¯ − 2 𝜔¯ 𝝆 −1/4 𝑺ˆ 𝑎 ( 𝜔) 𝜔∈𝑆 ¯ 𝜔0 (2.52) Let ⌊𝜔⌉ ¯ denote the rounding of 𝜔𝜇¯¯ to the closest integer, and suppose that 𝑷𝑖 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑗≠ 0. Then, there must exist some |𝜓𝑖 ⟩, 𝜓 𝑗 eigenvectors in the images of 𝑷𝑖 , 𝑷 𝑗 respectively such that ⟨𝜓𝑖 | 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝜓 𝑗 ≠ 0. Due to (2.47) we have that |𝐸𝑖 −𝐸 𝑗 −𝜔| ¯ ≤ 𝜇. ¯ 𝐸𝑗 𝐸𝑗 𝐸𝑖 𝐸𝑖 𝜔¯ 𝜔¯ 1 1 1 Then |𝑖 − 𝑗 − ⌊ 𝜔⌉| ¯ ≤ |𝑖 − 𝜇¯ | + | 𝜇¯ − 𝑗 | + | 𝜇¯ − ⌊ 𝜔⌉| ¯ + | 𝜇¯ − 𝜇¯ − 𝜇¯ | ≤ 2 + 2 + 2 + 1 < 3, thus we can define a bisection analogously to (2.49) as follows: 𝑺ˆ 𝑎 ( 𝜔) ¯ = ∑︁ 𝑷𝑖 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑗= ∑︁ 𝑖, 𝑗 : |𝑖− 𝑗−⌊𝜔⌉ ¯ |<3 𝑖, 𝑗 𝑷𝑖 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑗= 2 ∑︁ ∑︁ 𝑷𝑖 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑖−⌊𝜔⌉+ℓ , ¯ ℓ=−2 𝑖 which leads to the following “tensor-slicing” assuming that 𝑷𝑛 and 𝑴𝑚 commute: 𝑴1 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝑴2 ⊗ 𝑴3∗ 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ 𝑴4∗ 2   ∑︁ ∑︁ = (𝑷 𝑗 ⊗ 𝑷∗𝑗+𝑖 ) 𝑴1 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝑴2 ⊗ 𝑴3∗ 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ 𝑴4∗ (𝑷 𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′ ). ¯ ¯ ℓ,ℓ ′ =−2 𝑖, 𝑗 Using this we get the following decomposition analogously to (2.51) by expressing 𝝆 −1/4 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝝆 1/4 via (2.52) and (2.50) 𝑎 𝛿A = 2 ∑︁ ∑︁ 𝑎 𝛿A𝑖,ℓ,ℓ ′ = ℓ,ℓ ′ =−2 𝑖 ∑︁ 2 ∑︁ ∑︁ ℓ,ℓ ′ =−2 𝑖   𝛽 𝜇¯ ¯ ℓ+ℓ ′ ¯ 𝜔 ∗ 𝑎 𝑎 ∗ 𝑎 −𝛿𝑯 𝛿𝑯 ∗ ˆ 𝑎 ∗ −𝛿𝑯 ∗ ˆ ˆ ˆ 2 ( ⌊ 𝜔⌉− 𝜇¯ − 2 ) e𝛿𝑯 𝑺 𝛾( ˚ 𝜔)(𝑷 ¯ ⊗ 𝑷 ) 𝑺 ( 𝜔) ¯ ⊗ 𝑺 ( 𝜔) ¯ − e ( 𝜔)e ¯ ⊗e 𝑺 ( 𝜔) ¯ e 𝑗 𝑗+𝑖 𝑗,𝜔¯ · (𝑷 𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′ ). ¯ ¯ ′ 5 Since | 𝜔𝜇¯¯ − ⌊ 𝜔⌉| ¯ ≤ 12 and |ℓ|, |ℓ′ | ≤ 2, we have that |⌊ 𝜔⌉ ¯ − 𝜔𝜇¯¯ − ℓ+ℓ 2 | ≤ 2 and therefore the above factor is close to 1: 𝛽 𝜇¯ ¯ ℓ+ℓ ′ 𝜔¯ ℓ + ℓ′ 5 ¯ 𝜔 𝜇¯ − 2 ) − 1| ≤ 𝛽 𝜇 |e 2 ( ⌊𝜔⌉− ¯ ⌊𝜔⌉ ¯ − − ≤ 𝛽 𝜇. ¯ (2.53) 𝜇¯ 2 2 83 At this point, it seems intuitively clear that the error coming from the Boltzmann factor is small. However, we need to argue how the sum over 𝑗 does not blow up the error. We proceed by using the triangle inequality over ℓ, ℓ′ and then Lemma 2.7.1 over 𝑖 to get that ∥𝛿A∥ = ∥ ∑︁ ∑︁ 𝛿A 𝑎 ∥ ≤ 52 max′ 𝑎 𝛿A𝑖,ℓ,ℓ ′ . 𝑖,ℓ,ℓ 𝑎 𝑎∈𝐴 | {z 𝛿A 𝑖,ℓ,ℓ ′ := } The key for bounding the norm of 𝛿A𝑖,ℓ,ℓ′ is applying Lemma 2.7.2 to estimate the following for some weights 𝛼( 𝜔): ¯   ∑︁ ∗ 𝑎 𝑎 ∗ 2 ˆ ˆ ∥ 𝛼( 𝜔) ¯ 𝛾( ˚ 𝜔)(𝑷 ¯ ¯ ⊗ 𝑺 ( 𝜔) ¯ (𝑷 𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′ )∥ 𝑗 ⊗ 𝑷 𝑗+𝑖 ) 𝑺 ( 𝜔) ¯ ¯ 𝑗,𝑎,𝜔¯ =∥ ∑︁ √︁ √︁ 2 ¯ 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖 𝛾( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ ∗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ 𝛼( 𝜔)𝑷 ¯ 𝑗 𝛾(−𝜔) ′∥ ¯ ¯ 𝑗,𝑎,𝜔¯ ≤∥ ∑︁ 𝑺ˆ 𝑎 ( 𝜔)𝑷 ¯ 𝑗−⌊𝜔⌉+ℓ 𝑺ˆ 𝑎 ( 𝜔) ¯ † ¯ | {z } |𝛼( 𝜔)| ¯ 2 𝛾(−𝜔)𝑷 ¯ 𝑗 𝑗,𝑎,𝜔¯ ⪯ ·∥ 𝑷𝑗 ∥ Í ˆ𝑎 ¯ 𝑘 𝑺ˆ 𝑎 ( 𝜔) ¯ † = 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑘 𝑺 ( 𝜔)𝑷 ∑︁ ˆ 𝑎 ( 𝜔) ′𝑺 ′∥ 𝛾( 𝜔)𝑷 ¯ 𝑗+𝑖−⌊𝜔⌉+ℓ ¯ † 𝑷 𝑗+𝑖 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝑷 𝑗+𝑖−⌊𝜔⌉+ℓ ¯ ¯ | {z } 𝑗,𝑎,𝜔¯ (by Lemma 2.7.2) ⪯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ ≤ ∥𝛼∥ 2∞ ∑︁ 𝑗 𝑷𝑗 ∑︁ 𝛾(−𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ †𝑷𝑗 ∑︁ 𝑎,𝜔¯ | 𝑷𝑗 𝑺′′ := 𝛾( 𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † 𝑺ˆ 𝑎 ( 𝜔) ¯ 𝑷𝑗 𝑎,𝜔¯ 𝑗 {z ∑︁ } | {z =𝑺′ } since 0 ⪯ 𝑨 ⪯ 𝑩 ⇒ ∥ 𝑨∥ ≤ ∥ 𝑩∥ = ∥𝛼∥ 2∞ max 𝑷 𝑗 ′′ 𝑺′′ 𝑷 𝑗 ′′ · max 𝑷 𝑗 ′ 𝑺′ 𝑷 𝑗 ′ ≤ ∥𝛼∥ 2∞ ∥𝑺′ ∥∥𝑺′′ ∥. 𝑗 ′′ 𝑗′ | {z } (2.54) due to Lemma 2.7.1 Let us introduce a telescoping sum 𝛿A𝑖,ℓ,ℓ′ = e 𝛽 𝜇¯ ¯ ℓ+ℓ ′ ¯ 𝜔 2 (⌊ 𝜔⌉− 𝜇¯ − 2 ) (𝑠) ′ ¯ 𝑠=1 𝛿A 𝑖,ℓ,ℓ ′ , where 𝛾˚ ℓ,ℓ ( 𝜔) Í3 := 𝛾( ˚ 𝜔) ¯ and (1) 𝛿A𝑖,ℓ,ℓ ′   ∑︁ 𝛽 𝜇¯ ¯ ℓ+ℓ ′ ¯ 𝜔 𝑎 𝑎 ∗ ∗ 𝑎 𝑎 ∗ ˆ ˆ ˆ ˆ 2 ( ⌊ 𝜔⌉− 𝜇¯ − 2 ) 𝑺 := 𝛾( ˚ 𝜔)(𝑷 ¯ ⊗ 𝑷 ) 𝑺 ( 𝜔) ¯ ⊗ 𝑺 ( 𝜔) ¯ − e ( 𝜔) ¯ ⊗ 𝑺 ( 𝜔) ¯ (𝑷∗𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′ ), 𝑗 𝑗+𝑖 ¯ ¯ 𝑗,𝑎,𝜔¯ 84 (2) 𝛿A𝑖,ℓ,ℓ ′   ∑︁ ∗ 𝑎 𝑎 ∗ −𝛿𝑯 ˆ 𝑎 −𝛿𝑯 ∗ ˆ 𝑎 ∗ ˆ ˆ ′ := 𝛾˚ ℓ,ℓ ( 𝜔)(𝑷 ¯ ¯ ⊗ 𝑺 ( 𝜔) ¯ −e 𝑺 ( 𝜔) ¯ ⊗e 𝑺 ( 𝜔) ¯ 𝑗 ⊗ 𝑷 𝑗+𝑖 ) 𝑺 ( 𝜔) 𝑗,𝑎,𝜔¯ (𝑷∗𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′ ), ¯ ¯ (3) 𝛿A𝑖,ℓ,ℓ ′ := ∑︁ 𝑗,𝑎,𝜔¯   ∗ −𝛿𝑯 ˆ 𝑎 −𝛿𝑯 ∗ ˆ 𝑎 ∗ −𝛿𝑯 ˆ 𝑎 𝛿𝑯 −𝛿𝑯 ∗ ˆ 𝑎 ∗ 𝛿𝑯 ∗ 𝛾˚ ℓ,ℓ′ ( 𝜔)(𝑷 ¯ ⊗ 𝑷 ) e 𝑺 ( 𝜔) ¯ ⊗ e 𝑺 ( 𝜔) ¯ − e 𝑺 ( 𝜔)e ¯ ⊗ e 𝑺 ( 𝜔) ¯ e 𝑗 𝑗+𝑖 (𝑷∗𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′ ). ¯ ¯ √︁ (1) 5 Due to (2.53) we can bound ∥𝛿A𝑖,ℓ,ℓ ¯ ∥𝑺′ ∥∥𝑺′′ ∥ via (2.54). For bounding ′ ∥ ≤ 2 𝛽𝜇 (2) ∥𝛿A𝑖,ℓ,ℓ ′ ∥ observe that (2) ∥𝛿A𝑖,ℓ,ℓ ′∥   ∑︁ ∗ 𝑎 𝑎 ∗ −𝛿𝑯 ˆ 𝑎 −𝛿𝑯 ∗ ˆ 𝑎 ∗ ˆ ˆ =∥ 𝛾˚ ℓ,ℓ′ ( 𝜔)(𝑷 ¯ ⊗ 𝑷 ) 𝑺 ( 𝜔) ¯ ⊗ 𝑺 ( 𝜔) ¯ − e 𝑺 ( 𝜔) ¯ ⊗ e 𝑺 ( 𝜔) ¯ (𝑷∗𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′ )∥ 𝑗 𝑗+𝑖 ¯ ¯ 𝑗,𝑎,𝜔¯ ∗ = ∥( 𝑰 ⊗ 𝑰 − e−𝛿𝑯 ⊗ e−𝛿𝑯 ) ∑︁   ∗ ˆ 𝑎 ¯ ⊗ 𝑺ˆ 𝑎 ( 𝜔) 𝛾˚ ℓ,ℓ′ ( 𝜔)(𝑷 ¯ ¯ ∗ (𝑷∗𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′ )∥ 𝑗 ⊗ 𝑷 𝑗+𝑖 ) 𝑺 ( 𝜔) ¯ ¯ 𝑗,𝜔¯   ∑︁ ∗ 𝑎 𝑎 ∗ ∗ ˆ ˆ (𝑷∗𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ) 𝑺 ( 𝜔) ¯ ⊗ 𝑺 ( 𝜔) ¯ ≤ ∥ 𝑰 ⊗ 𝑰 − e−𝛿𝑯 ⊗ e−𝛿𝑯 ∥ ∥ 𝛾˚ ℓ,ℓ′ ( 𝜔)(𝑷 ¯ ⊗ 𝑷 ′ )∥ 𝑗 𝑗+𝑖 ¯ ¯ | {z } 𝑗,𝜔¯ | {z } ≤ 72 𝛽 𝜇¯ since ∥𝛿𝑯∥≤ 𝛽8𝜇¯ √ ′ ′′ 5 ≤(1+ 2 𝛽 𝜇) ¯ √︁ ≤ 𝛽 𝜇¯ ∥𝑺′ ∥∥𝑺′′ ∥. ∥𝑺 ∥ ∥𝑺 ∥ due to (2.53) and (2.54) 85 (3) Analogously we can bound ∥𝛿A𝑖,ℓ,ℓ ′ ∥ as follows: (3) ∥𝛿A𝑖,ℓ,ℓ ′∥ ∗ = ∥(e−𝛿𝑯 ⊗ e−𝛿𝑯 ) ∑︁   ∗ 𝑎 𝑎 ∗ ˆ ˆ 𝛾˚ ℓ,ℓ′ ( 𝜔)(𝑷 ¯ ⊗ 𝑷 ) 𝑺 ( 𝜔) ¯ ⊗ 𝑺 ( 𝜔) ¯ (𝑷∗𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′) 𝑗 𝑗+𝑖 ¯ ¯ 𝑗,𝜔¯ ∗ ( 𝑰 ⊗ 𝑰 − e𝛿𝑯 ⊗ e𝛿𝑯 )∥   ∑︁ ∗ ∗ ˆ 𝑎 ¯ ⊗ 𝑺ˆ 𝑎 ( 𝜔) ≤ ∥e−𝛿𝑯 ⊗ e−𝛿𝑯 ∥ ∥ 𝛾˚ ℓ,ℓ′ ( 𝜔)(𝑷 ¯ ¯ ∗ (𝑷∗𝑗−⌊𝜔⌉+ℓ ⊗ 𝑷∗𝑗+𝑖−⌊𝜔⌉+ℓ ′ )∥ 𝑗 ⊗ 𝑷 𝑗+𝑖 ) 𝑺 ( 𝜔) ¯ ¯ | {z } 𝑗,𝜔¯ | {z } ≤1+ 27 𝛽 𝜇¯ √ ′ ′′ 5 ≤(1+ 2 𝛽 𝜇) ¯ ∥𝑺 ∥ ∥𝑺 ∥ due to (2.53) and (2.54) ∗ ∥ 𝑰 ⊗ 𝑰 − e−𝛿𝑯 ⊗ e−𝛿𝑯 ∥ | {z } ≤ 27 𝛽 𝜇¯ since ∥𝛿𝑯∥≤ 𝛽8𝜇¯ ≤ 3 √︁ ′ 𝛽 𝜇¯ ∥𝑺 ∥∥𝑺′′ ∥. 2 Putting everything together, we get that   √︁ (1) (2) (3) ∥𝛿A∥ ≤ 25 max′ ∥𝛿A𝑖,ℓ,ℓ′ ∥ ≤ 25 max′ ∥𝛿A𝑖,ℓ,ℓ ∥ + ∥𝛿A ∥ + ∥𝛿A ∥ ≤ 125𝛽 𝜇¯ ∥𝑺′ ∥∥𝑺′′ ∥, ′ 𝑖,ℓ,ℓ ′ 𝑖,ℓ,ℓ ′ 𝑖,ℓ,ℓ 𝑖,ℓ,ℓ which concludes the proof for the first bound. √︁ √︁ ′ † Finally, if 𝛾(−𝜔) ¯ 𝑺ˆ 𝑎 ( 𝜔) ¯ † = 𝛾(−𝜔) ¯ 𝑺ˆ 𝑎 (−𝜔), ¯ then 𝑺′ = 𝑺′′ and D𝑠𝑒𝑐 = D𝑠𝑒𝑐 , so we easily get the other bound † † ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 ) † ∥ = ∥D𝑠𝑒𝑐 − D ( 𝝆, L 𝑠𝑒𝑐 )∥ 2−2 = ∥D𝑠𝑒𝑐 − D ( 𝝆, L 𝑠𝑒𝑐 )∥ 2−2 = ∥D𝑠𝑒𝑐 − D ( 𝝆, L 𝑠𝑒𝑐 )∥ 2−2 = ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 )∥ ≤ 132𝛽 𝜇∥𝑺 ¯ ′ ∥. ■ Bootstrapping the secular approximation Lemma 2.7.4 (Bootstrapping the secular approximation). Consider the decomÍ position 𝑓 = 𝑗 ∈𝐽 𝑓 𝑗 of the weight function, where 𝑓 𝑗 : 𝑆𝑡0 → C, and let 𝜇 𝑗 :=  min 𝜇 ≥ 0 : ∥ 𝑓ˆ𝑗 (𝜔) · 1 (|𝜔| > 𝜇)∥ = 0 . If the Hamiltonian 𝑯¯ has discretized spectrum so that 𝐵 ⊂ 𝜔0 Z, 𝛽𝜇 𝑗 ≤ 1, max𝜔∈𝐵, ¯ + 𝜇 𝑗 =: 𝜈 ∈ 𝑆𝜔0 , 𝛾 : 𝑆𝜔0 → R+ is ¯ 𝑗 ∈𝐽 𝜔 such that 𝛾( 𝜔)/𝛾(− ¯ 𝜔) ¯ = e−𝛽𝜔¯ for all 𝜔¯ ∈ [−𝜈, 𝜈] ∩ 𝑆𝜔0 , and the set of jumps is 86 self-adjoint { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴}, then43 ∑︁ ∑︁ ∑︁ ∥ D 𝑓 − D( 𝝆, L 𝑓 ) † ∥ = ∥ D 𝑓 − D( 𝝆, L 𝑓 )∥ ≤ ∥ 𝑓𝑖 ∥𝜇𝑖 ∥ 𝑓 𝑗 ∥∥𝛾∥ ∞ 1056𝛽∥ 𝑨𝑎† 𝑨𝑎 ∥, 𝑗 ∈𝐽 𝑖∈𝐽 𝑎∈𝐴 where 𝝆 = e−𝛽𝑯 /Tr[e−𝛽𝑯 ], ∑︁ √︁ D 𝑓 := 𝑰[·] 𝑰 + 𝑎,𝜔∈𝑆 ¯ 𝜔0 L 𝑓 := ∑︁ 𝛾( 𝜔) ¯ ˆ𝑎 { 𝑨 𝑓 ( 𝜔) ¯ † 𝑨ˆ 𝑎𝑓 ( 𝜔), ¯ ·}, and 𝛾( 𝜔)𝛾(− ¯ 𝜔) ¯ 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ [·] 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ †− 2 𝛾( 𝜔) ¯ 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ [·] 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ †− 𝑎∈𝐴,𝜔∈𝑆 ¯ 𝜔0 𝛾( 𝜔) ¯ ˆ𝑎 { 𝑨 𝑓 ( 𝜔) ¯ † 𝑨ˆ 𝑎𝑓 ( 𝜔), ¯ ·}. 2 Proof. The proof builds on the following “polarization” identity: for all 𝑴𝑖 matrices and 𝑐 1 , 𝑐 2 ∈ R     𝑐 1 𝑐 2 ∑︁ 𝑴1 𝑴2 𝑴4 𝑴3 𝑴1 ★ 𝑴4 + 𝑴2 ★ 𝑴3 = 𝑠 +𝑠 ★ +𝑠 , 2 𝑠=±1 𝑐 1 𝑐2 𝑐1 𝑐2 where ★ stands for any operation that is distributive with +, e.g., matrix product ★ = · or tensor product ★ = ⊗. Due to the linearity of the operator Fourier Transform, we have that 𝑨ˆ 𝑎𝑓 ( 𝜔) ¯ = Í 𝑎 𝑗 ∈𝐽 𝑨 𝑓 𝑗 ,𝜔¯ and consequently ∑︁   𝛿𝑖 𝑗 ∥ 𝑓𝑖 ∥∥ 𝑓 𝑗 ∥ ∑︁ D𝑓 = 1− 𝑠D 𝑓𝑖 𝑓𝑗 , 2 2 ∥ 𝑓𝑖 ∥ +𝑠 ∥ 𝑓 𝑗 ∥ 𝑖, 𝑗 ∈𝐽 𝑠=±1  ∑︁  𝛿𝑖 𝑗 ∥ 𝑓𝑖 ∥∥ 𝑓 𝑗 ∥ ∑︁ 𝑠D( 𝝆, L 𝑓𝑖 D( 𝝆, L 𝑓 ) = 1− 𝑓 𝑗 ). 2 2 ∥ 𝑓𝑖 ∥ +𝑠 ∥ 𝑓 𝑗 ∥ 𝑖, 𝑗 ∈𝐽 𝑠=±1 Due to the properties of the operator Fourier Transform (Proposition 2.5.1), we have that ∑︁ ∑︁ ∑︁ ∑︁ 𝑨𝑔𝑎 ( 𝜔) ¯ † 𝑨ˆ 𝑔𝑎 ( 𝜔)∥ ¯ ∥ 𝛾( 𝜔) ¯ 𝑨𝑔𝑎 ( 𝜔) ¯ † 𝑨ˆ 𝑔𝑎 ( 𝜔)∥ ¯ ≤ ∥𝛾∥ ∞ ∥ 𝑎∈𝐴 𝜔∈𝑆 ¯ 𝜔0 𝑎∈𝐴 𝜔∈𝑆 ¯ 𝜔0 =∥ ∑︁ ∑︁ 𝑎∈𝐴 𝑡¯∈𝑆𝑡0 43 Note that here we use notation D instead of 𝛽. ¯ ¯ |𝑔( 𝑡¯)| 2 ei 𝑯𝑡¯ 𝑨𝑎† 𝑨𝑎 e−i 𝑯𝑡¯ ∥ ≤ ∥ ∑︁ 𝑨𝑎† 𝑨𝑎 ∥∥𝑔∥ 2, 𝑎∈𝐴 𝑓 , L 𝑓 instead of D 𝛽 , L 𝛽 to spell out the dependence on 𝑓 87 𝑓 since ∥ ∥ 𝑓𝑓𝑖𝑖 ∥ + 𝑠 ∥ 𝑓 𝑗𝑗 ∥ ∥ ≤ 2, 𝛽𝜇 𝑗 ≤ 1 for all 𝑗 ∈ 𝐽, and42 𝜈 ∈ 𝑆𝜔0 (i.e., no wrapping around), by Lemma 2.7.3 we get ∥ D 𝑓𝑖 𝑓𝑗 ∥ 𝑓𝑖 ∥ +𝑠 ∥ 𝑓 𝑗 ∥ − D( 𝝆, L 𝑓𝑖 𝑓𝑗 ∥ 𝑓𝑖 ∥ +𝑠 ∥ 𝑓 𝑗 ∥ )∥ √︄ ∑︁ √︄ ∑︁ √︄ ∑︁ © ª 𝑎† 𝑎 𝑎† 𝑎 𝑨 𝑨 ∥ ­7 ∥ 𝑨 𝑨 ∥ + 125 ∥ 𝑨𝑎 𝑨𝑎† ∥ ® ≤ max{𝜇𝑖 , 𝜇 𝑗 }∥𝛾∥ ∞ 4𝛽 ∥ 𝑎∈𝐴 𝑎∈𝐴 𝑎∈𝐴 « ¬ ∑︁ 𝑎† 𝑎 ≤ max{𝜇𝑖 , 𝜇 𝑗 }∥𝛾∥ ∞ 528𝛽∥ 𝑨 𝑨 ∥. 𝑎∈𝐴 | {z } 𝐾:= The second line uses that { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴} thus ∥ Í ∥ 𝑎∈𝐴 𝑨𝑎 𝑨𝑎† ∥. Í 𝑎∈𝐴 𝑨 𝑎† 𝑨𝑎 ∥ = Finally, by the triangle inequality, we get that  ∑︁  𝛿𝑖 𝑗 ∥ 𝑓𝑖 ∥∥ 𝑓 𝑗 ∥ ∑︁ ∥ D 𝑓𝑖 ∥ D 𝑓 − D( 𝝆, L 𝑓 )∥ ≤ 1− 𝑓 𝑗 − D( 𝝆, L 𝑓𝑖 𝑓 𝑗 )∥ 2 2 ∥ 𝑓𝑖 ∥ +𝑠 ∥ 𝑓 𝑗 ∥ ∥ 𝑓𝑖 ∥ +𝑠 ∥ 𝑓 𝑗 ∥ 𝑖, 𝑗 ∈𝐽 𝑠=±1  𝛿𝑖 𝑗 ≤ ∥ 𝑓𝑖 ∥∥ 𝑓 𝑗 ∥ max{𝜇𝑖 , 𝜇 𝑗 }𝐾 1− 2 𝑖, 𝑗 ∈𝐽 ∑︁ ≤ ∥ 𝑓𝑖 ∥∥ 𝑓 𝑗 ∥(𝜇𝑖 + 𝜇 𝑗 )𝐾 ∑︁  𝑖, 𝑗 ∈𝐽 =2 ∑︁ 𝑖∈𝐽 ∥ 𝑓𝑖 ∥𝜇𝑖 ∑︁ ∥ 𝑓 𝑗 ∥𝐾. ■ 𝑗 ∈𝐽 Corollary 2.7.1 (Improved bounds for uniform weights). In the setting of Lemma 2.7.4, √︃ 𝑡0 consider the uniform weight function 𝑓 ( 𝑡¯) = 2𝑇 1 (−𝑇 ≤ 𝑡¯ < 𝑇) and the secular approximation with 𝑠( 𝜔) ¯ = 1 (| 𝜔| ¯ ≤ 𝜇) for some 𝜇 ≥ 𝜋/𝑇, then ! √︂ ∑︁ 𝜇 ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 ) † ∥ = ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 )∥ = O 𝛽∥𝛾∥ ∞ ∥ 𝑨𝑎† 𝑨𝑎 ∥ . 𝑇 𝑎∈𝐴 Proof. Let 𝑓𝑠 := F ∗−1 ( 𝑓ˆ · 𝑠); by Proposition 2.6.2, we know that D 𝑠𝑒𝑐 can be obtained by utilizing the weight function 𝑓𝑠 . We decompose 𝑠 into exponentially increasing intervals. We set 𝑠0 := 1 (| 𝜔| ¯ ≤ 𝜋/𝑇) and 𝑠 𝑗 := 1 (4 𝑗−1 𝜋/𝑇 < | 𝜔| ¯ ≤ min{4 𝑗 𝜋/𝑇, 𝜇}) ∀ 𝑗 ∈ Z+ . 88 Í ⌈log (𝜇𝑇/𝜋) ⌉ Í ⌈log (𝜇𝑇/𝜋) ⌉ Let 𝑓 𝑗 := F ∗−1 ( 𝑓ˆ · 𝑠 𝑗 ); since 𝑠 = 𝑗=0 4 𝑠 𝑗 we have 𝑓𝑠 = 𝑗=0 4 𝑓𝑗. Observe that due to Proposition 2.6.6 √︁ ∑︁ ∥ 𝑓 𝑗 ∥ = ∥ 𝑓ˆ · 𝑠 𝑗 ∥ ≤ ∥ 𝑓ˆ · 1 (4 𝑗−1 𝜋/𝑇 < | 𝜔|)∥ ¯ ≤ 41− 𝑗 = 21− 𝑗 , and thus ∥ 𝑓 𝑗 ∥ ≤ 4. 𝑗 The result follows from Lemma 2.7.4 since 𝑓ˆ𝑗 = 𝑓ˆ · 𝑠 𝑗 and by the definition of 𝑠 𝑗 we have 𝜇 𝑗 ≤ 4 𝑗 𝜋/𝑇 and thus (𝜇𝑇/𝜋) ⌉ ⌈log4∑︁ 𝑗=0 ∥ 𝑓 𝑗 ∥𝜇 𝑗 ≤ (𝜇𝑇/𝜋) ⌉ ⌈log4∑︁ 2𝜋 𝑗 4𝜋 ⌈log4 (𝜇𝑇/𝜋) ⌉ 8𝜋 log4 (𝜇𝑇/𝜋) 2 ≤ 2 ≤ 2 𝑇 𝑇 𝑇 𝑗=0 8𝜋 √︁ = 𝜇𝑇/𝜋 = 8 𝑇 √︂ 𝜋𝜇 . 𝑇 ■ Fourier Transform with uniform weights For simpler implementation, we can also work with the Fourier Transform with uniform weight (which is not smooth), leading to slightly worse bounds than the Gaussian damped case of Theorem 2.1.3. Theorem 2.7.1 (Uniform weight for Fourier Transform). Consider the discriminant Í proxy D 𝜷 (2.26) with the plain Fourier Transform 𝑨ˆ 𝑎 ( 𝜔) ¯ :∝ −𝑇 ≤𝑡¯<𝑇 𝑨𝑎 ( 𝑡¯)e−i𝜔¯ 𝑡¯. Let 𝜈 := 1𝛽 + max𝜔∈𝐵 𝜔 such that 𝜈 ≤ max𝜔∈𝑆 ¯ 𝜔0 , 𝛾 : 𝑆 𝜔0 → R+ is such that −𝛽 𝜔 ¯ 𝛾( 𝜔)/𝛾(− ¯ 𝜔) ¯ =e for all 𝜔¯ ∈ [−𝜈, 𝜈] ∩ 𝑆𝜔0 and the set of jumps are self-adjoint and normalized (2.10), then the (normalized) top eigenvector approximates the √ purified Gibbs state 𝝆 𝛽 such that 𝜆 1 (D 𝛽 ) − √ 𝝆𝛽 1 ≤O (𝜔0𝑇 + 𝜆 𝑔𝑎 𝑝 (D 𝛽 ) √︂ ! 𝛽 ) . 𝑇 The block-encoding for the discriminant proxy can be implemented exactly using Hamiltonian simulation time O (𝑇) using the construction outlined in section 2.3section 2.3. Even though with a worse asymptotic bound, the plain Fourier Transform is simpler to implement and closer to thermalization in nature (section 2.9). The proof is even simpler than the Gaussian case, partly because D = D𝑖𝑚 𝑝𝑙 as the uniform weights can be prepared exactly. 89 Proof of Theorem 2.7.1. We can assume without loss of generality that 𝑇/𝛽 ≥ 𝜋, since otherwise, the bound is vacuous. We bound the eigenvector distance by the operator norm using Proposition 2.3.3 and recall the secular approximation (Lemma 2.6.2, Proposition 2.6.6) and the improved bounds on approximate detailed balance (Corollary 2.7.1): 𝜆 1 (D 𝛽 ) − √ 𝝆 6 D 𝛽 − D( 𝝆, L 𝑠𝑒𝑐 ) † 𝜆 𝑔𝑎 𝑝 (D 𝛽 )   6 † ≤ ∥ D 𝛽 − D 𝑠𝑒𝑐 ∥ + ∥ D 𝑠𝑒𝑐 − D( 𝝆, L 𝑠𝑒𝑐 ) ∥ 𝜆 𝑔𝑎 𝑝 ( D 𝛽 ) √︂   1 1 𝜇 ≤O (𝜔0𝑇 + √ +𝛽 ) (now set 𝜇 := 1/𝛽) 𝜆 𝑔𝑎 𝑝 ( D 𝛽 ) 𝑇 𝜇𝑇 √︂ ! 𝛽 1 ■ (𝜔0𝑇 + ) . ≤O 𝜆 𝑔𝑎 𝑝 (D 𝛽 ) 𝑇 ≤ Unfortunately, the above result suggests that the Hamiltonian simulation time needs to scale with the inverse gap squared 𝜆−2 𝑔𝑎 𝑝 ; we do not know if better bounds are possible. 2.8 Appendix:Discretization error for Lindbladians and discriminant proxies In this appendix, we bound the discretization error for continuous Lindbladians. We use the notation established in section 2.5. In addition, for a function 𝑓 : R → C, by F̄ ( 𝑓 ( 𝑡¯)), we mean the discrete Fourier Transform of the vector obtained by evaluating ¯(𝑡) := √𝑡 0 𝑓 (𝑡) 𝑓 at the points 𝑡¯ ∈ 𝑆𝑡⌈𝑁⌋ . Also, we define the “discretized” version 𝑓 0 with a natural rescaling. We begin with a seemingly loose bound that will, however, be sufficient. Lemma 2.8.1. If 𝑓 , 𝑔, ℎ ∈ ℓ2 (R) and 𝛾 ∈ ℓ∞ (R), then for any norm |||·|||, we have that ∫ ∞ 𝛾(𝜔) 𝑨ˆ 𝑓 (𝜔) † ★ 𝑨ˆ 𝑓 (𝜔)d𝜔 − −∞ ≤ ∑︁ 𝑔( 𝜔) ¯ 𝑨ˆ ℎ ( 𝜔) ¯ † ★ 𝑨ˆ ℎ ( 𝜔) ¯ ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 ∑︁ ( 𝑨𝜈 ) † ★ 𝑨𝜈 ′ 𝜈,𝜈 ′ ∈𝐵 ∫ ∞ −∞ 𝛾(𝜔) 𝑓ˆ∗ (𝜔−𝜈) 𝑓ˆ(𝜔−𝜈′)d𝜔 − ∑︁ ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 ∗   (i𝜈 𝑡¯) (i𝜈 ′ 𝑡¯) ¯ ¯ 𝑔( 𝜔) ¯ F̄ ℎ( 𝑡 ) · e ( 𝜔) ¯ F̄ ℎ( 𝑡 ) · e ( 𝜔) ¯ ,  90 where ★ stands for any operation that is distributive with +, e.g., matrix product ★ = · or tensor product ★ = ⊗. Directly applying the above for the original Hamiltonian 𝑯 suffers from the number of the Bohr frequencies |𝐵(𝑯)|, which can generally scale with the Hilbert space dimension if the eigenvalue differences are nondegenerate. However, we will see that ¯ the above becomes sufficiently stringent if we consider a rounded Hamiltonian 𝑯, ¯ while staying substantially reducing the number of distinct Bohr frequencies 𝐵( 𝑯) close to the original Hamiltonian 𝑯. Proof. We use the defining decomposition of the continuous operator Fourier Transform from Proposition 2.5.2 ∑︁ 𝑨ˆ 𝑓 (𝜔) = 𝑓ˆ(𝜔 − 𝜈) 𝑨𝜈 , 𝜈∈𝐵 and its discrete counterpart 𝑨ˆ ℎ ( 𝜔) ¯ = ∑︁   F̄ ℎ( 𝑡¯) · ei𝜈𝑡¯ ( 𝜔) ¯ 𝑨𝜈 , 𝜈∈𝐵 ¯ = √1 where F̄ denotes the discrete Fourier Transform defined via F̄ (ℎ( 𝑡¯)) ( 𝜔) 𝑁 2𝜋 where 𝑡0 = 𝑁𝜔 . 0 −i𝜔¯ 𝑡¯, ⌈ 𝑁 ⌋ ℎ( 𝑡¯)e 𝑡¯∈𝑆𝑡 Í 0 Due to the distributivity of + and ★ we have ∫ ∞ ∑︁ 𝛾(𝜔) 𝑨ˆ 𝑓 (𝜔) † ★ 𝑨ˆ 𝑓 (𝜔)d𝜔 − 𝑔( 𝜔) ¯ 𝑨ˆ ℎ ( 𝜔) ¯ † ★ 𝑨ˆ ℎ ( 𝜔) ¯ −∞ = ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 ∑︁ ( 𝑨𝜈 ) † ★ 𝑨𝜈 ′ 𝜈,𝜈 ′ ∈𝐵 ∫  ∗   ª ∑︁ © ∞ (i𝜈 𝑡¯) (i𝜈 ′ 𝑡¯) ­ ˆ∗ (𝜔−𝜈) 𝑓ˆ(𝜔−𝜈′)d𝜔 − ¯ ¯ 𝛾(𝜔) 𝑓 𝑔( 𝜔) ¯ F̄ ℎ( 𝑡 ) · e ( 𝜔) ¯ F̄ ℎ( 𝑡 ) · e ( 𝜔) ¯ ®®. ­ −∞ ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 « ¬ We conclude the proof by using the triangle inequality. ■ We proceed by controlling the discretization error for the scalar integral. To do so, we also need to regularize the filter function and the transition weight 𝛾(𝜔) by truncations 𝑓𝑇 (𝑡) := 𝑓 (𝑡) · 1 (𝑡 ∈ [−𝑇/2, 𝑇/2)) 𝛾𝑊 (𝜔) := 𝛾(𝜔) · 1 (𝜔 ∈ [−𝑊/2, 𝑊/2)) . 91 Lemma 2.8.2 (Discretization error bounds for the integral). Let F̄ denote the discrete Fourier Transform with parameters 𝑁, 𝜔0 , 𝑡0 and consider 𝑓 ∈ ℓ2 (R) and 𝛾 ∈ ℓ∞ (R) with truncation parameters 𝑇, 𝑊 satisfying 𝑁 ≥ 𝑇/𝑡0 ∈ Z and 𝑁 ≥ 𝑊/𝜔0 ∈ Z. Then, for each 𝜈, 𝜈′ ∈ [−𝐾, 𝐾], ∫ ∞ 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ(𝜔 − 𝜈′) d𝜔 − {z } −∞ | 𝑔(𝜔):= ∑︁ ∗ √  √ (i𝜈 𝑡¯) (i𝜈 ′ 𝑡¯) ¯ ¯ ( 𝜔) ¯ F̄ 𝑡0 𝑓𝑇 ( 𝑡 ) · e ( 𝜔) ¯ ≤𝜖 𝛾𝑊 ( 𝜔) ¯ F̄ 𝑡 0 𝑓𝑇 ( 𝑡 ) · e ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 holds provided the following conditions: 44 ∑︁ 𝜖 𝜔0 ∥𝑔(𝜔) − 𝑔(𝑘𝜔0 )∥ [𝑘𝜔0 ,(𝑘+1)𝜔0 ) ≤ , 2 𝑘 ∈Z and ∑︁ 𝑡 0 ∥ 𝑓𝑇 (𝑡) − 𝑓𝑇 ( 𝑡¯)∥ [𝑡¯,𝑡¯+𝑡0 ) ≤ 𝛿, 𝑡0 ∥ 𝑓𝑇 (𝑡)∥ ∞𝑇 (𝑊 + 𝐾) ≤ 𝛿, (2.55) √ ∥ 𝑓 − 𝑓𝑇 ∥ 1 ≤ ( 2𝜋 − 2)𝛿, ⌈𝑁 ⌋ 𝑡¯∈𝑆𝑡 0 (2.56)   where 𝛿 = min 4𝑊 ∥𝛾𝑊 ∥ ∞𝜖 (∥ 𝑓 ∥ 1 +1) , 2 .45 Proof. By (2.55) we have that ∫ 𝑔(𝜔)d𝜔 − R ∑︁ 𝑔(𝑘𝜔0 )𝜔0 ≤ ∑︁ ∫ (𝑘+1)𝜔0 𝑘 ∈Z 𝑘∈Z ≤ ∑︁ 𝑘 ∈Z 𝑔(𝜔) − 𝑔(𝑘𝜔0 )d𝜔 (2.57) 𝑘𝜔0 𝜔0 ∥𝑔(𝜔) − 𝑔(𝑘𝜔0 )∥ [𝑘𝜔0 ,(𝑘+1)𝜔0 ) ≤ 𝜖 . 2 44 Note that the definition of 𝑔(𝜔) depends on the values 𝜈, 𝜈 ′ , however we do not explicitly indicate this dependence for ease of notation. 45 Note that in case 𝑁 is odd in order to match the spacing of the grid 𝑆 ⌈ 𝑁 ⌋ , the intervals [𝑘𝜔0 , (𝑘 + 1)𝜔0 ) should be changed to [(𝑘 − 12 )𝜔0 , (𝑘 + 21 )𝜔0 ) and analogously the endpoints of [𝑡¯, 𝑡¯ + 𝑡0 ) should be shifted to 𝑡¯ ± 𝑡0 /2. 92 Also observe that due to (2.56) we have for all 𝜔 ∈ R that ∑︁ 𝑡 0 ∥ 𝑓𝑇 (𝑡)e−i𝜔𝑡 − 𝑓𝑇 ( 𝑡¯)e−i𝜔𝑡¯ ∥ [𝑡¯,𝑡¯+𝑡0 ) ⌈𝑁 ⌋ 𝑡¯∈𝑆𝑡 0 ≤ ∑︁ 𝑡0 ∥ 𝑓𝑇 (𝑡)e−i𝜔𝑡 − 𝑓𝑇 ( 𝑡¯)e−i𝜔𝑡 ∥ [𝑡¯,𝑡¯+𝑡0 ) + 𝑡0 ∥ 𝑓𝑇 ( 𝑡¯)e−i𝜔𝑡 − 𝑓𝑇 ( 𝑡¯)e−i𝜔𝑡¯ ∥ [𝑡¯,𝑡¯+𝑡0 ) ⌈𝑁 ⌋ 𝑡¯∈𝑆𝑡 0 ≤ ∑︁ 𝑡0 ∥ 𝑓𝑇 (𝑡) − 𝑓𝑇 ( 𝑡¯)∥ [𝑡¯,𝑡¯+𝑡0 ) + 𝑡0 | 𝑓𝑇 ( 𝑡¯)|∥e−i𝜔𝑡 − e−i𝜔𝑡¯ ∥ [𝑡¯,𝑡¯+𝑡0 ) ⌈𝑁 ⌋ 𝑡¯∈𝑆𝑡 ≤ 0 ∑︁ 𝑡0 ∥ 𝑓𝑇 (𝑡) − 𝑓𝑇 ( 𝑡¯)∥ [𝑡¯,𝑡¯+𝑡0 ) + 𝑡0 | 𝑓𝑇 ( 𝑡¯)|𝑡 0 |𝜔| ⌈𝑁 ⌋ 𝑡¯∈𝑆𝑡 0 ≤ 𝛿 + 𝑡0 ∥ 𝑓𝑇 (𝑡)∥ ∞𝑇 |𝜔|. Í Next we define 𝑓˜ˆ𝑇 (𝜔) := √𝑡0 𝑡¯∈𝑆𝑡 𝑓𝑇 ( 𝑡¯)e−i𝜔𝑡¯. The above two inequalities imply 0 2𝜋 similarly to (2.57) that for all 𝜔 satisfying |𝜔| ≤ 𝑊 + 𝐾 we have ∫ ∫ 1 1 ˜ ˜ −i𝜔𝑡 ˆ ˆ ˆ 𝑓𝑇 (𝑡)e d𝑡 + √ ( 𝑓𝑇 (𝑡) − 𝑓 (𝑡))e−i𝜔𝑡 d𝑡 | 𝑓𝑇 (𝜔) − 𝑓 (𝜔)| ≤ 𝑓𝑇 (𝜔) − √ 2𝜋 2𝜋 𝑡0 1 ¯ ≤ √ 𝑓𝑇 ( 𝑡¯)e−i𝜔𝑡 − √ 2𝜋 2𝜋 𝑡¯∈𝑆𝑡 ∑︁ 0 ∫ 𝑡¯+𝑡0 𝑡¯ √ 𝛿 𝑓𝑇 (𝑡)e−i𝜔𝑡 d𝑡 + ( 2𝜋 − 2) √ 2𝜋 ∫ 𝑡¯+𝑡0 √ 𝛿 1 ∑︁ =√ 𝑓𝑇 ( 𝑡¯)e−i𝜔𝑡¯ − 𝑓𝑇 (𝑡)e−i𝜔𝑡 d𝑡 + ( 2𝜋 − 2) √ 2𝜋 𝑡¯∈𝑆𝑡 𝑡¯ 2𝜋 0 √ 2𝛿 𝛿 ≤ √ + ( 2𝜋 − 2) √ 2𝜋  2𝜋  𝜖 = 𝛿 = min ,2 . (2.58) 4𝑊 ∥𝛾𝑊 ∥ ∞ (∥ 𝑓 ∥ 1 + 1) Let 𝑔(𝜔) ˜ := 𝛾𝑊 (𝜔) 𝑓˜ˆ𝑇∗ (𝜔 − 𝜈) 𝑓˜ˆ𝑇 (𝜔 − 𝜈′). Considering that |𝜈|, |𝜈′ | ≤ 𝐾 we get that 93 for all 𝜔 ∈ R |𝑔(𝜔) − 𝑔(𝜔)| ˜ ≤ |𝛾𝑊 (𝜔)|   | 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ(𝜔 − 𝜈′) − 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓˜ˆ𝑇 (𝜔 − 𝜈′)| + | 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓˜ˆ𝑇 (𝜔 − 𝜈′) − 𝑓˜ˆ𝑇∗ (𝜔 − 𝜈) 𝑓˜ˆ𝑇 (𝜔 − 𝜈′)|   = |𝛾𝑊 (𝜔)| | 𝑓ˆ∗ (𝜔 − 𝜈)|| 𝑓ˆ(𝜔 − 𝜈′) − 𝑓˜ˆ𝑇 (𝜔 − 𝜈′)| + | 𝑓ˆ∗ (𝜔 − 𝜈) − 𝑓˜ˆ𝑇∗ (𝜔 − 𝜈)|| 𝑓˜ˆ𝑇 (𝜔 − 𝜈′)|   ≤ 𝛿|𝛾𝑊 (𝜔)| | 𝑓ˆ∗ (𝜔 − 𝜈)| + | 𝑓˜ˆ𝑇 (𝜔 − 𝜈′)|   ˜ ∗ ′ ′ ′ ˆ ˆ ˆ ˆ ≤ 𝛿|𝛾𝑊 (𝜔)| | 𝑓 (𝜔 − 𝜈)| + | 𝑓 (𝜔 − 𝜈 )| + | 𝑓𝑇 (𝜔 − 𝜈 ) − 𝑓 (𝜔 − 𝜈 )| (by (2.58)) ≤ 𝛿|𝛾𝑊 (𝜔)|(2∥ 𝑓 ∥ 1 + 2) 𝜖 . ≤ 2𝑊 (by (2.58)) (by (2.58)) This implies ∑︁ 𝜔0 (𝑔(𝑘𝜔0 ) − 𝑔(𝑘𝜔 ˜ 0 )) ≤ 𝜔0 𝑘∈Z ∑︁ |𝑔(𝑘𝜔0 ) − 𝑔(𝑘𝜔 ˜ 0 )| ≤ 𝑘∈Z ∑︁ 𝑘∈Z 𝜖𝜔0 𝜖 ≤ , 2𝑊 2 𝑘 𝜔0 ∈ [ −𝑊/2,𝑊/2) (2.59) showing that ∫ 𝑔(𝜔)d𝜔 − R ∑︁ 𝜔0 𝑔(𝑘𝜔 ˜ 0) 𝑘∈Z ∫ 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ(𝜔 − 𝜈′)d𝜔 − = R 𝜔0 𝛾𝑊 (𝑘𝜔0 ) 𝑓˜ˆ𝑇∗ (𝑘𝜔0 − 𝜈) 𝑓˜ˆ𝑇 (𝑘𝜔0 − 𝜈′) 𝑘 ∈Z ∫ 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ(𝜔 − 𝜈′)d𝜔 − ≤ R + 𝜔0 ∑︁ ∑︁ 𝜔0 𝛾𝑊 (𝑘𝜔0 ) 𝑓ˆ∗ (𝑘𝜔0 − 𝜈) 𝑓ˆ(𝑘𝜔0 − 𝜈′) 𝑘∈Z ∑︁ 𝛾𝑊 (𝑘𝜔0 ) 𝑓ˆ∗ (𝑘𝜔0 − 𝜈) 𝑓ˆ(𝑘𝜔0 − 𝜈′) − 𝛾𝑊 (𝑘𝜔0 ) 𝑓˜ˆ𝑇∗ (𝑘𝜔0 − 𝜈) 𝑓˜ˆ𝑇 (𝑘𝜔0 − 𝜈′) 𝑘∈Z ≤ 𝜖. (by (2.57) and (2.59)) 94 0 𝑡0 We conclude the proof by observing that due to 𝑊 ≤ 𝑁𝜔0 , 𝑇 ≤ 𝑁𝑡 0 , and 𝜔2𝜋 = 𝑁1 we have ∑︁ ∑︁ 𝜔0 𝑔(𝑘𝜔 ˜ ) = 𝜔 𝛾𝑊 ( 𝜔) ¯ 𝑓˜ˆ𝑇∗ ( 𝜔¯ − 𝜈) 𝑓˜ˆ𝑇 ( 𝜔¯ − 𝜈′) 0 0 𝜔∈Z𝜔 ¯ 0 𝑘∈Z ∗ 𝜔0 𝑡02 ∑︁ ∑︁ © ∑︁ ¯ 𝑡¯ª © ¯ ′ ) 𝑡¯ª 𝛾𝑊 ( 𝜔) ¯ ­ 𝑓𝑇 ( 𝑡¯)e−i(𝜔−𝜈) 𝑓𝑇 ( 𝑡¯)e−i(𝜔−𝜈 = ®­ ® 2𝜋 𝜔∈Z𝜔 ¯ ¯ ¯ 𝑡 ∈Z𝑡 𝑡 ∈Z𝑡 0 « 0 ¬« 0 ¬ ∗ ª © ∑︁ √ ª © ∑︁ √ ′ 1 ∑︁ 𝑡 0 𝑓𝑇 ( 𝑡¯)ei𝜈𝑡¯e−i𝜔¯ 𝑡¯®® ­­ 𝑡0 𝑓𝑇 ( 𝑡¯)ei𝜈 𝑡¯e−i𝜔¯ 𝑡¯®® 𝛾𝑊 ( 𝜔) ¯ ­­ = 𝑁 ⌈𝑁 ⌋ ⌈𝑁 ⌋ ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 ¬ «𝑡¯∈𝑆𝑡0 ¬ «𝑡¯∈𝑆𝑡0 = ∑︁ 𝛾𝑊 ( 𝜔) ¯ F̄ ∗ √  √ ′ ¯ 𝑡0 𝑓𝑇 ( 𝑡¯) · e (i𝜈𝑡¯) ( 𝜔) ¯ F̄ 𝑡0 𝑓𝑇 ( 𝑡¯) · e (i𝜈 𝑡¯) ( 𝜔). ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 ■ As a sanity check, the above implies that the discretized Lindbladian convergences to the continuum in the limit. Theorem 2.8.1 (Discretizations converge to the continuum). If 𝛾 ∈ ℓ∞ (R), 𝑓 ∈ ℓ2 (R), and 𝛾, 𝑓 are continuous almost everywhere (i.e., the set of points of discontinuity has measure zero) while 𝑓 is bounded on every finite interval, then   ∑︁ 1 ˆ † ˆ † ˆ ˆ ¯ 𝑨 𝑓¯𝑇 ( 𝜔), ¯ ·} lim lim 𝛾𝑊 ( 𝜔) ¯ 𝑨 𝑓¯𝑇 ( 𝜔) ¯ [·] 𝑨 𝑓¯𝑇 ( 𝜔) ¯ − { 𝑨 𝑓¯𝑇 ( 𝜔) 𝑊,𝑇→∞ 𝑁→∞ 2 ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 ∫ ∞   1 ˆ † † ˆ ˆ ˆ = 𝛾(𝜔) 𝑨 𝑓 (𝜔) [·] 𝑨 𝑓 (𝜔) − { 𝑨 𝑓 (𝜔) 𝑨 𝑓 (𝜔), ·} d𝜔, 2 −∞ and if 𝛾 ≥ 0, then also ∑︁ √︁ lim lim 𝛾𝑊 ( 𝜔)𝛾 ¯ 𝑊 (−𝜔) ¯ 𝑨ˆ 𝑓¯𝑇 ( 𝜔) ¯ ⊗ 𝑨ˆ 𝑓¯𝑇 ( 𝜔) ¯ ∗ 𝑊,𝑇→∞ 𝑁→∞ ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0  𝛾( 𝜔) ¯ ˆ − 𝑨 𝑓¯𝑇 ( 𝜔) ¯ † 𝑨ˆ 𝑓¯𝑇 ( 𝜔) ¯ ⊗ 𝑰 + 𝑰 ⊗ 𝑨ˆ 𝑓¯𝑇 ( 𝜔) ¯ †∗ 𝑨ˆ 𝑓¯𝑇 ( 𝜔) ¯ ∗ ∫ ∞2√︁ 𝛾(𝜔)𝛾(−𝜔) 𝑨ˆ 𝑓 (𝜔) ⊗ 𝑨ˆ 𝑓 (𝜔) ∗ = −∞  𝛾(𝜔)  ˆ − 𝑨 𝑓 (𝜔) † 𝑨ˆ 𝑓 (𝜔) ⊗ 𝑰 + 𝑰 ⊗ 𝑨ˆ 𝑓 (𝜔) †∗ 𝑨ˆ 𝑓 (𝜔) ∗ d𝜔, 2 √︁ √ where 𝜔0 = 𝑡0 = 2𝜋/𝑁 and 𝑓¯𝑇 (𝑡) := 𝑡0 𝑓𝑇 (𝑡). 95 Proof. Due to Lemma 2.8.1, it suffices to prove for all 𝜈, 𝜈′ ∈ 𝐵 that  ∗   ∑︁ ′ lim lim 𝛾𝑊 ( 𝜔) ¯ F̄ 𝑓¯𝑇 ( 𝑡¯) · e (i𝜈𝑡¯) ( 𝜔) ¯ F̄ 𝑓¯𝑇 ( 𝑡¯) · e (i𝜈 𝑡¯) ( 𝜔) ¯ 𝑊,𝑇→∞ 𝑁→∞ ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 ∫ ∞ lim = 𝛾(𝜔) 𝑓ˆ∗ (𝜔−𝜈) 𝑓ˆ(𝜔−𝜈′)d𝜔, −∞     ∑︁ √︁ ′ ∗ lim 𝛾𝑊 ( 𝜔)𝛾 ¯ 𝑊 (−𝜔) ¯ F̄ 𝑓¯𝑇 ( 𝑡¯) · e (i𝜈𝑡¯) ( 𝜔) ¯ F̄ 𝑓¯𝑇 ( 𝑡¯) · e (i𝜈 𝑡¯) ( 𝜔) ¯ 𝑊,𝑇→∞ 𝑁→∞ ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 ∫ ∞ √︁ = 𝛾(𝜔)𝛾(−𝜔) 𝑓ˆ(𝜔−𝜈) 𝑓ˆ∗ (𝜔−𝜈′)d𝜔. (2.60) −∞ Since 𝑓 ∈ ℓ2 (R), we have 𝑓ˆ ∈ ℓ2 (R) and therefore by Hölder’s inequality we get that 𝛾(𝜔) 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ(𝜔 − 𝜈′) ∈ ℓ1 (R), which then implies ∫ ∞ ∫ 𝑊 ∗ ′ ˆ ˆ 𝛾(𝜔) 𝑓 (𝜔 − 𝜈) 𝑓 (𝜔 − 𝜈 )d𝜔 = lim 𝛾(𝜔) 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ(𝜔 − 𝜈′)d𝜔. −∞ 𝑊→∞ −𝑊 Since 𝑓 = lim𝑇→∞ 𝑓𝑇 , Parseval’s Theorem implies 𝑓ˆ = lim𝑇→∞ 𝑓ˆ𝑇 and so by Hölder’s inequality we get ∫ 𝑊 ∫ 𝑊 ∗ ′ 𝛾(𝜔) 𝑓ˆ (𝜔 − 𝜈) 𝑓ˆ(𝜔 − 𝜈 )d𝜔 = lim 𝛾(𝜔) 𝑓ˆ𝑇∗ (𝜔 − 𝜈) 𝑓ˆ𝑇 (𝜔 − 𝜈′)d𝜔. −𝑊 𝑇→∞ −𝑊 As 𝑓𝑇 is bounded, continuous almost everywhere, and has compact support, the Lebesgue-Vitali Theorem [137, 154, Theorem 11.33] implies that it is Riemann integrable. Therefore, 𝑓ˆ𝑇 is bounded and continuous which similarly implies that 𝑔(𝜔) := 𝛾𝑊 (𝜔) 𝑓ˆ𝑇∗ (𝜔 − 𝜈) 𝑓ˆ𝑇 (𝜔 − 𝜈′) is Riemann integrable on [−𝑊, 𝑊]. Since 𝑓 ← 𝑓𝑇 , lim𝑁→0 𝜔0 = lim𝑁→0 𝑡0 = 0, and 𝑓𝑇 (𝑡), 𝑔(𝜔) are Riemann integrable (c.f. [48, Theorem 8.26.]), for every 𝜖 > 0 (2.55)-(2.56) are satisfied for large enough 𝑁. Thus Lemma 2.8.2 implies that ∫ 𝑊  ∗   ∑︁ ′ 𝛾(𝜔) 𝑓ˆ𝑇∗ (𝜔 − 𝜈) 𝑓ˆ𝑇 (𝜔 − 𝜈′)d𝜔 = lim 𝛾𝑊 ( 𝜔) ¯ F̄ 𝑓¯𝑇 ( 𝑡¯) · e (i𝜈𝑡¯) ( 𝜔) ¯ F̄ 𝑓¯𝑇 ( 𝑡¯) · e (i𝜈 𝑡¯) ( 𝜔). ¯ −𝑊 𝑁→∞ ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 √︁ Equation (2.60) can be analogously proven after replacing 𝛾(𝜔) with 𝛾(𝜔)𝛾(−𝜔) throughout the argument. ■ We believe that a similar result can be shown for any 𝛾 ∈ ℓ∞ (R), 𝑓 ∈ ℓ2 (R) without the other assumptions in Theorem 2.8.1 by applying a further approximation with 96 the help of mollifiers. While such an asymptotic result is conceptually elegant, for quantum algorithm implementation, we need quantitative, nonasymptotic error bounds for the particular functions we encounter. Proposition 2.8.1 (Discretization error of Lindbladians and discriminant proxies). In the setting of Lemma 2.8.2, assume continuity and boundedness assumption for 𝑓 , 𝛾 as in Theorem 2.8.1 with normalization ∥𝛾∥ ∞ , ∥ 𝑓 ∥ 2 ≤ 1. Consider a single jump √︁ operator 𝑨 with ∥ 𝑨∥ ≤ 1, assume that 𝛾(𝜔) and 𝛾(𝜔)𝛾(−𝜔) are 𝐶-Lipschitz continuous, 𝑓ˆ is (𝐷 · ∥ 𝑓ˆ∥ ∞ )-Lipschitz continuous, 𝑓𝑇 is (𝐿 · ∥ 𝑓𝑇 ∥ ∞ )-Lipschitz continuous on [−𝑇/2, 𝑇/2), and assume the following conditions: 𝜖 ∥ 𝑓𝑇 − 𝑓 ∥ 2 ≤ , 64 ∑︁ 𝑡0 | 𝑓 ( 𝑡¯) − 𝑓𝑇 ( 𝑡¯)| 2 ≤  𝜖 2 ⌈𝑁 ⌋ 0 64 , 𝜔∈𝑆 ¯ 𝑡 ∑︁ 𝑡 0 | 𝑓 ( 𝑡¯)| 2 ≤ 1 ⌈𝑁 ⌋ 0 𝜔∈𝑆 ¯ 𝑡 ∫ ∞ 𝜖 | 𝑓ˆ(𝜔) 1 (|𝜔| ≥ 𝑊 − 2∥𝑯∥)| 2 d𝜔 ≤  2 , 256∥𝑯∥𝑇 −∞ 8 +1 𝜖 √ and ∥ 𝑓 − 𝑓𝑇 ∥ 1 ≤ ( 2𝜋 − 2)𝛿 for 𝛿 = min   𝜖 , 2 , and  2 32 256∥𝜖𝑯 ∥𝑇 +1 𝑊 (∥ 𝑓 ∥ 1 +1)   1 1 𝛿 min , , 𝑡0 ≤ 𝑇 ∥ 𝑓𝑇 ∥ ∞ 𝑊 + 2∥𝑯∥ 𝐿 𝜔0 ≤ ª © 1 1 𝜖 ®. · min­­ ,   2® 16 8𝑇 256∥𝑯∥𝑇 2 𝑊 (𝐶 + 2𝐷)∥ 𝑓ˆ∥ ∞ +1 ¬ 𝜖 « (2.61) Then, in the notation of Corollary 2.6.1-2.6.2, ∥L ( 𝑓 ,𝑯) − L̄ ((𝑊) ∥ ≤ 𝜖, 𝑓 ,𝑯) 1−1 (𝑊) ∥ D ( 𝑓 ,𝑯) − D̄ ( 𝑓 ,𝑯) ∥ ≤ 𝜖, where the discretized Lindbladian and discriminant proxy uses 𝛾𝑊 (𝜔) instead of 𝛾(𝜔). Moreover, if   2 ′ 𝜖 F̄ 𝑓¯( 𝑡¯)e (i𝜈 𝑡¯) ( 𝜔) ¯ 1 (| 𝜔| ¯ ≥ 𝑊) ≤  2 ⌈𝑁 ⌋ 8 256∥𝑯∥𝑇 + 1 𝜔∈𝑆 ¯ 𝜔0 𝜖 ∑︁ for each |𝜈| ≤ 2∥𝑯∥, (2.62) then ∥L ( 𝑓 ,𝑯) − L̄ ( 𝑓 ,𝑯) ∥ 1−1 ≤ 𝜖, ∥ D ( 𝑓 ,𝑯) − D̄ ( 𝑓 ,𝑯) ∥ ≤ 𝜖 . 97 Proof. Let 𝑯¯ be the Hamiltonian obtained by rounding the spectrum of 𝑯 (down in absolute value) to Z𝜂; this is crucial before we invoke Lemma 2.8.1. Our proof proceeds using the following triangle inequalities: ∥L ( 𝑓 ,𝑯) − L̃ ( 𝑓 ,𝑯) ∥ 1−1 ≤ ∥L ( 𝑓 ,𝑯) − L ( 𝑓 , 𝑯) ¯ ∥ 1−1 + ∥L ( 𝑓 , 𝑯) ¯ − L̃ ( 𝑓 , 𝑯) ¯ ∥ 1−1 + ∥ L̃ ( 𝑓 , 𝑯) ¯ − L̃ ( 𝑓 ,𝑯) ∥ 1−1 ∥ D ( 𝑓 ,𝑯) − D̃ ( 𝑓 ,𝑯) ∥ ≤ ∥ D ( 𝑓 ,𝑯) − D ( 𝑓 , 𝑯) ¯ ∥ + ∥ D ( 𝑓 , 𝑯) ¯ − D̃ ( 𝑓 , 𝑯) ¯ ∥ + ∥ D̃ ( 𝑓 , 𝑯) ¯ − D̃ ( 𝑓 ,𝑯) ∥, (𝑊) where L̃ ( 𝑓 ,𝑯) stands for either L̄ ((𝑊) or L̄ ( 𝑓 ,𝑯) and D̃ ( 𝑓 ,𝑯) for either D̄ ( 𝑓 ,𝑯) or 𝑓 ,𝑯) D̄ ( 𝑓 ,𝑯) . 𝜖 Choosing 𝜂 := 64𝑇 , we get by Corollary 2.6.2 that ∥L ( 𝑓 ,𝑯) − L ( 𝑓 , 𝑯) ¯ ∥ 1−1 ≤ ∥L ( 𝑓 ,𝑯) − L ( 𝑓𝑇 , 𝑯) ¯ ∥ 1−1 + ∥L ( 𝑓𝑇 , 𝑯) ¯ − L ( 𝑓 , 𝑯) ¯ ∥ 1−1 𝜖 ≤ 8(𝑇𝜂 + ∥ 𝑓𝑇 − 𝑓 ∥ 2 ) ≤ , 8 ∥ D ( 𝑓 ,𝑯) − D ( 𝑓 , 𝑯) ¯ ∥ ≤ ∥ D ( 𝑓 ,𝑯) − D ( 𝑓𝑇 , 𝑯) ¯ ∥ + ∥ D ( 𝑓𝑇 , 𝑯) ¯ − D ( 𝑓 , 𝑯) ¯ ∥ 𝜖 ≤ 8(𝑇𝜂 + ∥ 𝑓𝑇 − 𝑓 ∥ 2 ) ≤ , 8 and similarly by Corollary 2.6.1 that 𝜖 ∥ L̃ ( 𝑓 ,𝑯) − L̃ ( 𝑓 , 𝑯) ¯ ∥ 1−1 ≤ ∥ L̃ ( 𝑓 ,𝑯) − L̃ ( 𝑓𝑇 , 𝑯) ¯ ∥ 1−1 + ∥ L̃ ( 𝑓𝑇 , 𝑯) ¯ − L̃ ( 𝑓 , 𝑯) ¯ ∥ 1−1 ≤ , 8 𝜖 ∥ D̃ ( 𝑓 ,𝑯) − D̃ ( 𝑓 , 𝑯) ¯ ∥ ≤ ∥ D̃ ( 𝑓 ,𝑯) − D̃ ( 𝑓𝑇 , 𝑯) ¯ ∥ + ∥ D̃ ( 𝑓𝑇 , 𝑯) ¯ − D̃ ( 𝑓 , 𝑯) ¯ ∥ ≤ . 8 3𝜖 Therefore, it suffices to show that ∥L ( 𝑓 , 𝑯) ¯ − L̃ ( 𝑓 , 𝑯) ¯ ∥ 1−1 , ∥ D ( 𝑓 , 𝑯) ¯ − D̃ ( 𝑓 , 𝑯) ¯ ∥ ≤ 4. Let 𝛾˜ be either 𝛾 or 𝛾𝑊 matching the definition of L̃, and D̃. We apply Lemma 2.8.1 with ★ = ·𝝆· and ★ = ·, showing that the difference between the discrete and continuous generators ∥L ( 𝑓 , 𝑯) ¯ − L̃ ( 𝑓 , 𝑯) ¯ ∥ 1−1 , ∥ D ( 𝑓 , 𝑯) ¯ − D̃ ( 𝑓 , 𝑯) ¯ ∥ can be bounded by  256∥𝑯∥𝑇 2 +1 𝜖 2 ∫ ∞ 𝛾(𝜔) 𝑓ˆ∗ (𝜔−𝜈) 𝑓ˆ(𝜔−𝜈′)d𝜔 − −∞ ∑︁  ∗   ′ 𝛾( ˜ 𝜔) ¯ F̄ 𝑓¯𝑇 ( 𝑡¯) · e (i𝜈𝑡¯) ( 𝜔) ¯ F̄ 𝑓¯𝑇 ( 𝑡¯) · e (i𝜈 𝑡¯) ( 𝜔) ¯ , ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 (2.63) since the number of Bohr frequencies for the discretized Hamiltonian 𝑯¯ satisfies ¯ ¯ ≤ 4∥ 𝑯∥ 𝐵( 𝑯) + 1 ≤ 4∥𝑯∥ + 1 = 256∥𝑯∥𝑇 + 1. 𝜂 𝜂 𝜖 98 We further bound ∥L ( 𝑓 , 𝑯) ¯ − L̃ ( 𝑓 , 𝑯) ¯ ∥ 1−1 in three steps according to the following triangle inequality: ∫ ∞ 𝛾(𝜔) 𝑓ˆ∗ (𝜔−𝜈) 𝑓ˆ(𝜔−𝜈′)d𝜔 − −∞ ∑︁  ∗   ′ 𝛾( ˜ 𝜔) ¯ F̄ 𝑓¯( 𝑡¯)e (i𝜈𝑡¯) ( 𝜔) ¯ F̄ 𝑓¯( 𝑡¯)e (i𝜈 𝑡¯) ( 𝜔) ¯ ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 ∫ ∞ ≤ 𝛾(𝜔) 𝑓ˆ∗ (𝜔−𝜈) 𝑓ˆ(𝜔−𝜈′)d𝜔 − ∫ ∞ −∞ −∞ ∫ ∞ ∑︁ + 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔−𝜈) 𝑓ˆ(𝜔−𝜈′)d𝜔 − −∞ 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔−𝜈) 𝑓ˆ(𝜔−𝜈′)d𝜔 (2.64)  ∗   ′ 𝛾𝑊 ( 𝜔) ¯ F̄ 𝑓¯( 𝑡¯)e (i𝜈𝑡¯) ( 𝜔) ¯ F̄ 𝑓¯( 𝑡¯)e (i𝜈 𝑡¯) ( 𝜔) ¯ . ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 (2.65) + ∑︁  ∗   ′ (𝛾𝑊 ( 𝜔) ¯ − 𝛾( ˜ 𝜔)) ¯ F̄ 𝑓¯( 𝑡¯)e (i𝜈𝑡¯) ( 𝜔) ¯ F̄ 𝑓¯( 𝑡¯)e (i𝜈 𝑡¯) ( 𝜔) ¯ . (2.66) ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 Considering that |𝜈|, |𝜈′ | ≤ 2∥𝑯∥ and using Hölder’s inequality, we can see that truncation at threshold 𝑊 introduces error ≤ 256∥ 𝑯𝜖 ∥𝑇
2 ≤ 8|𝐵(𝜖𝑯)| ¯ 2 in (2.64), 8 𝜖 +1 inducing no more than 4𝜖 error in (2.63). If 𝛾˜ := 𝛾, and (2.62) holds, then the analogous argument shows the same bound for (2.66). 𝜖 We complete our proof by showing that (2.65) is bounded by 8
, by 2 256∥ 𝑯 ∥𝑇 +1 𝜖 applying Lemma 2.8.2, i.e., showing that (2.55)-(2.56) are fulfilled. In Lemma 2.8.2 we set 𝐾 = 2∥𝑯∥ and our accuracy goal 𝜖 ← 256∥ 𝑯𝜖 ∥𝑇
2 . To bound (2.55) we 8 𝜖 +1 observe that 𝑔(𝜔) := 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ(𝜔 − 𝜈′) is (𝐶 + 2𝐷)∥ 𝑓ˆ∥ 2∞ -Lipschitz continuous: |𝑔(𝜔 + 𝛿) − 𝑔(𝜔)| ≤ 𝛾𝑊 (𝜔 + 𝛿) 𝑓ˆ∗ (𝜔 + 𝛿 − 𝜈) 𝑓ˆ𝑇 (𝜔 + 𝛿 − 𝜈′) − 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔 + 𝛿 − 𝜈) 𝑓ˆ𝑇 (𝜔 + 𝛿 − 𝜈′) + 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔 + 𝛿 − 𝜈) 𝑓ˆ𝑇 (𝜔 + 𝛿 − 𝜈′) − 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ𝑇 (𝜔 + 𝛿 − 𝜈′) + 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ𝑇 (𝜔 + 𝛿 − 𝜈′) − 𝛾𝑊 (𝜔) 𝑓ˆ∗ (𝜔 − 𝜈) 𝑓ˆ𝑇 (𝜔 − 𝜈′) ≤(𝐶 + 2𝐷)∥ 𝑓ˆ∥ 2∞ · 𝛿, therefore by (2.61) we can upper bound the left-hand side of (2.55) as 𝜖 𝑊𝜔0 (𝐶 + 2𝐷)∥ 𝑓ˆ∥ 2∞ ≤
2 . 256∥𝑯∥𝑇 16 +1 𝜖 99 Now observe that due to the (𝐿 · ∥ 𝑓𝑇 ∥ ∞ )-Lipschitz continuity of 𝑓𝑇 we have ∑︁ ∑︁ 𝑡 0 ∥ 𝑓𝑇 (𝑡) − 𝑓𝑇 ( 𝑡¯)∥ [𝑡¯,𝑡¯+𝑡0 ) ≤ 𝑡 02 (𝐿 · ∥ 𝑓𝑇 ∥ ∞ ) = 𝑇𝑡0 (𝐿 · ∥ 𝑓𝑇 ∥ ∞ ) ≤ 𝛿. ⌈𝑁 ⌋ 𝑡¯∈𝑆𝑡 ⌈𝑁 ⌋ 𝑡¯∈𝑆𝑡 ∩[−𝑇/2,𝑇/2) 0 0 Repeating the above argument replacing 𝛾(𝜔) by bound for ∥ D ( 𝑓 , 𝑯) ¯ − D̃ ( 𝑓 , 𝑯) ¯ ∥. √︁ 𝛾(𝜔)𝛾(−𝜔) proves the same ■ Lemma 2.8.3. Both functions 𝛾𝐺 (𝜔) := (e4𝛽𝜔 +1) −1 and 𝛾 𝑀 (𝜔) := min(1, e−𝛽𝜔 ) are √︁ √︁ 𝛽-Lipschitz continuous for all 𝛽 ≥ 0. Moreover, 𝛾𝐺 (𝜔)𝛾𝐺 (−𝜔) and 𝛾 𝑀 (𝜔)𝛾 𝑀 (−𝜔) are 2𝛽 -Lipschitz continuous for all 𝛽 ≥ 0. Proof. A simple calculation shows that the absolute value of the derivative of (e4𝛽𝜔 + 1) −1 is largest at 0, where it is 𝛽, therefore it is 𝛽-Lipschitz continuous. Similarly, since min(1, e−𝛽𝜔 ) is continuous, and the absolute value of the (right) derivative of min(1, e−𝛽𝜔 ) is bounded by 𝛽 for every 𝜔 ∈ R, it is also 𝛽-Lipschitz continuous. Similar elementary calculation shows the 2𝛽 -Lipschitz continuity of √︁ √︁ 𝛾𝐺 (𝜔)𝛾𝐺 (−𝜔) and 𝛾 𝑀 (𝜔)𝛾 𝑀 (−𝜔). ■ Note that since we apply our generic bound Proposition 2.8.1 to the following nice functions, we get rather loose estimates of 𝑁, which are certainly off by polynomial factors from the tight values. However, since algorithmically, we only pay (poly)logarithmic cost in 𝑁, the looseness of our bounds probably only results in constant overheads. Corollary 2.8.1 (Discretization error of “finite-time” Davies generators). Let 𝑓 (𝑡) = √︃ 1 𝑇 1 (𝑡 ∈ [−𝑇/2, 𝑇/2)) and 𝜖 > 0. Assuming the normalization condition (2.10), √︁ if ∥𝛾∥ ∞ ≤ 1 and 𝛾(𝜔)𝛾(−𝜔) are 𝛽-Lipschitz continuous (e.g., Metropolis    or 2| 𝐴|(∥𝑯∥+1) (𝑇+1) (𝛽+1) 𝑛 Glauber), ∥𝛾∥ ∞ ≤ 1, then setting 𝑁 = 2 for 𝑛 = Θ log with 𝜖 appropriate constants ensures that ∥L ( 𝑓 ,𝑯) − L̄ ( 𝑓 ,𝑯) ∥ 1−1 ≤ 𝜖, where 𝜔0 = 𝑇2 √︃ 2𝜋 𝑇 (𝐶+1)𝑁 , 𝑡 0 = 2 √︃ ∥ D ( 𝑓 ,𝑯) − D̄ ( 𝑓 ,𝑯) ∥ ≤ 𝜖 . 2𝜋(𝐶+1) . 𝑁 𝜖 Proof. We use Proposition 2.8.1 for each 𝑨𝑎 setting 𝜖 ′ ← | 𝐴| to prove the claim. First Í 2 observe that 𝑓𝑇 = 𝑓 , and ∥ 𝑓 ∥ 2 = 1 = 𝜔∈𝑆 ⌈ 𝑁 ⌋ 𝑡 0 | 𝑓 ( 𝑡¯)| = 1. As 𝑓ˆ(𝑤) = sinc(𝑇𝜔/2), ¯ 𝑡0 100 𝜔 and | sinc(𝑇𝜔/2)| ≤ 1, and | d𝜔 sinc(𝑇𝜔/2)| ≤ 𝑇/4 we get that 𝐷 ≤ 𝑇4 and trivially 𝐿 = 0. Let us recall the tail bound on the discrete Fourier Transform from Proposition 2.6.6   ∑︁ 2 𝜋 . F̄ 𝑓¯( 𝑡¯) · e (i𝜈𝑡¯) ( 𝜔¯ − 𝜈) ≤ 𝑇 𝑚𝜔0 | 𝜔|>𝑚𝜔 ¯ 0 √︃ 𝑇 ˆ Similar bound holds for the continuous Fourier Transform 𝑓 (𝜔) = 2𝜋 sinc(𝑇𝜔/2) of the uniform weight function: ∫ ∞ 𝜋 | 𝑓ˆ(𝜔) 1 (|𝜔| ≥ 𝑊)| 2 d𝜔 ≤ , 𝑇𝑊 −∞  3   2 (∥𝑯∥+1) (𝑇+1) 𝑇 256∥𝑯∥𝑇 so choosing 𝑊 := Θ ensures the nec≥ 2∥𝑯∥ + 8𝜋 + 1 𝜖′ 𝜖′ 𝜖′  O (1)  (∥𝑯∥+1) (𝑇+1) (𝛽+1) essary tail bounds. Therefore it is easy to see that setting 𝑁 = Θ | 𝐴| 𝜖 satisfies all requirements of Proposition 2.8.1. ■ Corollary 2.8.2 (Discretization error of Gaussian Lindbladians and discriminant − 𝑡 2 proxies). Let 𝑓 (𝑡) = √ 1√ 𝑒 4𝜎2 and 𝜖 > 0. Assuming the normalization condition 𝜎 √︁2𝜋 (2.10), if ∥𝛾∥ ∞ ≤ 1 and 𝛾(𝜔)𝛾(−𝜔) are 𝛽-Lipschitz   continuous (e.g., Metropolis  (𝛽+1) or Glauber), , then setting 𝑁 = 2𝑛 for 𝑛 = Θ log 2| 𝐴|(∥𝑯∥+1) (𝜎+1/𝜎) with 𝜖 appropriate constants ensures ∥L ( 𝑓 ,𝑯) − L̄ ( 𝑓 ,𝑯) ∥ 1−1 ≤ 𝜖, √︃ √︃ 1 2𝜋 where 𝜔0 = 𝜎 𝑁 , 𝑡 0 = 𝜎 2𝜋 𝑁. ∥ D ( 𝑓 ,𝑯) − D̄ ( 𝑓 ,𝑯) ∥ ≤ 𝜖, 𝜖 to prove the claim. Proof. We use Proposition 2.8.1 for each 𝑨𝑎 setting 𝜖 ′ ← | 𝐴| First observe that ∥ 𝑓 ∥ 2 = 1 and since 𝑡0 = O (𝜎) we have ∑︁ 𝑡0 | 𝑓 ( 𝑡¯)| 2 = Θ(1). (2.67) ⌈𝑁 ⌋ 0 𝜔∈𝑆 ¯ 𝑡 √︂ √︃ 2 2 𝜔 ˆ As 𝑓ˆ(𝑤) = 𝜎 𝜋2 𝑒 −𝜎 𝜔 , we also get | d𝜔 𝑓 (𝑤)|/∥ 𝑓ˆ∥ ∞ ≤ 𝜎, so 𝐷 ≤ 𝜎. Since 𝜔 1 1 ∥ 𝑓 ∥ ∞ = ∥ 𝑓𝑇 ∥ ∞ and | d𝑡 𝑓 (𝑡)|/∥ 𝑓 ∥ ∞ ≤ 2𝜎 , we get 𝐿 ≤ 2𝜎 . Standard Gaussian tail bound tells us that ∥ 𝑓 − 𝑓𝑇 ∥ 22 ≤ 2 2 −𝑇 √ 𝑒 2𝜎2 , 𝑇 2𝜋 101 from which it follows that   ∑︁ 2 2𝑡0 2 −𝑇 2 𝑡0 | 𝑓 ( 𝑡¯) − 𝑓𝑇 ( 𝑡¯)| ≤ √ + √ 𝑒 2𝜎2 , 𝑇 2𝜋 𝜎 2𝜋 ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝑡 0   √︁ implying that it suffices to choose 𝑇 = Θ 𝜎 log(1/𝜖 ′) + 1 . By (2.67) and Proposition 2.6.7 the Fourier-transformed tail satisfies ! v u u ∑︁ 2 u 2 2 2 2 2 2 2 2 1 1 1 u t 𝑓ˆ¯( 𝜔) ¯ ≤O √ e−𝑁 𝜔0 𝜎 /2 + √︁ e−𝑊 𝜎 , e−𝑁 𝑡0 /16𝜎 + √ 𝑁𝜔 𝜎 𝑊 𝜎 𝑁𝑡 0 /𝜎 0 ⌈𝑁 ⌋ 𝜔∈𝑆 ¯ 𝜔0 | 𝜔¯ | ≥𝑊 √︁   √︁  log(1/𝜖 ′) + 1 and 𝑊 = Θ 𝜎1 log(1/𝜖 ′) + 1 + 2∥𝑯∥ .  O (1)  (∥𝑯∥+1) (𝜎+1/𝜎) (𝛽+1) Therefore, it is easy to see that setting 𝑁 = Θ | 𝐴| satisfies 𝜖 so it suffices to choose 𝑁 = Ω all requirements of Proposition 2.8.1. 2.9 ■ Appendix:Implications for Lindbladians from system-bath interaction Our algorithmic constructions are closely related to their physical origins, and the analytic framework conversely sheds light on the thermalization of open quantum systems. Under physical assumptions, one can microscopically derive a Lindbladian for a system coupled weakly to a bath (See, e.g., [152]). Among many candidates [56, 58, 139, 150, 170], we mainly focus on the Coarsed Grained Master Equation [121, 134] that enjoys transparent nonasymptotic error bounds and nicely connects to our algorithmic construction. Recall   ∑︁ ∫ ∞ 1 ˆ𝑎 𝑎 † † ˆ𝑎 𝑎 ˆ ˆ L (𝐶𝐺 𝑀 𝐸) [𝝆] := −i[𝑯 𝐿𝑆 , 𝝆] + 𝛾(𝜔) 𝑨 (𝜔) 𝝆 𝑨 (𝜔) − { 𝑨 (𝜔) 𝑨 (𝜔), 𝝆} d𝜔 2 𝑎∈𝐴 −∞ (2.68) =: L𝑢𝑛𝑖 [𝝆] + L 𝑑𝑖𝑠𝑠 [𝝆] and the correlation function 𝛾(𝜔) satisfying the symmetry 𝛾(𝜔)/𝛾(−𝜔) = e−𝛽𝜔 . The Lamb-shift term 46 ∫ 𝑇/2 ∫ 𝑇/2 ∑︁ i 𝑯 𝐿𝑆 := sgn(𝑡 1 − 𝑡2 )𝑐(𝑡 2 − 𝑡1 ) 𝑨𝑎† (𝑡2 ) 𝑨𝑎 (𝑡 1 )d𝑡2 d𝑡(2.69) √ 1 𝑎∈𝐴 2 2𝜋𝑇 −𝑇/2 −𝑇/2 46 Compared with [134, Eq.24], the factor of √1 is due to our Fourier Transform convention. 2𝜋 102 depends on the inverse Fourier Transform 𝑐(𝑡) of 𝛾(𝜔), i.e., ∫ ∞ 1 𝑐(𝑡) = √ 𝛾(𝜔)ei𝜔𝑡 d𝜔. 2𝜋 −∞ Let us also impose the normalization convention ∥𝑐∥ 1 ≤ √ 2𝜋 (2.70) √ 1 ≤ 1. Nicely, the strength which also controls the frequency domain by ∥𝛾∥ ∞ ≤ ∥𝑐∥ 2𝜋 of the Lamb-shift term is also suitably normalized: Lemma 2.9.1 (Norm of Lamb-shift term). For 𝑯 𝐿𝑆 as in (2.69), ∥𝑐∥ 1 ∑︁ 𝑎† 𝑎 𝑨 𝑨 . ∥𝑯 𝐿𝑆 ∥ ≤ √ 2 2𝜋 𝑎∈𝐴 For our normalization conventions (Eq. (2.10), Eq. (2.70)), the RHS would be 12 , which is comparable to the super-operator norm of the dissipative part (2.13). Note that if 𝑐(𝑡) is sufficiently smooth (for example, 𝑙-Lipschitz continuous with a not √︁ too large 𝑙), and we can efficiently prepare discretized states proportional to |𝑐(𝑡)| and √𝑐(𝑡) , then we can get a block-encoding of a good approximation of 1 ∫𝑯𝑇 𝐿𝑆 |𝑐(𝑡)| √ 2𝜋 −𝑇 |𝑐(𝑡)|d𝑡 by using Hamiltonian simulation time O (𝑇). The key is to prepare states proportional √︁ to |𝑐(𝑡 2 − 𝑡1 )| and √𝑐(𝑡2 −𝑡1 ) over the domain [−𝑇/2, 𝑇/2] × [−𝑇/2, 𝑇/2]. This |𝑐(𝑡 2 −𝑡1 )| can be done by first preparing a uniform superposition over discretized values of √︁ 𝑡1 on the interval [−𝑇/2, 𝑇/2], and also a state proportional to |𝑐(𝑡2 )| on the interval [−𝑇, 𝑇]. This is a product state, but then we add the first variable to the second 𝑡˜2 ← 𝑡 2 + 𝑡1 (which we implement in superposition on the register containing the discretized values of 𝑡2 ). The resulting new variables 𝑡1 , 𝑡˜2 restricted to the domain [−𝑇/2, 𝑇/2] × [−𝑇/2, 𝑇/2] have the desired amplitudes. The case of √𝑐(𝑡2 −𝑡1 ) is completely analogous. Now the block-encoding is simple: prepare a state |𝑐(𝑡2 −𝑡1 )| ¯ ¯ proportional √𝑐(𝑡2 −𝑡1 ) | 𝑡¯1 ⟩| 𝑡¯2 ⟩, apply sgn( 𝑡¯2 − 𝑡¯1 ) 𝑨𝑎 ( 𝑡¯2 ) 𝑨𝑎 ( 𝑡¯1 ) and finally unprepare |𝑐( 𝑡¯2 −𝑡¯1 )| √︁ the state proportional to |𝑐( 𝑡¯2 − 𝑡¯1 )|| 𝑡¯1 ⟩| 𝑡¯2 ⟩ (run the preparation in reverse). This block-encoding ensures that we can accurately simulate the above Master Equation using Theorem 2.3.2. Compared with our algorithmic construction, the CGME differs in the following ways. First, it contains a unitary part L𝑢𝑛𝑖 , especially the Lamb-shift term 𝑯 𝐿𝑆 . This requires additional technical tools to handle, so we temporarily drop this term and 103 postpone its discussion to section 2.9. The second difference is less essential: instead of discrete Fourier Transforms, the Kraus operators are labeled by continuous Bohr frequencies with appropriate normalizations. √︂ ˆ 𝑨(𝜔) := 1 2𝜋𝑇 ∫ 𝑇/2 √︂ e−i𝜔𝑡 𝑨(𝑡)d𝑡 = −𝑇/2 1 ∑︁ e−i(𝜔−𝜈)𝑇/2 − ei(𝜔−𝜈)𝑇/2 𝑨𝜈 . 2𝜋𝑇 i𝜔 𝜈∈𝐵(𝑯) Note the Fourier Transform convention. The time scale 𝑇 depends on parameters of the open system, such as the bath correlation function and the coupling strength [134]; in our error bounds, we will keep 𝑇 as a tunable abstract parameter. Still, using a similar argument for analyzing the algorithm, we control the fixed point error for the Lindbladian. In fact, the bounds and the presentation simplify as discretization errors vanish. The main idea is to introduce the secular approximation (using notations in section 4.15)   ∑︁ ∫ ∞ 1 𝑎 † † 𝑎 L 𝑠𝑒𝑐 := 𝛾(𝜔) 𝑺ˆ (𝜔) [·] 𝑺ˆ𝑎 (𝜔) − { 𝑺ˆ𝑎 (𝜔) 𝑺ˆ (𝜔), ·} d𝜔. (2.71) 2 −∞ 𝑎∈𝐴 Theorem 2.9.1 (Fixed point of the dissipative part). The dissipative part of the CGME Lindbladian (2.68) (satisfying normalization and symmetry conditions (2.10),(2.12),(2.70)), has an approximate Gibbs fixed point ! √︂ 𝛽 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠 ) . ∥ 𝝆 𝑓 𝑖𝑥 (L 𝑑𝑖𝑠𝑠 ) − 𝝆 𝛽 ∥ 1 ≤ O 𝑇 Since the proof structure is analogous, we present the altogether bounds and derive them in the following sections. Proof. Telescope for the fixed points ∥ 𝝆 𝑓 𝑖𝑥 (L 𝑑𝑖𝑠𝑠 ) − 𝝆 𝛽 ∥ 1 ≤ ∥ 𝝆 𝑓 𝑖𝑥 (L 𝑑𝑖𝑠𝑠 ) − 𝝆 𝑓 𝑖𝑥 (L 𝑠𝑒𝑐 )∥ 1 + ∥ 𝝆 𝑓 𝑖𝑥 (L 𝑠𝑒𝑐 ) − 𝝆 𝛽 ∥ 1    ≤ O ∥L − L 𝑠𝑒𝑐 ∥ 1−1 + D ( 𝝆, L 𝑠𝑒𝑐 ) − D ( 𝝆, L 𝑠𝑒𝑐 ) † 2−2 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠 )  ≤O 1 +𝛽 √ 𝜇𝑇 √︂   𝜇 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠 ) . 𝑇 The second inequality uses identical arguments as (2.20) from the proof of Theorem 2.1.3. The third inequality plugs in bounds for the secular approximation (the 104 continuous case is the limit of the discrete case (Lemma 2.6.2)) and approximate detailed balance (Proposition 2.9.1). Optimize the free parameter 𝜇 = 1𝛽 to conclude the proof. ■ The mixing time can be bounded by the gap (Proposition 2.2.3) if needed, but for conceptual simplicity, we stuck to the mixing time for the main presentation. Bounds for approximate detailed balance Here, we show approximate detailed balance for the Lindbladian of interest L 𝑠𝑒𝑐 . Proposition 2.9.1 (Approximate detailed balance). Suppose the secular approximation for L (𝐶𝐺 𝑀 𝐸) (2.71) is truncated at energy 𝜇. Then,  √︂  𝜇 1 † D ( 𝝆, L 𝑠𝑒𝑐 ) − D ( 𝝆, L 𝑠𝑒𝑐 ) 2−2 ≤ O 𝛽 . 2 𝑇 Proof. We simply telescope by inserting the algorithmically constructed discriminant D𝑠𝑒𝑐 (Corollary 2.7.1) D ( 𝝆, L 𝑠𝑒𝑐 ) − D ( 𝝆, L 𝑠𝑒𝑐 ) † 2−2 ≤ ∥D ( 𝝆, L 𝑠𝑒𝑐 ) − D𝑠𝑒𝑐 ∥ 2−2 + ∥D𝑠𝑒𝑐 − D ( 𝝆, L 𝑠𝑒𝑐 ) † ∥ 2−2 ≤ 2∥D𝑠𝑒𝑐 − D ( 𝝆, L 𝑠𝑒𝑐 )∥ 2−2 . † The last inequality uses that D𝑠𝑒𝑐 = D𝑠𝑒𝑐 and that ∥A ∥ 2−2 = ∥A † ∥ 2−2 . Note that the algorithmic discussion considered discrete energy labels, so we have to take a continuum limit for the bilinear sum47 ∫ ∞ ∑︁ 𝑎 † 𝑎 𝛾( 𝜔) ¯ 𝑺ˆ ( 𝜔) ¯ 𝑺ˆ ( 𝜔) ¯ → 𝛾(𝜔) 𝑺ˆ 𝑎† (𝜔) 𝑺ˆ 𝑎 (𝜔)d𝜔. 𝜔∈𝑆 ¯ 𝜔0 −∞ Indeed, the bound does not depend on the discretization scale. ■ Effects of the Lamb-shift term In this section, we include the unitary part of the CGME generator. The resulting bounds now depend on two mixing times, and we do not have a desirable conversion between the two mixing times. Still, one can upper bound both via the spectral gap of the Hermitian part of the dissipative part H𝑑𝑖𝑠𝑠 . 47 Formally speaking, the correctness in this limit can be derived by using our discrete results and taking their limit as in 2.8.1. 105 Theorem 2.9.2 (Fixed point of CGME). For the full CGME generator (2.68), which satisfy the symmetry and normalization conditions (2.10),(2.11),(2.12),(2.70), we have that ! √︂ 𝛽 (𝑡 𝑚𝑖𝑥 (L𝐶𝐺 𝑀 𝐸 ) + 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠 )) . ∥ 𝝆 𝑓 𝑖𝑥 (L𝐶𝐺 𝑀 𝐸 ) − 𝝆 𝛽 ∥ 1 ≤ O 𝑇 We have already calculated the errors for the dissipative part; here we only study errors for the Lamb-shift term as in the following sections and combine them at section 2.9. The strategy is similar; we discretize the Hamiltonian, truncate the operator via the secular approximation, and then argue that the resulting operator nearly commutes with the Gibbs state 𝑯 𝐿𝑆 ≈ 𝑯 𝐿𝑆,𝑠𝑒𝑐 ≈ √ √ −1 𝝆𝑯 𝐿𝑆,𝑠𝑒𝑐 𝝆 . Formally, we introduce the intermediate constructs L 𝑠𝑒𝑐 := L𝑢𝑛𝑖,𝑠𝑒𝑐 + L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 where and L ′ := √ √ −1 √ −1 √ 𝝆(L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 − L𝑢𝑛𝑖,𝑠𝑒𝑐 ) [ 𝝆 · 𝝆 ] 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 := −i[𝑯 + 𝑯 𝐿𝑆,𝑠𝑒𝑐 , ·]. Note that in L ′ we had to manually flip the sign of the coherent part to ensure L 𝑠𝑒𝑐 ≈ L ′. One may add any coherent term −i[𝑽, ·] as long as [𝑽, 𝝆] = 0 (most notably the Hamiltonian 𝑯); it would not contribute to the error bounds. Secular approximation for the Lamb-shift Rewrite the integral by change-of-variable 𝑠 := 𝑡2 − 𝑡1 and apply the secular approximation to the inner integral ! ! ∫ 𝑇 ∫ min(𝑇/2−𝑠,𝑇/2) ∑︁ i 𝑯 𝐿𝑆 := sgn(−𝑠)𝐶 (𝑠) 𝑨𝑎† (𝑠) 𝑨𝑎 (𝑡1 )d𝑡1 d𝑠 2𝑇 −𝑇 max(−𝑇/2,−𝑇/2−𝑠) 𝑎∈𝐴 The secular approximation in this context differs from the one we used for the bilinear expressions (Lemma 2.6.3). We define and analyze the error as follows, inspired by [40]. Lemma 2.9.2 (Secular approximation for time average). Consider an operator 𝑨 and a Hermitian operator 𝑯. Then, for any unitarily invariant norm ∥·∥ ∗ and times 𝑡1 , 𝑡2 , there exists a secular approximated operator 𝑺 𝜇 such that 𝜓 𝑗 𝑺 𝜇 |𝜓𝑖 ⟩ = 0 if 𝐸𝑖 − 𝐸 𝑗 ≤ 𝜇 106 and ∫ 𝑡2 ei𝑯𝑠 𝑨e−i𝑯𝑠 d𝑠 − 𝑺 𝜇 ≤ O ∥ 𝑨∥ ∗ 1 + log( 𝜇|𝑡21−𝑡1 | ) . 𝜇 ∗ 𝑡1 ! ∫𝑡 See section 2.9 for the proof. Intuitively, the time average 𝑡 2 𝑨(𝑠)d𝑠 weakens the 1 off-diagonal entries (in the 𝑯 eigenbasis) with a large Bohr frequency. Dropping them incurs an error depending on the truncation value 𝜇. Applying the secular approximation for the Lamb-shift Hamiltonian yields the following bound. Corollary 2.9.1 (Secular approximation for the Lamb-shift term). In the setting of Theorem 2.9.2, the exists a Hermitian operator 𝑯 𝐿𝑆,𝑠𝑒𝑐 such that 𝜓 𝑗 𝑯 𝐿𝑆,𝑠𝑒𝑐 |𝜓𝑖 ⟩ = 0 if 𝐸𝑖 − 𝐸 𝑗 ≤ 𝜇 and     Í ∫ ∥ 𝑎∈𝐴 𝑨𝑎† 𝑨𝑎 ∥ 𝑇 1 |𝐶 (𝑠)|d𝑠 = Õ ∥𝑯 𝐿𝑆 − 𝑯 𝐿𝑆,𝑠𝑒𝑐 ∥ = Õ . 𝜇𝑇 𝜇𝑇 −𝑇 Proof. Apply secular approximation to the inner integral (which depends on 𝑠) to obtain ∫ 𝑇 i 𝑯 𝐿𝑆,𝑠𝑒𝑐 := sgn(−𝑠)𝐶 (𝑠)𝑺 𝜇(𝑠) d𝑠 2𝑇 −𝑇 and calculate 1 ∥𝑯 𝐿𝑆 − 𝑯 𝐿𝑆,𝑠𝑒𝑐 ∥ ≤ 2𝑇 ∫ 𝑇 ∫ min(𝑇/2−𝑠,𝑇/2) ∑︁ |𝐶 (𝑠)| ( 𝑨𝑎† (𝑠) 𝑨𝑎 )(𝑡1 )d𝑡1 − 𝑺 𝜇(𝑠) d𝑠. −𝑇 max(−𝑇/2,−𝑇/2−𝑠) 𝑎∈𝐴 Use the secular approximation (Lemma 2.9.2) for the integral over 𝑡 1 to conclude the proof. ■ Approximate detailed balance Thirdly, we also control the error for approximate detailed balance. Lemma 2.9.3 (Apprximate detailed balance for the unitary part). In the setting of Theorem 2.9.2, if 𝛽𝜇 ≤ 1,  ∫ 𝑇  √ √ −1 √ −1 √ † |𝐶 (𝑠)| d𝑠 = Õ (𝛽𝜇). L𝑢𝑛𝑖 + 𝝆L𝑢𝑛𝑖 [ 𝝆 · 𝝆 ] 𝝆 ≤ Õ 𝛽𝜇 2−2 −𝑇 107 We present the superoperator form to feed into our existing fixed-point analysis. Still, the calculation essentially reduces to the operator norm. We will need the following proposition, whose proof is reminiscent of the arguments (section 2.7) analyzing the 𝛿R part of Lemma 2.7.3. Proposition 2.9.2. Suppose an operator 𝑨 satisfies ⟨𝐸𝑖 | 𝑨 𝐸 𝑗 = 0 whenever 𝐸𝑖 − 𝐸 𝑗 ≤ 𝜇 and 𝛽𝜇 ≤ 1, then √ √ −1 ∥ 𝑨 − 𝝆 𝑨 𝝆 ∥ = O (∥ 𝑨∥ 𝛽𝜇). Proof. Consider nearby energy projectors at energy resolution 𝜇. ∑︁ ∑︁ 𝑰= 𝑷𝑎𝜇 where 𝑷𝑎𝜇 := 𝑷𝐸 . (𝑎+ 12 )𝜇>𝐸 ≥(𝑎− 21 )𝜇 𝑎∈Z Then, the matrix 𝑨 is tri-block-diagonal 𝑨 = 𝑼 + 𝑳 + 𝑫 with blocks labeled by Í integer multiples of 𝜇. For the lower-diagonal-blocks 𝑳 = 𝑎 𝑷 (𝑎+1)𝜇 𝑨𝑷𝑎𝜇 , we evaluate the commutator for each term √ −1 √ ′ ′ 𝑷 (𝑎+1)𝜇 𝑨𝑷𝑎𝜇 − 𝝆𝑷 (𝑎+1)𝜇 𝑨𝑷𝑎𝜇 𝝆 = 𝑷 (𝑎+1)𝜇 𝑨𝑷𝑎𝜇 − e−𝛽𝑯 /2 𝑷 (𝑎+1)𝜇 𝑨𝑷𝑎𝜇 e 𝛽𝑯 /2 , where 1 𝑯′ = 𝑯 − (𝑎 + ) 𝑰 2 such that ∥𝑯′ ∥ ≤ 𝜇. Therefore, √ √ −1 √ √ −1 ∥ 𝑳 − 𝝆 𝑳 𝝆 ∥ ≤ max ∥𝑷 (𝑎+1)𝜇 𝑨𝑷𝑎𝜇 − 𝝆𝑷 (𝑎+1)𝜇 𝑨𝑷𝑎𝜇 𝝆 ∥ 𝑎 (By Lemma 2.7.1) = O (𝛽𝜇). (By |e𝑥 − 1| ≤ 2|𝑥| for |𝑥| ≤ 1) The bounds on 𝑫 and 𝑼 are analogous. Proof of Lemma 2.9.3. Expand √ −1 √ −1 √ √ † L𝑢𝑛𝑖 [ 𝑨] + 𝝆L𝑢𝑛𝑖 [ 𝝆 𝑨 𝝆 ] 𝝆 √ √ −1 √ −1 √ = i𝑯 𝐿𝑆,𝑠𝑒𝑐 𝑨 − i𝑨𝑯 𝐿𝑆,𝑠𝑒𝑐 − i 𝝆𝑯 𝐿𝑆,𝑠𝑒𝑐 𝝆 𝑨 + i𝑨 𝝆 𝑯 𝐿𝑆,𝑠𝑒𝑐 𝝆   √ √ −1  √ −1 √  = i 𝑯 𝐿𝑆,𝑠𝑒𝑐 − 𝝆𝑯 𝐿𝑆,𝑠𝑒𝑐 𝝆 𝑨 − i𝑨 𝑯 𝐿𝑆,𝑠𝑒𝑐 − 𝝆 𝑯 𝐿𝑆,𝑠𝑒𝑐 𝝆 . ■ 108 The Hamiltonian term 𝐻 disappears because it commutes with the Gibbs state. Now, Holder’s inequality reduces the superoperator norm to the operator norm, which can be controlled by Proposition 2.9.2
√ √ −1 𝑯 𝐿𝑆,𝑠𝑒𝑐 − 𝝆𝑯 𝐿𝑆,𝑠𝑒𝑐 𝝆 ≤ O (∥𝑯 𝐿𝑆,𝑠𝑒𝑐 ∥ 𝛽𝜇) = O (∥𝑯 𝐿𝑆 ∥ + ∥𝑯 𝐿𝑆 − 𝑯 𝐿𝑆,𝑠𝑒𝑐 ∥) 𝛽𝜇 Use Corollary 2.9.1 and Lemma 2.9.2 to conclude the proof. ■ Altogether: Proof of fixed point correctness (Theorem 2.9.2) We now put together the error bounds for the full CGME Lindbladian. Proof of Theorem 2.9.2. Recall the bound on the fixed point error ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆∥ 1 = ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆 𝑓 𝑖𝑥 (L 𝑠𝑒𝑐 )∥ 1 + ∥ 𝝆 𝑓 𝑖𝑥 (L 𝑠𝑒𝑐 ) − 𝝆 𝑓 𝑖𝑥 (L ′)∥ 1   ∥D ( 𝝆, L 𝑠𝑒𝑐 ) − D ( 𝝆, L ′)∥ 2−2 ≤ O ∥L − L 𝑠𝑒𝑐 ∥ 1−1 𝑡 𝑚𝑖𝑥 (L) + 𝜍 −2 (D ( 𝝆, L 𝑠𝑒𝑐 ))  ≤ O ∥L − L 𝑠𝑒𝑐 ∥ 1−1 𝑡 𝑚𝑖𝑥 (L) + L 𝑑𝑖𝑠𝑠 − L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 1−1 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠 )  † †   + ∥D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) + D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) ∥ 2−2 + ∥D ( 𝝆, L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 ) − D ( 𝝆, L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 ) ∥ 2−2 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠 ) , (2.72) where we compare the fixed points of L 𝑠𝑒𝑐 and L ′ by eigenvector perturbation (Proposition 2.10.2, noting that L 𝑠𝑒𝑐 contains an eigenvalue zero as it generates a CPTP map; L ′ has the Gibbs state as its fixed point, which has eigenvalue zero). The third inequality bound the singular value by the mixing time: apply Fan-Hoffman [21, Proposition III.5.1] and use perturbation bounds for sorted singular values   2𝜍 −2 (D ( 𝝆, L 𝑠𝑒𝑐 )) ≥ 𝜍 −2 D ( 𝝆, L 𝑠𝑒𝑐 ) + D ( 𝝆, L 𝑠𝑒𝑐 ) †   † ≥ 𝜆 2 D ( 𝝆, L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 ) + D ( 𝝆, L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 ) − D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) + D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) † 2−2 ≥ Ω( 1 ) − D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) + D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) † 2−2 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 ) ≥ Ω( 1 ) − D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) + D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) † 2−2(2.73) . 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠 ) 109 The rest are analogous to (2.20), except that we have to manually include the term L 𝑑𝑖𝑠𝑠 − L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 1−1 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠 ) to ensure Eqn. (2.73) holds. By secular approximation with truncation energy 𝜇 and 𝜇′ (which will be set to different values to minimize the error bounds), ∥𝑯 𝐿𝑆 − 𝑯 𝐿𝑆,𝑠𝑒𝑐 ∥ = Õ ( 1 ) 𝜇𝑇 and 1 ∥L 𝑑𝑖𝑠𝑠 − L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 ∥ 1−1 = O ( √ ′ ) 𝜇𝑇 which combines to ∥L − L 𝑠𝑒𝑐 ∥ 1−1 = i[𝑯 𝐿𝑆 , ·] + L 𝑑𝑖𝑠𝑠 − i[𝑯 𝐿𝑆,𝑠𝑒𝑐 , ·] − L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 1−1   1 1 = Õ +√ ′ . 𝑇𝜇 𝜇𝑇 The second inequality reduces the superoperator norm ∥·∥ 1−1 to operator norm by ∥ 𝑨𝝆∥ 1 ≤ ∥ 𝑨∥∥ 𝝆∥ 1 . Next, we combine the approximate detailed balance-type errors from the Lamb-shift term (Lemma 2.9.3) and the dissipative term (Proposition 2.9.1) ∥D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) + D ( 𝝆, L𝑢𝑛𝑖,𝑠𝑒𝑐 ) † ∥ 2−2 + ∥D ( 𝝆, L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 ) − D ( 𝝆, L 𝑑𝑖𝑠𝑠,𝑠𝑒𝑐 ) † ∥ 2−2 √︂ = Õ 𝛽𝜇 + 𝛽 ! 𝜇′ . 𝑇 Altogether, choose 𝜇 = √︃ 1 1 1 ′ 𝛽𝑇 ≤ 𝛽 and 𝜇 = 𝛽 so that √︂ ! 𝛽 (2.72) ≤ Õ max(𝑡 𝑚𝑖𝑥 (L), 𝑡 𝑚𝑖𝑥 (L 𝑑𝑖𝑠𝑠 )) · , 𝑇 which concludes the proof. ■ Proof for secular approximation for time average (Lemma 2.9.2) Intuitively, we want to truncate the Bohr frequency far from zero. Unfortunately, the sharp truncation from section 4.15 does not seem to work here because the truncation error is related to the 1-norm ∥ 𝑓 ∥ 1 (instead of 2-norm ∥ 𝑓 ∥ 2 ). The 1-norm is more delicate to handle, forcing us to smoothly truncate the tail and explicitly evaluate the Fourier Transform in the time domain. Pictorially, the time domain function becomes a smeared version of the sharp window function 1 (|𝑡| ≤ 𝑇) where the discontinuity is smoothed out due to convolution with a smooth bump function. 110 Proof of Lemma 2.9.2. Without loss of generality, we can conjugate with time evolution to shift the integral so that 𝑡2 = −𝑡1 = 𝑇/2. Let 𝑓 (𝑡) = 1 (|𝑡| ≤ 𝑇/2), then 1 √ 2𝜋 with 𝑓ˆ(𝜔) = e ∫ 𝑇/2 1 𝑨(𝑠)d𝑠 = √ −𝑇/2 2𝜋 −i𝜔𝑇/2 −ei𝜔𝑇/2 √ 2𝜋𝜔 1 √ 2𝜋 ∫ ∞ 𝑓 (𝑠) 𝑨(𝑠)d𝑠 = −∞ ∑︁ 𝑨𝜈 𝑓ˆ(−𝜈) 𝜈 . Let us truncate the frequency domain function ∫ ∞ (𝑏 ∗ 𝑓 )(𝑠) 𝑨(𝑠)d𝑠 = −∞ ∑︁ 𝑨𝜈 𝑏ˆ · 𝑓ˆ(−𝜈) =: 𝑺 𝜇 𝜈 by multiplying with a carefully chosen smooth bump function    0 if |𝑥| ≥ 𝜇     ˆ 𝑏(𝑥) := 1 if 𝑥 = 0 .      ≤ 1 else  Then, for any unitarily invariant norm, ∫ 𝑇/2 𝑨(𝑠)d𝑠 − 𝑺 𝜇 −𝑇/2 ∗ 1 ≤ √ ∥ 𝑓 − 𝑏 ∗ 𝑓 ∥1 2𝜋   ∫ ∫ 1 =√ ( 𝑓 − 𝑏 ∗ 𝑓 )(𝑡)d𝑡 + ( 𝑓 − 𝑏 ∗ 𝑓 )(𝑡)d𝑡 𝑊𝜖 2𝜋 𝑊 𝜖𝑐 1 ! 1 1 + log( 𝜇𝑇 ) ≤O + . 𝜇 𝜇 (2.74) The second equality separately evaluates the integral around 𝜖-balls near ±𝑇 𝑊𝜖 := [ 𝑇 𝑇 𝑇 𝑇 − 𝜖, + 𝜖] ∪ [− − 𝜖, − + 𝜖] 2 2 2 2 for 𝜖= 1 . 𝜇 For each 𝑡 ∈ 𝑊𝜖𝑐 , the convolution is point-wise close to the original value ∫ ∞  ∫ ∞ 𝑓 (𝑡 − 𝑠)𝑏(𝑠)d𝑠 − 𝑓 (𝑡) = 𝑓 (𝑡) 𝑏(𝑠)d𝑠 − 1 + 𝑟 (𝜇|𝑡 − 𝑇 |) −∞ −∞ up to an error 𝑟 (𝑥) falling super-polynomially with |𝑥|. Thus, the integral over 𝑊𝜖𝑐 (2.74) is then bounded by ∫ ∫ ∞ 4 1 |𝑟 (𝜇|𝑡 − 𝑇 |)|d𝑡 = |𝑟 (𝑥)|d𝑥 = O ( ). 2 𝜇 1 𝜇 |𝑡−𝑇 |≥𝜇 111 The main error arises from the sharp edge at ±𝑇; we invoke general norm bounds ∥ 𝑓 ∥ ∞ = 1 and 1 ∥𝑏 ∗ 𝑓 ∥ ∞ ≤ √ ∥ 𝑏ˆ · 𝑓ˆ∥ 1 2𝜋 ∫ 1/𝑇  ∫ 𝜇 ∫ −1/𝑇 1 ˆ ≤1) ≤ √ 𝑓ˆ(𝜔) d𝜔 + ( + ) 𝑓ˆ(𝜔) d𝜔 (By ∥ 𝑏∥ 1/𝑇 −𝜇 2𝜋 −1/𝑇   1 (By 𝑓ˆ(𝜔) ≤ min( √𝑇 , √ 1 ) ) = O 1 + log( ) 2𝜋 2𝜋𝜔 𝜇𝑇 and integrate over 𝑊𝜖 to obtain the bound. 2.10 ■ Appendix:Spectral bounds and mixing times In this section, we present missing proofs for lemmas and propositions. While some arguments are standard and included merely for completeness, controlling the spectrum of nearly Hermitian matrices requires a substantial linear algebraic argument. We begin with eigenvalue and eigenvector perturbation theory (section 2.10), which is crucial for establishing mixing time bounds (section 2.10) and the correctness of fixed points (section 2.10). Perturbation bounds for eigenvalues and eigenvectors In this section, we present some useful bounds for eigenvalue and eigenvector perturbation. Proposition 2.10.1 (Bauer-Fike Theorem with multiplicity, cf. [21, Theorem VI.3.3 & Problem VI.8.6]). Perturb a normal matrix 𝑵 by an arbitrary matrix 𝑨. Then, the spectrum of 𝑵 and 𝑵 + 𝑨 are ∥ 𝑨∥-close to each other: Spec(𝑵 + 𝑨) ⊂ ∪𝑠∈Spec(𝑵) 𝐷 (𝑠, ∥ 𝑨∥), Spec(𝑵) ⊂ ∪𝑠∈Spec(𝑵+𝑨) 𝐷 (𝑠, ∥ 𝑨∥), and where 𝐷 (𝑠, 𝜖) = {𝑧 ∈ C : |𝑧 − 𝑠| ≤ 𝜖 }. Moreover, the connected components of ∪𝑠∈Spec(𝑵) 𝐷 (𝑠, ∥ 𝑨∥) contain an equal number of eigeinvalues of 𝑵 and 𝑵 + 𝑨 when counted with algebraic multiplicity. Proof. The first half of the statement is the Bauer-Fike Theorem [21, Theorem VI.3.3]. To study the number of eigenvalues per connected component, we consider an interpolation path 𝑿 (𝑡) = 𝑵 + 𝑡 𝑨 for 0 ≤ 𝑡 ≤ 1. 112 The Bauer-Fike Theorem applied to 𝑵 + 𝑡 𝑨 implies for every 0 ≤ 𝑡 ≤ 1 that Spec( 𝑿 (𝑡)) ⊂ ∪𝑠∈Spec(𝑵) 𝐷 (𝑠, 𝑡 ∥ 𝑨∥) ⊂ ∪𝑠∈Spec(𝑵) 𝐷 (𝑠, ∥ 𝑨∥). Then, by continuity of eigenvalues along the path, cf. [21, Corollary VI.1.6], no eigenvalues enter or exit the connected components of ∪𝑠∈Spec(𝑵) 𝐷 (𝑠, ∥ 𝑨∥), therefore their number (counted with algebraic multiplicity) is the same for 𝑿 (0) = 𝑵 and 𝑿 (1) = 𝑵 + 𝑨. ■ Corollary 2.10.1 (Eigenvalue perturbation for discriminants). If D is Hermitian and has norm bounded by ∥ D∥ ≤ 1, and D′ has a right eigenvector |𝜓⟩ with eigenvalue 1, then the top eigenvalue of D satisfies |𝜆 1 (D) − 1| ≤ ∥ D − D′ ∥. Eigenvector perturbation bounds Intuitively, perturbing a matrix yields small changes in eigenvectors with wellisolated eigenvalues — we prove this below rigorously under suitable but quite general conditions using a simple linear algebraic argument. Proposition 2.10.2 (Eigenvector perturbation). Perturb a matrix 𝑴 by another matrix 𝑨. Let |𝑣⟩ be a normalized right eigenvector 𝑴 |𝑣⟩ = 𝜆|𝑣⟩, and 𝜆′ an eigenvalue of 𝑴 + 𝑨. Then the corresponding right eigenvector ( 𝑴 + 𝑨)|𝑣 ′⟩ = 𝜆′ |𝑣 ′⟩ can be normalized such that √ 2 2(∥ 𝑨∥ + |𝜆′ − 𝜆|) ⟨𝑣|𝑣⟩ = ⟨𝑣 ′ |𝑣 ′⟩ = 1 and ∥|𝑣 ′⟩ − |𝑣⟩∥ ≤ , 𝜍 −2 ( 𝑴 − 𝜆𝑰) where 𝜍 −2 (·) denotes the second-smallest singular value (with multiplicity). Due to Fan-Hoffman [21, Proposition III.5.1] the singular value 𝜍 −2 ( 𝑴 − 𝜆𝑰) ≥  † −𝜆 2 𝑴+𝑴 − Re(𝜆) 𝑰 can be bounded in terms of the Hermitian part. 2 Proof. We can assume without loss of generality that |𝑣 ′⟩ ∝ |𝑣⟩ +𝜖 |𝑣 ⊥ ⟩ for 𝜖 ∈ [0, ∞) and ⟨𝑣 ⊥ |𝑣 ⊥ ⟩ = 1, yielding ( 𝑴 + 𝑨)(|𝑣⟩ + 𝜖 𝑣 ⊥ ) = 𝜆′ 𝑰(|𝑣⟩ + 𝜖 𝑣 ⊥ ) ⇕ 𝑨(|𝑣⟩ + 𝜖 𝑣 ⊥ ) + ( 𝑴 − 𝜆′ 𝑰)|𝑣⟩ = 𝜖 (𝜆′ 𝑰 − 𝑴) 𝑣 ⊥ . 113 Taking the norms above on both sides and defining 𝜅 := |𝜆′ − 𝜆| we get 𝜖= ∥ 𝑨(|𝑣⟩ + 𝜖 |𝑣 ⊥ ⟩) + (𝜆 − 𝜆′)|𝑣⟩∥ (1 + 𝜖)∥ 𝑨∥ + 𝜅 ≤ ′ ⊥ ∥( 𝑴 − 𝜆 𝑰)|𝑣 ⟩∥ ⇓ 𝜍 −2 ( 𝑴 − 𝜆𝑰) − 𝜅 ∥ 𝑨∥ + 𝜅 𝜍 −2 ( 𝑴 − 𝜆𝑰) − ∥ 𝑨∥ − 𝜅 ∥ 𝑨∥ + 𝜅 1 = · 𝜍 −2 ( 𝑴 − 𝜆𝑰) 1 − ∥ 𝑨∥+𝜅 𝜖≤ . (2.75) 𝜍 −2 ( 𝑴−𝜆𝑰) The last inequality is a rearrangement. The first inequality uses the triangle inequality for the numerator, and for the denominator that ∥( 𝑴 − 𝜆′ 𝑰)|𝑣 ⊥ ⟩∥ ≥ ∥ ( 𝑴 − 𝜆𝑰)|𝑣 ⊥ ⟩∥ − ∥(𝜆′ − 𝜆) 𝑰|𝑣 ⊥ ⟩∥ = ∥( 𝑴 − 𝜆𝑰)|𝑣 ⊥ ⟩∥ − 𝜅 and ∥( 𝑴 − 𝜆𝑰) 𝑣 ⊥ ∥ ≥ 𝜍 −2 ( 𝑴 − 𝜆𝑰). We conclude by setting the appropriate normalization |𝑣 ′⟩ = √ 1 2 (|𝑣⟩ + 𝜖 |𝑣 ⊥ ⟩) and 1+𝜖 utilizing the above bound (2.75): √︁ 1 ∥|𝑣⟩ − |𝑣 ′⟩∥ = √ ∥ 1 + 𝜖 2 |𝑣⟩ − (|𝑣⟩ + 𝜖 𝑣 ⊥ )∥ + 𝜖2 √︃1 √ ! √ √ 2 2(∥ 𝑨∥ + 𝜅) ( 1 + 𝜖 2 − 1) 2 + 𝜖 2 √ ≤ 2𝜖 ≤ min 2, = . √ 𝜍 −2 ( 𝑴 − 𝜆′ 𝑰) 1 + 𝜖2 √ The first inequality uses 1 + 𝜖 2 − 1 ≤ 𝜖. The last inequality uses that the bound √ is vacuous at ∥|𝑣⟩ − |𝑣 ′⟩∥ ≤ 2 and combines (2.75) with the elementary estimate √ √ 𝑥 ≤ 2 2𝑥 after substituting 𝑥 ← 𝜍 −2∥(𝑨∥+𝜅 ∀𝑥 ∈ [0, 1/2] : 2 1−𝑥 ■ 𝑴−𝜆𝑰) . When we apply the above perturbation bound for a Hermitian 𝑴, we can set 𝜆 to be the top eigenvalue such that 𝜍 −2 ( 𝑴 − 𝜆𝑰) will be the gap 𝜆 𝑔𝑎 𝑝 ( 𝑴) = 𝜆1 ( 𝑴) − 𝜆 2 ( 𝑴). In that case, Proposition 2.10.1 guarantees the existence of a nearby eigenvalue 𝜆′ of 𝑴 + 𝑨 such that |𝜆′ − 𝜆| ≤ ∥ 𝑨∥. When we handle the Lamb-shift term, we need to consider a nonHermitian 𝑴. There, bounding eigenvalue perturbation is not generally obvious, so we will simply assume that there is a nearby eigenvalue 𝜆′ of 𝑴 + 𝑨. Approximate detailed balance implies approximately correct fixed point When detailed balance holds approximately for 𝝆, we still expect the fixed point to be approximately 𝝆; we provide a proof of this in this section, which relies on the matrix perturbation results (section 2.10). Recall that in section 2.2, we defined the 114 Hermitian and anti-Hermitian parts (under similarity transformation) as follows: D ( 𝝆, L) = 𝝆 −1/4 L [𝝆 1/4 · 𝝆 1/4 ] 𝝆 −1/4 = H + A, D ( 𝝆, L) † = 𝝆 1/4 L † [𝝆 −1/4 · 𝝆 −1/4 ] 𝝆 1/4 = H − A. √ Observe that D ( 𝝆, L) † [ 𝝆] = 0, but it needs not be the case for D ( 𝝆, L), which we care about. We recall some facts: every Lindbladian satisfies that Spec(L) ⊆ {𝑧 ∈ C : Re(𝑧) ≤ √ 0} [181, Proposition 6.1]. As D ( 𝝆, L) † [ 𝝆] = 0 we also have that 0 is an element of the spectrums Spec(D ( 𝝆, L) † ), Spec(D ( 𝝆, L)). Since D ( 𝝆, L) is defined by a similarity transformation we have that Spec(L) = Spec(D ( 𝝆, L)) and due to Proposition 2.10.1 this implies |𝜆 1 (H )| ≤ ∥A ∥ 2−2 . (2.76) Proposition 2.10.3 (Fixed point accuracy). Suppose a Lindbladian L satisfies the 𝜖-approximate 𝝆-detailed balance condition. If 𝜆 𝑔𝑎 𝑝 (H ) > 2𝜖, then there is a unique state as its fixed point 𝝆 𝑓 𝑖𝑥 (L) ≻ 0 and its deviation from 𝝆 is bounded by 𝝆 𝑓 𝑖𝑥 (L) − 𝝆 1 ≤ 14𝜖 . 𝜆 𝑔𝑎 𝑝 (H ) The RHS indicates that the fixed point accuracy may deteriorate if the map has a large anti-Hermitian component or if the gap closes. Proof. Every CPTP map has at least a stationary state [181, Theorem 6.11], and thus there is a fixed point 𝝆 𝑓 𝑖𝑥 (L) ⪰ 0 of unit trace. The condition 𝜆 𝑔𝑎 𝑝 (H ) > 2∥A ∥ 2−2 translates to 𝜆 2 (H ) < −∥A ∥ 2−2 implying that 0 has algebraic multiplicity 1 in Spec(D ( 𝝆, L)) due to Proposition 2.10.1, which then proves the uniqueness of the fixed point. By our eigenvector perturbation bound (Proposition 2.10.2), we get that there is a matrix 𝑹 of unit Frobenius norm in the kernel of D ( 𝝆, L) such that √ √ 4 2∥A ∥ 2−2 ∥ 𝝆 − 𝑹∥ 2 ≤ , −𝜆 2 (H ) where we used that the Frobenius norm of a matrix is equal to the Euclidean norm of its vectorization ∥ 𝑨∥ 2 = ∥| 𝑨⟩∥. This in turn means that 𝝆 1/4 𝑹𝝆 1/4 is in the kernel 115 of L, thus 𝝆 𝑓 𝑖𝑥 (L) = 𝝆 1/4 𝑹𝝆 1/4 /Tr( 𝝆 1/4 𝑹𝝆 1/4 ), moreover ∥ 𝝆 1/4 𝑹𝝆 1/4 − 𝝆 𝑓 𝑖𝑥 (L)∥ 1 = Tr( 𝝆 1/4 𝑹𝝆 1/4 ) − 1 ∥ 𝝆 𝑓 𝑖𝑥 (L)∥ 1 = Tr( 𝝆 1/4 𝑹𝝆 1/4 − 𝝆) ≤ ∥ 𝝆 1/4 𝑹𝝆 1/4 − 𝝆∥ 1 , where in the last step we used the trace-norm inequality |Tr( 𝑨)| ≤ ∥ 𝑨∥ 1 . We can further bound √ √ √ ∥ 𝝆 1/4 𝑹𝝆 1/4 − 𝝆∥ 1 = ∥ 𝝆 1/4 ( 𝑹 − 𝝆) 𝝆 1/4 ∥ 1 ≤ ∥ 𝝆 1/4 ∥ 24 · 𝑹 − 𝝆 2 = ∥ 𝑹 − 𝝆∥ 2 , where we used Hölder’s inequality ∥ 𝑩 𝑨𝑩∥ 1 ≤ ∥ 𝑩∥ 24 ∥ 𝑨∥ 2 . Combining the above three inequalities, we get ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆∥ 1 ≤ ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆 1/4 𝑹𝝆 1/4 ∥ 1 + ∥ 𝝆 1/4 𝑹𝝆 1/4 − 𝝆∥ 1 √ √ 8 2∥A ∥ 2−2 ≤ 2∥ 𝑹 − 𝝆∥ 2 ≤ . −𝜆 2 (H ) Finally, we convert 𝜆 2 to 𝜆 𝑔𝑎 𝑝 in the above bound. Due to (2.76) we have −𝜆 2 (H ) = 𝜆 𝑔𝑎 𝑝 (H ) − 𝜆 1 (H ) ≥ 𝜆 𝑔𝑎 𝑝 (H ) − ∥A ∥ 2−2 so we can further bound the above by √ 8 2∥A ∥ 2−2 /(𝜆 𝑔𝑎 𝑝 (H ) − ∥A ∥ 2−2 ). But this bound is vacuous at ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆∥ 1 ≤ √ √ 2, i.e., when 𝜆 𝑔𝑎 𝑝 (H ) < √(4 2 + 1)∥A ∥ 2−2 . If 𝜆 𝑔𝑎 𝑝 (H ) ≥ (4 2 + 1)∥A ∥ 2−2 , then 𝜆 𝑔𝑎 𝑝 (H ) − ∥A ∥ 2−2 ≥ √4 2 𝜆 𝑔𝑎 𝑝 (H ), yielding 4 2+1 √ (8 2 + 2)∥A ∥ 2−2 ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆∥ 1 ≤ . 𝜆 𝑔𝑎 𝑝 (H ) ■ Corollary 2.2.1 (Fixed point accuracy). If a Lindbladian L satisfies the 𝜖-approximate 𝝆-detailed balance condition, then its fixed point 𝝆 𝑓 𝑖𝑥 (L) deviates from 𝝆 by at most ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆∥ 1 ≤ 20𝑡 𝑚𝑖𝑥 (L)𝜖 . Proof. In the proof of Proposition 2.10.3 we got the ultimate bound by converting 𝜆 2 to 𝜆 𝑔𝑎 𝑝 by the observation −𝜆 2 (H ) ≥ 𝜆 𝑔𝑎 𝑝 (H ) − ∥A ∥ 2−2 . Due to Proposition 2.10.1 the same bound −𝜆 2 (H ) ≥ 𝜆 Re(𝑔𝑎 𝑝) (L) − ∥A ∥ 2−2 also holds for 𝜆 Re(𝑔𝑎 𝑝) (L) (c.f. Proposition 2.10.5) since 0 is an eigenvalue of L. Hence the conversion combined with Proposition 2.10.5 shows √ √ (8 2 + 2)∥A ∥ 2−2 (8 2 + 2)∥A ∥ 2−2 ∥ 𝝆 𝑓 𝑖𝑥 (L) − 𝝆∥ 1 ≤ ≤ 𝑡 𝑚𝑖𝑥 (L). ■ 𝜆 Re(𝑔𝑎 𝑝) (L) ln(2) 116 Perturbation bounds for Lindbladians regarding gaps and mixing times This section provides proof for scattered statements circling spectral gaps and mixing time. Most results are standard, except maybe the most technical result (Lemma 2.10.2). Lemma 2.2.1 (Fixed point difference). For any two Lindbladians L1 and L2 , the difference of their fixed points (in the Schrödinger picture) is bounded by ∥ 𝝆 𝑓 𝑖𝑥 (L1 ) − 𝝆 𝑓 𝑖𝑥 (L2 )∥ 1 ≤ 4∥L1 − L2 ∥ 1−1 · 𝑡 𝑚𝑖𝑥 (L1 ). Proof. We begin by recalling Duhamel’s identity. We use its integral form derived in, e.g., [81, Eq. (40)]: ∫ 1 𝑨 𝑩 e −e = e𝑠 𝑨 ( 𝑨 − 𝑩)e (1−𝑠) 𝑩 d𝑠. 0 We apply the above identity with 𝑨 ← 𝑡L1 , 𝑩 ← 𝑡L2 and take the 1 − 1 operator norm on both sides ∫ 1 𝑡L1 𝑡L2 ∥e − e ∥ 1−1 ≤ ∥e𝑠𝑡L1 (𝑡L1 − 𝑡L2 )e (1−𝑠)𝑡L2 ∥ 1−1 d𝑠 0 ∫ 1 ≤𝑡 ∥e𝑠𝑡L1 ∥ 1−1 ∥𝑡L1 − 𝑡L2 ∥ 1−1 ∥e (1−𝑠)𝑡L2 ∥ 1−1 d𝑠 = 𝑡 ∥L1 − L2 ∥ 1−1 . 0 (2.77) The last equality uses that e𝑥L is a CPTP map and so ∥e𝑥L ∥ 1−1 = 1 [181, Theorem 8.16].48 Now, let 𝝆1 := 𝝆 𝑓 𝑖𝑥 (L1 ), 𝝆2 := 𝝆 𝑓 𝑖𝑥 (L2 ), and 𝑡 𝑚𝑖𝑥 := 𝑡 𝑚𝑖𝑥 (L1† ). Then ∥ 𝝆1 − 𝝆2 ∥ 1 = eL1 ·𝑡 𝑚𝑖 𝑥 [𝝆1 ] − eL2 𝑡 𝑚𝑖 𝑥 [𝝆2 ] 1 ≤ eL1 ·𝑡 𝑚𝑖 𝑥 [𝝆1 ] − eL1 𝑡 𝑚𝑖 𝑥 [𝝆2 ] 1 + eL1 𝑡 𝑚𝑖 𝑥 [𝝆2 ] − eL2 𝑡 𝑚𝑖 𝑥 [𝝆2 ] 1 1 ≤ · ∥ 𝝆1 − 𝝆2 ∥ 1 + 2𝑡 𝑚𝑖𝑥 ∥L1 − L2 ∥ 1−1 . (2.78) 2 The second inequality follows from (2.77) and that ∥ 𝝆1 − 𝝆2 ∥ 1 ≤ 2. Rearrange (2.78) to conclude the proof. ■ Proposition 2.10.4 (Mixing time difference). The mixing times of two Lindbladians L1 , L2 are related by   ln(1/2) 1 𝑡 𝑚𝑖𝑥 (L2 ) ≤ 𝑡 𝑚𝑖𝑥 (L1 ) if 𝑡 𝑚𝑖𝑥 (L1 )∥L1 − L2 ∥ 1−1 < . ln(1/2 + 𝑡 𝑚𝑖𝑥 (L1 )∥L1 − L2 ∥ 1−1 ) 2 48 Here, by the 1 − 1 norm, we mean the 1 − 1 norm of the operators restricted to the subspace of Hermitian matrices. 117 Proof. Let 𝑡 𝑚𝑖𝑥 := 𝑡 𝑚𝑖𝑥 (L1 ) and 𝑹 be a traceless Hermitian matrix with ∥ 𝑹∥ 1 = 1 maximizing ∥eL2 𝑡 𝑚𝑖 𝑥 [𝑹] ∥ 1 , then ∥eL2 𝑡 𝑚𝑖 𝑥 [𝑹] ∥ 1 ≤ ∥eL1 𝑡 𝑚𝑖 𝑥 [𝑹] ∥ 1 + ∥eL2 𝑡 𝑚𝑖 𝑥 [𝑹] − eL1 𝑡 𝑚𝑖 𝑥 [𝑹] ∥ 1   1 1 ≤ ∥ 𝑹∥ 1 + 𝑡 𝑚𝑖𝑥 ∥L1 − L2 ∥ 1−1 ∥ 𝑹∥ 1 = + 𝑡 𝑚𝑖𝑥 ∥L1 − L2 ∥ 1−1 ∥ 𝑹∥ 1 . 2 2 The second inequality follows from (2.77). Set 𝑹 = 𝝆1 − 𝝆2 , rearrange, and take the logarithm to conclude. ■ Relating the mixing time to the spectral gap using exact detailed balance The mixing time of a general Lindbladian may be difficult to analyze. Fortunately, many handy bounds exist, especially circling the spectral gap when detailed balance holds. Proposition 2.2.2 (Mixing time from spectral gap [97]). If a Lindbladian L satisfies 𝝆-detailed balance, then ln(2∥ 𝝆 −1/2 ∥) , 𝜆 𝑔𝑎 𝑝 (L) 𝑡 𝑚𝑖𝑥 (L) ≤ where 𝜆 𝑔𝑎 𝑝 (L) is the eigenvalue gap of the Lindbladian, and the mixing time 𝑡 𝑚𝑖𝑥 is the smallest time for which eL𝑡 𝑚𝑖 𝑥 [𝝆1 − 𝝆2 ] 1 ≤ 1 ∥ 𝝆1 − 𝝆2 ∥ 1 2 for any states 𝝆1 , 𝝆2 . Proof. Write 𝑹 = 𝝆1 − 𝝆2 , then eL𝑡 [𝑹] 1 = 𝝆 1/4 eD𝑡 [𝝆 −1/4 𝑹𝝆 −1/4 ] 𝝆 1/4 ≤ 𝝆 ≤e ≤e 1/4 · e 4 −𝜆 𝑔𝑎 𝑝 (H )𝑡 −𝜆 𝑔𝑎 𝑝 (H )𝑡 D𝑡 𝝆 [𝝆 −1/4 ∥𝝆 −1/4 𝑹𝝆 𝑹𝝆 −1/4 1 ] 2 · 𝝆 1/4 4 −1/4 2 −1/4 2 ∥ ∥ 𝑹∥ 2 ≤ e−𝜆𝑔𝑎 𝑝 (H )𝑡 ∥ 𝝆 −1/4 ∥ 2 ∥ 𝑹∥ 1 † = e−𝜆𝑔𝑎 𝑝 (L )𝑡 ∥ 𝝆 −1/2 ∥∥ 𝑹∥ 1 . The first inequality uses Hölder’s inequality. The second inequality uses the √ orthogonality to the leading eigenvector such that Tr[ 𝝆 · 𝝆 −1/4 𝑹𝝆 −1/4 ] = Tr[𝑹] = 0. Take the logarithm to conclude the proof. ■ 118 Relating the mixing time to the Hermitian gap (and approximate detailed balance) Proposition 2.10.5 (Spectral gap from mixing time). For any Lindbladian L, let −𝜆 Re(𝑔𝑎 𝑝) (L) be the second largest real part in its spectrum (counted by algebraic multiplicity), then 𝜆 𝑔𝑎 𝑝 (H ) + 2∥A ∥ 2−2 ≥ ∥A ∥ 2−2 − 𝜆 2 (H ) ≥ 𝜆 Re(𝑔𝑎 𝑝) (L) ≥ ln(2) . 𝑡 𝑚𝑖𝑥 (L) Moreover, if 𝜆 Re(𝑔𝑎 𝑝) (L) ≥ 2∥A ∥ 2−2 , then there is unique eigenvalue 𝜆 1 (H ) ≥ −∥A ∥ 2−2 and 𝜆 𝑔𝑎 𝑝 (H ) + 2∥A ∥ 2−2 ≥ ∥A ∥ 2−2 − 𝜆 2 (H ) ≥ 𝜆 Re(𝑔𝑎 𝑝) (L). (2.79) Proof. We know any Lindbladian has at least a stationary state of eigenvalue 0, and each eigenvalue which has no real part has a trivial Jordan block [181, Theorem 6.11 & Proposition 6.2]. Therefore, if 𝜆 Re(𝑔𝑎 𝑝) (L) = 0, then 𝑡 𝑚𝑖𝑥 (L) = ∞. If 𝜆Re(𝑔𝑎 𝑝) (L) > 0, take any eigenvalue 𝜆 such that −𝜆 Re(𝑔𝑎 𝑝) (L) = Re(𝜆), and let 𝑹 be a corresponding right eigenvector of L, which is then necessarily traceless. We can assume without loss of generality that the Hermitian part 𝑹𝐻 := ( 𝑹 + 𝑹 † )/2 is nonzero (otherwise, we can just take 𝑹 ← i𝑹). Since L is Hermiticity preserving, ln(2) we get that 𝑹𝐻 is also a right eigenvector with eigenvalue 𝜆. For 𝑡 < 𝜆Re(𝑔𝑎 we 𝑝) (L) have 1 ∥eL𝑡 [𝑹𝐻 ] ∥ 1 = e𝜆𝑡 · ∥ 𝑹𝐻 ∥ 1 > ∥ 𝑹𝐻 ∥ 1 2 implying that 𝑡 𝑚𝑖𝑥 (L) ≥ ln(2) . 𝜆 Re(𝑔𝑎 𝑝) (L) We conclude using Bauer-Fike Proposition 2.10.1 to show 𝜆 Re(𝑔𝑎 𝑝) (L) ≤ −𝜆 2 (H ) + ∥A ∥ 2−2 , combined with (2.76). ■ What if the detailed balance condition is violated? In the worst case, the conversion from spectral gap to mixing time can be poor. However, the Lindbladians we consider are “sufficiently” detailed balanced so that essentially the same consequences hold. The following two scenarios cover our main use cases D ( 𝝆, L) = H + A where ∥A ∥ 2−2 ≪ 𝜆 𝑔𝑎 𝑝 (H ), (approximate detailed balance) or 𝜆 1 (H ) ≪ 𝜆 𝑔𝑎 𝑝 (H ). (nonperturbative A) 119 The first case should be understood as nonHermitian eigenvalue perturbation, consistent with the framework of approximate detailed balance. The second case is less intuitive, as the anti-Hermitian part can be arbitrarily large. Intriguingly, the spectral properties of the Hermitian part H suffice to control convergence even in the presence of a large perturbation H + A. In fact, the second case is strictly more general since 𝜆 1 (H ) ≤ ∥A ∥ 2−2 (2.76), therefore we will only analyze the second scenario. Intuitively, when the Lindbladian is exactly detailed balanced A = 0, we have that 𝜆 1 (H ) = 0; the 𝜆 1 (H ) ≪ 𝜆 𝑔𝑎 𝑝 (H ) condition is essentially the requirement that 𝜆 1 (H ) ≈ 0 in spite of a large anti-hermitian component ∥A ∥ 2−2 ≫ 0. Before proving our result, we need a few lemmas. Lemma 2.10.1 (Norm of matrix exponential). For a Hermitian matrix 𝑯 and an anti-Hermitian matrix 𝑩, we have ∥e (𝑯+𝑩)𝑡 ∥ ≤ e𝜆1 (𝑯)𝑡 for each 𝑡 ≥ 0. If 𝑩 is an arbitrary matrix, we alternatively get ∥e (𝑯+𝑩)𝑡 ∥ ≤ e (𝜆1 (𝑯)+∥ 𝑩∥)𝑡 . Proof. The claim follows from the Trotter representation of exponential and triangle inequality  𝑟𝑡  𝑟𝑡  𝑟𝑡 (𝑯+𝑩)𝑡 𝑯/𝑟 𝑩/𝑟 𝑯/𝑟 𝑩/𝑟 𝑯/𝑟 𝑩/𝑟 ∥e ∥ = ∥ lim e e ∥ = lim ∥ e e ∥ ≤ lim inf ∥e ∥∥e ∥ 𝑟→∞ 𝑟→∞ 𝑟→∞  𝑟𝑡 = lim inf ∥e 𝑯/𝑟 ∥ = e𝜆1 (𝑯)𝑡 , 𝑟→∞ where we used that e 𝑩/𝑟 is unitary so that ∥e 𝑩/𝑟 ∥ = 1. The second claim follows from isolating the Hermitian part 21 (𝑩 + 𝑩† ), whose norm is bounded by ∥ 𝑩∥. ■ Lemma 2.10.2 (Hermitian gap controls decay). Consider a Hermitian matrix 𝑯 and an anti-Hermitian 𝑨. If 𝑟 := 𝜆 1 (𝑯) 1 ≤ , 𝜆 𝑔𝑎 𝑝 (𝑯) 100 and 𝑯 + 𝑨 has an eigenvalue 0 with left and right eigenvectors ⟨𝐿|(𝑯 + 𝑨) = 0 and (𝑯 + 𝑨)|𝑅⟩ = 0, then e (𝑯+𝑨)𝑡   𝜆 2 (𝑯)𝑡 3 ( 𝑰 − 𝑷0 ) ≤ exp 2 2 where 𝑷0 := 1 |𝑅⟩⟨𝐿|. ⟨𝐿|𝑅⟩ 120 Intuitively speaking, the conditions above ensure that even if the anti-Hermitian 𝑨 is large, it mainly introduces “rotations” and the spectral properties of 𝑯 still guarantee fast convergence to 𝑷0 . Proof. Let |𝜓1 ⟩ be the eigenvector of 𝑯 corresponding to its top eigenvalue 𝜆 1 (𝑯). We can assume without loss of generality that ⟨𝐿|, |𝑅⟩, and |𝜓1 ⟩ have unit norm ∥|𝜓1 ⟩∥ = ∥⟨𝐿|∥ = ∥|𝑅⟩∥ = 1, and we are free to choose the phase of these vectors as the projector 𝑷0 is invariant under changing the phase of ⟨𝐿|, |𝑅⟩. The guiding intuition behind the proof is that if 𝑟 is small, then (up to a phase) |𝐿⟩ ≈ |𝜓1 ⟩ ≈ |𝑅⟩. To show this, take the real part of ⟨𝑅|(𝑯 + 𝑨)|𝑅⟩ = 0 to obtain49 ∑︁ 2 ⟨𝑅|𝑯|𝑅⟩ = 0 = 𝜆 1 (𝑯)|⟨𝑅|𝜓1 ⟩| + 𝜆𝑖 (𝑯)|⟨𝑅|𝜓𝑖 ⟩| 2 𝑖=2 2 ≤ 𝜆 1 |⟨𝑅|𝜓1 ⟩| + 𝜆 2 (1 − |⟨𝑅|𝜓1 ⟩| 2 ), 𝜆 1 (𝑯) = 𝑟, implying 1 − |⟨𝑅|𝜓1 ⟩| 2 ≤ 𝜆 𝑔𝑎 𝑝 (𝑯) √ and thus ∥|𝑅⟩ − |𝜓1 ⟩∥ ≤ 1.01 𝑟, where in the last step, we assumed without loss of generality that the phase of |𝑅⟩ is such that ⟨𝑅|𝜓1 ⟩ is nonnegative real, so we get ⟨𝑅|𝜓1 ⟩ = cos(𝜃) for some √ 𝜃 ∈ [0, 𝜋/2]. Then sin(𝜃) ≤ 𝑟, and since for every 𝜃 ∈ [0, arcsin(1/10)] we have 101 √ 1 − cos(𝜃) ≤ sin(𝜃)/10, we get the norm bound 101 100 sin(𝜃) ≤ 100 𝑟. Similarly, ⟨𝐿|(𝑯 + 𝑨)|𝐿⟩ = 0 implies √ ∥|𝐿⟩ − |𝜓1 ⟩∥ ≤ 1.01 𝑟. √ As a direct consequence, using that 𝑟 ≤ 1/10, we get |⟨𝐿|𝑅⟩ − 1| = |⟨𝐿|𝑅⟩ − ⟨𝜓1 |𝜓1 ⟩| ≤ |⟨𝐿|𝑅⟩ − ⟨𝐿|𝜓1 ⟩| + |⟨𝐿|𝜓1 ⟩ − ⟨𝜓1 |𝜓1 ⟩| √ 1 ≤ ∥|𝑅⟩ − |𝜓1 ⟩∥ + ∥|𝐿⟩ − |𝜓1 ⟩∥ ≤ 2.02 𝑟 ≤ , 4 and similarly that √ 1 1 − ⟨𝐿|𝑅⟩ 4 −1 = ≤ |1 − ⟨𝐿|𝑅⟩| ≤ 2.73 𝑟. ⟨𝐿|𝑅⟩ ⟨𝐿|𝑅⟩ 3 49 Note that ℜ(⟨𝑅|(𝑯 + 𝑨)|𝑅⟩) = 0 implies 𝜆 (𝑯) ≥ 0, and therefore 𝑟 ≥ 0. This then further 1 implies 𝜆2 (𝑯) ≤ 0 due to 𝑟 ≤ 0.01. 121 Therefore, ∥𝑷0 − |𝜓1 ⟩⟨𝜓1 |∥ ≤ ∥ = |𝑅⟩⟨𝐿| − |𝑅⟩⟨𝐿|∥ + ∥|𝑅⟩⟨𝐿| − |𝑅⟩⟨𝜓1 |∥ + ∥|𝑅⟩⟨𝜓1 | − |𝜓1 ⟩⟨𝜓1 |∥ ⟨𝐿|𝑅⟩ √ 1 − 1 + ∥|𝐿⟩ − |𝜓1 ⟩∥ + ∥|𝑅⟩ − |𝜓1 ⟩∥ ≤ 4.75 𝑟. ⟨𝐿|𝑅⟩ (2.80) We will use the following properties of the projector 𝑷0 ; one can easily see that it commutes with (𝑯 + 𝑨): 1 |𝑅⟩⟨𝐿|(𝑯 + 𝑨) = 0 = (𝑯 + 𝑨)𝑷0 , ⟨𝐿|𝑅⟩ 𝑷0 (𝑯 + 𝑨) = (2.81) and since it is a projector, it satisfies the algebraic identity 𝑷0 ( 𝑰 − 𝑷0 ) = 0 so that e𝑥𝑷0 ( 𝑰 − 𝑷0 ) = 𝑰 − 𝑷0 for any 𝑥 ∈ C. (2.82) The above properties streamline the rest of the proof of our bound ∥e (𝑯+𝑨)𝑡 ( 𝑰 − 𝑷0 )∥ = ∥e (𝑯+𝑨)𝑡 · e−𝜆𝑔𝑎 𝑝 (𝑯)𝑷0 𝑡 ( 𝑰 − 𝑷0 )∥ (by (2.82)) = ∥e (𝑯+𝑨−𝜆𝑔𝑎 𝑝 (𝑯)𝑷0 )𝑡 ( 𝑰 − 𝑷0 )∥ (by (2.81)) ≤ ∥e (𝑯+𝑨−𝜆𝑔𝑎 𝑝 (𝑯)𝑷0 )𝑡 ∥ · ∥ 𝑰 − 𝑷0 ∥ 𝑯 ′ := 𝑩:= }| { z }| {ª ©z ­ ® = exp­ (𝑯 − 𝜆 𝑔𝑎 𝑝 (𝑯)|𝜓1 ⟩⟨𝜓1 |)𝑡 + 𝑨𝑡 + 𝜆 𝑔𝑎 𝑝 (𝑯)(|𝜓1 ⟩⟨𝜓1 | − 𝑷0 )𝑡 ® · ∥ 𝑰 − 𝑷0 ∥ ® ­ ¬ « ′ ≤ e (𝜆1 (𝑯 )+∥ 𝑩∥)𝑡 (∥ 𝑰 − |𝜓1 ⟩⟨𝜓1 |∥ + ∥|𝜓1 ⟩⟨𝜓1 | − 𝑷0 ∥) (by Lemma 2.10.1)  √  √ 4.75 𝑟 ≤ exp 𝜆 2 (𝑯)(1 − )𝑡 (1 + 4.75 𝑟). (by Eq. (2.80)) 1−𝑟   3 1 ≤ · exp 𝜆 2 (𝑯)𝑡 . (by 𝑟 ≤ 1/100) ■ 2 2 Now, we can specialize the above to the case of Lindbladians to obtain mixing times. Proposition 2.2.3 (Mixing time from Hermitian gap). For any Lindbladian L and a full-rank state 𝝆, suppose the self-adjoint component H = H ( 𝝆, L) satisfies 𝜆 1 (H ) 1 ≤ , 𝜆 𝑔𝑎 𝑝 (H ) 100 then 𝑡 𝑚𝑖𝑥 (L) ≤ 3 ln(3∥ 𝝆 −1/2 ∥) . 𝜆 𝑔𝑎 𝑝 (H ) 122 Proof. We proceed as in the proof of Proposition 2.2.2. For any traceless Hermitian 𝑹, we have eL𝑡 [𝑹] 1 = 𝝆 1/4 e𝑡 (H +A) [𝝆 −1/4 ( 𝑹) 𝝆 −1/4 ] 𝝆 1/4 ≤ 𝝆 1/4 4 · e𝑡 (H +A) [𝝆 −1/4 ( 𝑹) 𝝆 −1/4 ] (by definition) 1 2 · 𝝆 1/4 4 (by Hölder’s inequality) 3 𝜆2 (𝑯)𝑡/2 −1/4 −1/4 e 𝝆 𝑹𝝆 2 2 3 𝜆2 (𝑯)𝑡/2 −1/4 2 ≤ e ∥𝝆 ∥ ∥ 𝑹∥ 2 2 3 ≤ e𝜆2 (𝑯)𝑡/2 ∥ 𝝆 −1/2 ∥∥ 𝑹∥ 1 . 2 ≤ (by Lemma 2.10.2) (by Hölder’s inequality) (since 𝝆 ≻ 0 and ∥·∥ 2 ≤ ∥·∥ 1 ) The second inequality uses Lemma 2.10.2 since lim𝑡→∞ eL𝑡 [𝑹] = 0 (or in the notation of Lemma 2.10.2, 𝑷0 | 𝑹⟩ = 0). By assumption, 𝜆 2 (H ) = 𝜆 1 (H ) − 𝜆 𝑔𝑎 𝑝 (H ) ≤ −0.99𝜆 𝑔𝑎 𝑝 (H ); take the logarithm to conclude the proof. ■ 2.11 Appendix:Improved incoherent Lindbladian simulation The improved algorithm builds on the circuit in Figure 2.10, which is similar to that of our weak measurement scheme in Figure 2.3, with a key technical difference: here, √ √ a specific ancilla state 𝒀𝛿 |0⟩ = 1 − 𝛿|0⟩ + 𝛿|1⟩ “triggers” the appropriate weak measurement. Thus, the circuit is “idle” if we remove the 𝒀𝛿 gate, which is essential for our “compression” argument. However, this also makes the circuit post-selective, which could exponentially decrease the probability of success if we were to run for longer times naively. Thus, for 𝑡 > 2 we decompose the simulation to ⌈𝑡/2⌉ equal segments, each of which can be amplified with constant (coherent) repetitions by oblivious amplitude amplification. The main conceptual ingredient leading to the substantial improvement is compression [49]: instead of naively running 𝑟-repetitions of the circuit from Figure 2.10 as outlined in Figure 2.11, we compress the circuit such that it uses only ℎ ∼ 𝑡 log(𝑡/𝜖) ≪ 𝑟 = Θ(𝑡 2 /𝜖) repetitions. The compression technique relies on
⊗𝑟 understanding the joint initial state 0𝑐+1 𝒀𝛿 |0⟩ of the circuit in Figure 2.11 after ⊗𝑟 𝑟 the single-qubit gates. The upshot is that 𝒀𝛿 |0 ⟩ is concentrated on strings with Hamming weight ≤ ℎ, thus ℎ repetitions will suffice for the mass of the amplitudes. Before diving into the proof, we explain the intuitive compression strategy in more detail; the proof closely follows this, but with technical changes. Let 𝑡 ≤ 2 and let † 𝑪 ′ be the circuit that we get from 𝑪 by removing the two single qubit gates 𝒀𝛿 , 𝒀𝛿/4 123 𝒀 †𝛿 𝒀𝛿 |0⟩ 4 |0⟩ accept the all-zero outcome and when the second qubit is 1 0𝑏 0𝑎−𝑏 𝑼† 𝑼 = (1 − 4𝛿 )e𝛿L[𝝆] + O 𝛿2 𝝆
Figure 2.10: Alternative quantum circuit implementation of an approximate 𝛿time step via a postselective weak measurement scheme. Let 𝑪 ′ be the circuit † that we get by removing the two single qubit rotation gates 𝒀𝛿 , 𝒀𝛿/4 from the first qubit. For our compression argument it is of paramount importance that 𝑪 ′ · |0𝑎+2 ⟩⟨0𝑎+2 | ⊗ 𝑰 = |0𝑎+2 ⟩⟨0𝑎+2 | ⊗ 𝑰. |0⟩ 𝒀𝛿 𝑪′ 0𝑐+1 .. . .. |0⟩ 𝒀𝛿 † 𝒀𝛿/4 † 𝒀𝛿/4 |0⟩ 𝒀𝛿 0𝑐+1 𝝆 𝑪′ 𝑪′ accept the all-zero outcome and when the second qubit is 1 .. . . 𝑪′ 0𝑐+1 † 𝒀𝛿/4 𝑪′ accept the all-zero outcome and when the second qubit is 1 accept the all-zero outcome and when the second qubit is 1
= (1 − 4𝛿 ) 𝑟 e𝑟𝛿L[𝝆] + O 𝑟𝛿2 Figure 2.11: 𝑟 subsequent repetitions of the circuit 𝑪 ′ from Figure 2.10. The circuits 𝑪 ′ act on potentially nonadjacent qubits, which is indicated by the vertical curly connection between the visually split “halves” of the affected 𝑪 ′ circuits. from the first qubit in Figure 2.10. Let 𝑋𝑖 be the random variable representing the measurement outcome of a computational basis measurement of the 𝑖-th qubit in 𝑡
Í𝑟 e𝑦 𝒀𝛿⊗𝑟 |0𝑟 ⟩. Then, the Chernoff bound tells us that Pr 𝑖=1 𝑋𝑖 > (1 + 𝑦)𝑡 ≤ (1+𝑦) 1+𝑦 √  ⊗𝑟 √ so that the probability that the Hamming weight of 1 − 𝛿|0⟩ + 𝛿|1⟩ is greater 124 ℎ−𝑡 ℎ ℎ than ℎ := (1 + 𝑦)𝑡 is at most e ℎ ℎ𝑡 ≤ ( 𝑒𝑡ℎ ) ℎ ≤ ( 2𝑒 ℎ ) . In particular, choosing   log(1/𝜖) ℎ=Θ log log(1/𝜖)
ensures that this probability is at most O 𝜖 2 . Therefore, the initial state can be replaced by its (normalized) projection |𝜙0 ⟩ to the subspace of Hamming-weight ≤ ℎ states while inflicting an error that is bounded by ∥𝒀𝛿⊗𝑟 |0𝑟 ⟩ − |𝜙0 ⟩∥ ≤ 𝜖 . This bound on Hamming-weights translates into a reduction of applications of the circuit 𝑪 ′. Since |𝜙0 ⟩ is a superposition of bitstrings of Hamming-weights at most ℎ, in all branches of the superposition all but ℎ applications of 𝑪 ′ can be neglected, crucially because it acts trivially when the ancilla register is in state 0𝑎+2 : 𝑪 ′ · |0𝑎+2 ⟩⟨0𝑎+2 | ⊗ 𝑰 = |0𝑎+2 ⟩⟨0𝑎+2 | ⊗ 𝑰. (2.83) Now we define a compression scheme for the 𝑟 ancilla registers, each containing (𝑎+2) qubits50 in Figure 2.11, inspired by [18, 49]. The compression scheme can represent the ≤ ℎ Hamming-weight states of the ancilla registers on just ℎ · (log(𝑟 + 1) + 𝑎 + 2) qubits (concerning the 𝑟 registers to be compressed, by Hamming weight, we mean the number of registers that do not contain the state 0𝑎+2 ). Marking the register state 0𝑎+2 by 0· and the content of the 𝑖-th nonzero register by 𝑑𝑖 the encoding works as follows:  𝑟  ℎ {0, 1} 𝑎+2 ∋ 0· 𝑠1 𝑑1 0· 𝑠2 𝑑2 . . . → (𝑠1 , 𝑠2 , . . .) × (𝑑1 , 𝑑2 , . . .) ∈ {0, 1, 2, . . . , 𝑟 } ℎ × {0, 1} 𝑎+2 . (2.84) The compressed representation’s first “compression” register contains ℎ blocks of log(𝑟 + 1) qubits, designated to store a sequence 𝑠 ∈ {0, 1, 2, . . . , 𝑟 } ℎ , where 𝑠𝑖 is the number of consecutive ancilla registers containing 0𝑎+2 before the 𝑖-th ancilla register that is not in state 0· ; if 𝑖 exceeds the Hamming weight, then we set 𝑠𝑖 = 𝑟. The second “data” register consists of ℎ blocks of (𝑎 + 2) qubits, where the 𝑖-th block represents the qubits of the 𝑖-th nonzero register of the uncompressed state; if the Hamming weight is less than 𝑖, then the block is set to 0· . The property (2.83) means that the 𝑪 ′ gates can be applied “transversally” on the second “data” register of the encoded scheme because they do not change the location of the nonzero registers of the uncompressed state. 50 Note that the first qubit is redundant in this encoding, but we add it here for clarity of the presentation. 125 Theorem 2.3.2 (Compressed incoherent Lindbladian simulation algorithm). Suppose 𝑼 is a block-encoding of the Lindblad operators of a purely irreversible Lindbladian L as in Definition 2.1.2. Let 𝜖 ≤ 1/2, then we can simulate the action of the superoperator e𝑡L to precision 𝜖 in diamond norm using O ((𝑐 + log((𝑡 + 1)/𝜖)) log((𝑡 + 1)/𝜖))   log((𝑡 + 1)/𝜖) O (𝑡 + 1) log log((𝑡 + 1)/𝜖) and O ((𝑡 + 1)(𝑐 + 1)polylog((𝑡 + 1)/𝜖)) (resettable) ancilla qubits, (controlled) uses of 𝑼 and 𝑼 † , other two-qubit gates. If the Lindbladian has a coherent part −i[𝑯, 𝝆], and we have access  to a block log((𝑡+1)/𝜖) 𝑐 𝑐 𝑡L encoding of 𝑯 = (⟨0 |⊗𝑰)𝑽 (|0 ⟩⊗𝑰), then we can simulate e with O (𝑡 + 1) log log((𝑡+1)/𝜖) additional (controlled) uses of 𝑽 and 𝑽 † . Proof of Theorem 2.3.2. We begin with analyzing the (modified) weak-measurement scheme using similar calculations to Theorem 2.3.1, and then compress it. We focus on the purely irreversible scenario and, at the end, handle the general case. (Postselected weak-measurement.) The circuit 𝑪 from Figure 2.10 on a pure input state |𝜓⟩ acts as √ (1) √ 0𝑎+2 |𝜓⟩ → ( 1 − 𝛿|0⟩ + 𝛿|1⟩)|0⟩𝑼|0𝑐 ⟩|𝜓⟩   √ (2) √ → 1 − 𝛿|00⟩𝑼|0𝑐 ⟩|𝜓⟩ + 𝛿|11⟩ |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ √ + 𝛿|10⟩( 𝑰 − |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰)𝑼|0𝑐 ⟩|𝜓⟩ √ √ √ = ( 1 − 𝛿|00⟩ + 𝛿|10⟩)𝑼|0𝑐 ⟩|𝜓⟩ + 𝛿|11⟩ 0𝑏 ( 0𝑏 ⊗ 𝑰)𝑼|0𝑐 ⟩|𝜓⟩ | {z } ′ | 𝜓0 ⟩ :=   √ − 𝛿|10⟩ |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩   √ √ √ (3) † → 𝒀 𝛿 ⊗ 𝑰 ( 1 − 𝛿|00⟩ + 𝛿|10⟩)|0𝑐 ⟩|𝜓⟩ + 𝛿|11⟩ 0𝑏 𝜓0′ 4    √ − 𝛿|10⟩𝑼 † |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ (2.85) 126 Now, let us compute the part of 𝑪 0𝑎+2 |𝜓⟩ starting with 0𝑎+2 ( 0𝑎+2 ⊗ 𝑰)𝑪 0𝑎+2 |𝜓⟩ √︂    𝛿 𝛿 𝛿 = 1 − (1 − 𝛿) + |𝜓⟩ − (⟨0𝑐 | ⊗ 𝑰)𝑼 † |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ 4 2 2 | {z } | {z } Í √ † 𝑗 ∈ 𝐽 𝑳 𝑗 𝑳 𝑗 |𝜓⟩ =1− 8𝛿 +O ( 𝛿2 ) = 1− 4𝛿 +O ( 𝛿2 ) ! ! √︂   𝛿 𝛿 ∑︁ † = 1− 𝑰− 𝑳 𝑗 𝑳 𝑗 |𝜓⟩ + O 𝛿2 , 4 2 𝑗 ∈𝐽 and the part where second qubit is |1⟩ ∑︁ √ √ | 𝑗⟩ 𝑳 𝑗 |𝜓⟩. ( 𝑰 ⊗ ⟨1| ⊗ 𝑰)𝑪 0𝑎+2 |𝜓⟩ = 𝛿𝒀 †𝛿 |1⟩ 0𝑏 𝜓0′ = 𝛿𝒀 †𝛿 |1⟩ 0𝑏 4 4 𝑗 ∈𝐽
e := √ 1 |0𝑎+2 ⟩⟨0𝑎+2 | ⊗ 𝑰 + 𝑰 ⊗ |1⟩⟨1| ⊗ 𝑰 , the above implies that Let 𝚷 𝛿 1− 4     e |0𝑎+2 ⟩⟨0𝑎+2 | ⊗ |𝜓⟩⟨𝜓| 𝐶 † 𝚷 e Tr𝑎+2 𝚷𝐶 ! !   ∑︁ ∑︁ ∑︁ 𝛿 𝛿 † † † = 𝑰− 𝑳 𝑗 𝑳 𝑗 |𝜓⟩⟨𝜓| 𝑰 − 𝑳𝑗𝑳𝑗 + 𝛿 𝑳 𝑗 |𝜓⟩⟨𝜓| 𝑳 𝑗 + O 𝛿2 2 𝑗 ∈𝐽 2 𝑗 ∈𝐽 𝑗 ∈𝐽   = e𝛿L[|𝜓⟩⟨𝜓|] + O 𝛿2 . Similarly to the proof of Theorem 2.3.1, it is easy to see that this implies       𝑎+2 𝑎+2 †e 𝛿L e ∥Tr𝑎+2 𝚷𝐶 |0 ⟩⟨0 | ⊗ [·] 𝐶 𝚷 − e [·] ∥ ^ = O 𝛿2 . Like in Theorem 2.3.1, choosing 𝛿 = Θ(𝜖/𝑡) and repeating the process 𝑟 := 𝑡/𝛿 times (every time using 𝑎 + 2 fresh ancillas) yields an 𝜖-accurate simulation (Figure 2.11). This gives similar circuit complexity as Theorem 2.3.1, except that the resulting 𝑡2 postselective protocol has success probability about (1 − 𝜖𝑡 ) 𝜖 = exp(−Θ(𝑡)). More 𝑡 precisely, the square of the subnormalization factor is (1 − 4𝛿 ) 𝛿 , which is at least 12 𝑡 2 for 𝑡 ≤ 2 and 𝛿 ∈ (0, 1] since (1 − 4𝛿 ) 𝛿 ≥ (1 − 4𝛿 ) 𝛿 ≥ ( 43 ) 2 > 12 . (Compression.) While the above discussion is largely similar to Theorem 2.3.1, we now further compress to obtain the desired complexity. We analyze the case when 𝑡 ≤ 2 and later show how to bootstrap the results for arbitrary large 𝑡. Instead of running the verbose circuit of Figure 2.11, we directly prepare the compressed version of the state |𝜙0 ⟩ using the techniques of [18]. Using the algorithm of [18], we can directly prepare the “compression” register corresponding 127 to |𝜙0 ⟩, then we can initialize the “data” register by looping through all blocks and applying an 𝑿 gate on the first qubit conditioned on the corresponding “compression” block containing a number less than 𝑟. Then we apply the 𝑪 ′ gates “transversally”, looping through each of the ℎ blocks of the “data” register of the compressed state in increasing order starting from the first block. The last and technically most challenging difficulty that we face is to evaluate the post-selection criterion in Figure 2.11. We could, of course, uncompress the † ⊗𝑟 state, apply the single qubit gates (𝒀𝛿/4 ) and perform the measurement literally as depicted on Figure 2.11, however it is possible to evaluate this criterion while keeping the compressed representation.51 We use a slightly modified variant of the compressed measurement scheme outlined in [18, Section 5] that enables us to perform a measurement of the form
⊗𝑟 † 𝑷0 ⊗ |0𝑐+1 ⟩⟨0𝑐+1 |, 𝑰 − 𝑷0 ⊗ |0𝑐+1 ⟩⟨0𝑐+1 | , where 𝑷0 = 𝒀𝛿/4 |0⟩⟨0|𝒀𝛿/4 . This compressed measurement procedure reports the result also in a compressed form by listing the (uncompressed) indices (𝑖 1 , 𝑖2 , . . . , 𝑖ℓ ) where the measurement outcome is 𝑰 − 𝑷0 ⊗ |0𝑐+1 ⟩⟨0𝑐+1 |. 
⊗𝑟
⊗𝑟  𝑐+1 𝑐+1 𝑐+1 𝑐+1 The initial observation of [18] is that the measurement 𝑷0 ⊗ |0 ⟩⟨0 | , 𝑰 − 𝑷0 ⊗ |0 ⟩⟨0 | can be approximately performed by using the compressed state preparation circuit
⊗𝑟 approximately preparing the compressed version of the state 0𝑐+1 𝒀𝛿/4 |0⟩ . Indeed, we can just run this compressed state preparation in reverse and verify that we get the all-zero state. A similar procedure can be devised for performing the compressed measurement   ⊗ℓ   ⊗ℓ  𝑐+1 𝑐+1 𝑐+1 𝑐+1 𝑷0 ⊗ |0 ⟩⟨0 | , 𝑰 − 𝑷0 ⊗ |0 ⟩⟨0 | (2.86) 51The compression and compact verification procedure for checking whether all segments were applied successfully in the encoded scheme (and the resulting reflection operator) does not appear to work as described in [49]. The issue is manifest in their description of the reflection operator about accepted outcomes: “Therefore, the corresponding operation in the encoded representation is first applying e† , then applying the reflection about the encoded state |0𝑐 ⟩|0𝑏 ⟩ on the first two registers, and last applying 𝐸.” However, this approach does not seem to work, as noted in [18] (here 𝑈𝑚 is the analog of 𝐸 in the above quote from [49]): “At first glance, one might imagine that applying 𝑈𝑚 in place of 𝑅 ⊗𝑚 would yield a succinct representation of the final outcome state, so measuring in the computational basis would provide the correct result. Unfortunately, this does not accurately simulate the final measurement except in the case where the all-zero string is obtained.” For completeness, we include and analyze our modified compression / verification scheme. There may be a simpler fix for the above issue [176], or alternatively, the techniques of the recent work [112] — circumventing compression — could also be compatible with our improvements [176], which would in turn probably also simplify our circuits. 128 for any given consecutive ℓ (uncompressed) registers 𝑑𝑖 , 𝑑𝑖+1 , . . . , 𝑑𝑖+ℓ−1 . Towards this, observe that we can efficiently convert an encoded string 𝑒 1;𝑟 of the form (2.84) to a new form (𝑒 1;𝑖−1 , 𝑒𝑖;𝑖+ℓ−1 , 𝑒𝑖+ℓ;𝑟 ) where 𝑒 𝑗;𝑘 is an encoding of the uncompressed block-string 𝑢 𝑗 , 𝑢 𝑗+1 , . . . , 𝑢 𝑘 . The measurement is then performed by applying this conversion 𝑒 1;𝑟 → (𝑒 1;𝑖−1 , 𝑒𝑖;𝑖+ℓ−1 , 𝑒𝑖+ℓ;𝑟 ) in superposition, then applying the “full” measurement (2.86) on 𝑒𝑖;𝑖+ℓ−1 , and finally reversing the conversion (𝑒 1;𝑖−1 , 𝑒𝑖;𝑖+ℓ−1 , 𝑒𝑖+ℓ;𝑟 ) → 𝑒 1;𝑟 . This way, using binary search, we can, e.g., locate the first (uncompressed) index where the measurement result is 𝑰 − 𝑷0 ⊗ |0𝑐+1 ⟩⟨0𝑐+1 | using log(𝑟) such compressed measurements. With very high probability at most O (ℎ) indices will result in outcome 𝑰 − 𝑷0 ⊗ |0𝑐+1 ⟩⟨0𝑐+1 |, so this binary-search-based compressed measurement scheme will terminate after O (ℎ log(𝑟)) steps with very high probability. We refer the reader to [18, Section 5] for further details about the precise error and run-time bounds. Once we obtained the list of indices (𝑖1 , 𝑖2 , . . . , 𝑖ℓ ) where the uncompressed measurement would have resulted in 𝑰 − 𝑷0 ⊗ |0𝑐+1 ⟩⟨0𝑐+1 |, we need to check the alternative acceptance condition, i.e., whether the second qubit is in state 1 or not, completing the verification whether all circuit segments were applied successfully. Given such an index 𝑖 we first use the first “compression” register of the compressed encoding to identify how many nonzero (uncompressed) registers are before 𝑖, and then look up the corresponding block in the “data” register to check whether the second qubit is in state 1; if the 𝑖-th uncompressed register contains 0𝑎+2 according to the “compression” register then we conclude that the second qubit is in state 0 without looking at the “data” register. All of these operations can be performed in time that is polynomial in the size of the “compression” register, which is O (polylog(1/𝜖)). This completes the description of how to simulate e𝑡L[·] to precision O (1/𝜖) with success probability ≥ 1/2 when 𝑡 ≤ 2 using O (ℎ(𝑎 + log((𝑟 + 1)))) = O (log(1/𝜖)(𝑎 + log(1/𝜖))) ancilla qubits,   log(1/𝜖) ℎ=O (controlled) uses of 𝑼 and 𝑼 † , log log(1/𝜖) O ((𝑎 + 1)polylog(1/𝜖)) other two-qubit gates. The success probability can be improved to 1 while keeping the precision O (1/𝜖), using 3-steps of oblivious amplitude amplification, cf. [49, 73]. For 𝑡 > 2, we divide up the evolution to ⌈𝑡/2⌉ equal segments and repeat the process ⌈𝑡/2⌉ times, setting the precision to 𝜖/𝑡 in each segment. This gives the stated final complexity. 129 With a slight modification, we can make the above algorithm work for general Lidbladians as well, assuming that we have an (at most) 𝑎-qubit block-encoding 𝑽 of the driving Hamiltonian term 𝑯. One just needs to modify the circuit of Figure 2.10, sketched in Figure 2.12. |0⟩ |0⟩ 𝒀 5† 𝒀 2𝛿 4𝛿 𝒀1 −i arcsin( e 2 √︃ 4 5)𝑿 accept the all-zero outcome and when the third qubit is 1 |0⟩ 0𝑏 0𝑎−𝑏 𝑽 𝑼 𝑼† = (1 − 54 𝛿)e𝛿L[𝝆] + O 𝛿2 𝝆
Figure 2.12: Alternative quantum circuit implementation of an approximate 𝛿-time step via a postselective weak measurement scheme including the coherence term −𝑖[𝑯, 𝝆] for the block-encoded Hamiltonian 𝑯 = (⟨0𝑐 | ⊗ 𝑰)𝑽 (|0𝑐 ⟩ ⊗ 𝑰). For completeness, we include the analysis of the circuit 𝑪 𝑯 from Figure 2.12. Let us define   √︃ −i arcsin( 45 ) 𝑿 † 𝑹 := (𝒀 5 ⊗ 𝑰) · |0⟩⟨0| ⊗ 𝑰 + |1⟩⟨1| ⊗ e . 4𝛿 130 Similarly to (2.85) we analyze the action of 𝑪 𝑯 on a pure state |𝜓⟩: 0𝑎+3 |𝜓⟩ √ (1) √ → ( 1 − 2𝛿|0⟩ + 2𝛿|1⟩)|00⟩|0𝑐 ⟩|𝜓⟩ √ √ (2) √ → ( 1 − 2𝛿|00⟩ + 𝛿|10⟩ + 𝛿|11⟩)|0⟩|0𝑐 ⟩|𝜓⟩ √ √ (3) √ → ( 1 − 2𝛿|000⟩ + 𝛿|100⟩)|0𝑐 ⟩|𝜓⟩ + 𝛿|110⟩𝑽 |0𝑐 ⟩|𝜓⟩ √ √ (4) √ → ( 1 − 2𝛿|000⟩ + 𝛿|100⟩)𝑼|0𝑐 ⟩|𝜓⟩ + 𝛿|110⟩𝑼𝑽 |0𝑐 ⟩|𝜓⟩   √ (5) √ → 1 − 2𝛿|000⟩𝑼|0𝑐 ⟩|𝜓⟩ + 𝛿|101⟩ |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ √ √ + 𝛿|100⟩( 𝑰 − |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰)𝑼|0𝑐 ⟩|𝜓⟩ 𝛿|110⟩𝑼𝑽 |0𝑐 ⟩|𝜓⟩ √ √ √ = ( 1 − 2𝛿|000⟩ + 𝛿|100⟩)𝑼|0𝑐 ⟩|𝜓⟩ + 𝛿|101⟩ 0𝑏 ( 0𝑏 ⊗ 𝑰)𝑼|0𝑐 ⟩|𝜓⟩ | {z } ′ | 𝜓0 ⟩ :=   √ √ − 𝛿|100⟩ |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ + 𝛿|110⟩𝑼𝑽 |0𝑐 ⟩|𝜓⟩  √ √ √ (6−7) → ( 𝑹 ⊗ 𝑰) ( 1 − 𝛿|000⟩ + 𝛿|100⟩)|0𝑐 ⟩|𝜓⟩ + 𝛿|101⟩ 0𝑏 𝜓0′    √ √ − 𝛿|100⟩𝑼 † |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ + ( 𝑹 ⊗ 𝑰) 𝛿|110⟩𝑽 |0𝑐 ⟩|𝜓⟩. Considering that  † † ⟨000|( 𝑹 ⊗ 𝐼) = 𝑹 |00⟩ ⟨0| = √︂ 5 1 − 𝛿⟨000| + 4 √︂ ! √ 𝛿 ⟨100| − 𝑖 𝛿⟨110| , 4 let us compute the part of 𝑪 𝑯 0𝑎+3 |𝜓⟩ starting with 0𝑎+3 : ( 0𝑎+3 ⊗ 𝑰)𝑪 𝑯 0𝑎+3 |𝜓⟩ √︂     𝛿 𝛿 5  = 1 − 𝛿 (1 − 𝛿) + |𝜓⟩ − (⟨0𝑐 | ⊗ 𝑰)𝑼 † |0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰 𝑼|0𝑐 ⟩|𝜓⟩ 4 2 2 | {z } | {z } √︃ =1− 85 𝛿+O ( 𝛿2 ) = 1− 45 𝛿+O ( 𝛿2 ) √︂ = 5 1− 𝛿 4 Í † 𝑗 ∈ 𝐽 𝑳 𝑗 𝑳 𝑗 |𝜓⟩ − 𝑖𝛿(⟨0𝑐 | ⊗ 𝑰)𝑽 |0𝑐 ⟩|𝜓⟩ ! !   𝛿 ∑︁ † 𝑰 − 𝑖𝛿𝑯 − 𝑳 𝑳 𝑗 |𝜓⟩ + O 𝛿2 , 2 𝑗 ∈𝐽 𝑗 and the part where third qubit is |1⟩: ∑︁ √ √ | 𝑗⟩ 𝑳 𝑗 |𝜓⟩. ( 𝑰 ⊗ ⟨1| ⊗ 𝑰)𝑪 𝑯 0𝑎+3 |𝜓⟩ = 𝛿𝑹|10⟩ 0𝑏 𝜓0′ = 𝛿𝑹|10⟩ 0𝑏 𝑗 ∈𝐽 131 e := √︃ 1 Let 𝚷 5
|0𝑎+3 ⟩⟨0𝑎+3 | ⊗ 𝑰 + 𝑰 ⊗ |1⟩⟨1| ⊗ 𝑰 , the above implies that 1− 4 𝛿     𝑎+3 𝑎+3 †e e Tr𝑎+3 𝚷𝐶 |0 ⟩⟨0 | ⊗ |𝜓⟩⟨𝜓| 𝐶 𝚷 ! ! 𝛿 ∑︁ † 𝛿 ∑︁ † = 𝑰 − 𝑖𝛿𝑯 − 𝑳 𝑳 𝑗 |𝜓⟩⟨𝜓| 𝑰 + 𝑖𝛿𝑯 − 𝑳 𝑳𝑗 2 𝑗 ∈𝐽 𝑗 2 𝑗 ∈𝐽 𝑗   ∑︁ +𝛿 𝑳 𝑗 |𝜓⟩⟨𝜓| 𝑳 †𝑗 + O 𝛿2 𝑗 ∈𝐽   = e𝛿L[|𝜓⟩⟨𝜓|] + O 𝛿2 . Similarly to the proof of Theorem 2.3.1, it is easy to see that this implies       e |0𝑎+3 ⟩⟨0𝑎+3 | ⊗ [·] 𝐶 † 𝚷 e − e𝛿L [·] ∥ ^ = O 𝛿2 . ∥Tr𝑎+2 𝚷𝐶 2.12 ■ Appendix:Quantum simulated annealing A subroutine for a coherent Gibbs sampler is to prepare the top eigenvector of the discriminant. In semi-group settings, one simply iterates the map for an arbitrary initial state to find its fixed point; given coherent access to some discriminant D (which is not quite a CPTP map), the standard approach is quantum simulated annealing [24, 179, 186]. To keep this section self-contained, in the following, we assume coherent access to some discriminants D 𝛽 𝑗 . First, we use QSVT to boost the gap; this is the origin of the quadratic speedup. Proposition 2.12.1 (Quadratic speedup [73, 118]). Given a block-encoding 𝑼D of a Hermitian matrix 𝑰 + D with eigenvalue gap 𝜆 𝑔𝑎 𝑝 (D) and 𝜆 1 (D) ≥ −𝜆 𝑔𝑎 𝑝 (D), we can construct a unitary 𝑼′ block-encoding a matrix 𝑝( 𝑰 + D) that has the same top eigenvector as D but with O (1) eigenvalue gap, with O ( √ 1 ) uses of 𝑼D and 𝜆 𝑔𝑎 𝑝 (D) † 𝑼D . Second, following [24], we consider the discretized adiabatic paths through temperatures 𝜆 1 (D 𝛽0 ) → · · · 𝜆 1 (D 𝛽 𝑗 ) → · · · 𝜆 1 (D 𝛽 𝑘 ) 𝑼D𝛽0 → · · · 𝑼D𝛽 𝑗 → · · · 𝑼D𝛽𝑘 where 𝛽𝑗 = 𝑗 𝛽. 𝑘 (2.87) While a more refined annealing schedule is possible, we consider the above linear schedule for simplicity. In particular, the initial state is the maximally entangled state 𝛽 = 0. In the following, we show that choosing 𝑘 = Θ(𝛽∥𝑯∥) ensures that the 132 consecutive overlaps remain constant large, allowing us to jump between consecutive states using a few steps of (fixed-point) amplitude amplification. Proposition 2.12.2 (Consecutive overlaps). Suppose the discriminants have a top eigenvector close to the purified Gibbs states √︁ E 1 . (2.88) ∥ 𝜆 1 (D 𝛽 𝑗 − 𝝆 𝛽 𝑗 ∥ ≤ 10 Let 𝛿𝛽 := 𝛽 𝑗+1 − 𝛽 𝑗 , then the consecutive overlaps are large 7 2 ⟨𝜆 1 (D 𝛽 𝑗 )|𝜆1 (D 𝛽 𝑗+1 )⟩ ≥ − O ((𝛿𝛽) 2 ∥𝑯 2 e−𝛿𝛽𝑯 ∥). (2.89) 10 Proof. Let us evaluate the overlap between the ideal Gibbs states and rewrite using the Hilbert-Schmidt inner product  2 ⟨e−𝛿𝛽𝑯/2 ⟩ 2𝛽 𝑗 2 Tr e−𝛽 𝑗 𝑯/2 e−𝛽 𝑗 𝑯/2 e−𝛿𝛽𝑯/2 √︁ √︁ ⟨ 𝝆 𝛽 𝑗 | 𝝆 𝛽 𝑗+1 ⟩ = = ⟨𝑰⟩ 𝛽 𝑗 ⟨e−𝛿𝛽𝑯 ⟩ 𝛽 𝑗 Tr[e−𝛽 𝑗 𝑯 ]Tr[e−𝛽 𝑗 𝑯 e−𝛿𝛽𝑯 ] = 1 − O ((𝛿𝛽) 2 ⟨𝑯 2 e−𝛿𝛽𝑯 ⟩ 𝛽 𝑗 ) ≥ 1 − O ((𝛿𝛽) 2 ∥𝑯 2 e−𝛿𝛽𝑯 ∥), where we denote the thermal expectation by ⟨𝑨⟩ 𝛽 = Tr[𝝆 𝛽 𝑨]. The last equality expands the exponential e−𝛿𝛽𝑯/2 = 𝑰 − 𝛿𝛽𝑯/2 + (e−𝛿𝛽𝑯/2 − 𝑰 + 𝛿𝛽𝑯/2). The small discrepancy (2.88) between 𝜆 1 (D 𝛽 𝑗 and change in the overlaps thus proving (2.89). √ 𝝆 𝛽 𝑗 only induces a minor ■ Proposition 2.12.3 (Simulated annealing). In the setting of Proposition 2.12.2, following the discretized adiabatic path (2.87) with 𝑘 = ⌈𝑐𝛽∥𝑯∥⌉ prepares a state |𝜓⟩ such that ∥|𝜓⟩ − 𝜆 1 ( D 𝛽 𝑗+1 ∥ ≤ 𝛿 using ª 𝛽∥𝑯∥ log2 (𝛽∥𝑯∥/𝛿) ®®. √︃ min 𝑗 𝜆 𝑔𝑎 𝑝 ( D 𝛽 𝑗 ) ¬ « total calls for the oracle for discriminants D 𝛽 𝑗 . © O ­­ Proof. Use fixed-point amplitude amplification [185] to “jump” between the eigen√ 𝛿 vectors 𝜆 1 ( D 𝛽 𝑗 . We can implement a 𝑘 -approximate projector to each eigenvector log(𝑘/𝛿) with O ( √ ) calls to the block-encoded discriminants D 𝛽 𝑗 . To ensure that 𝜆 𝑔𝑎 𝑝 (D 𝛽 𝑗 ) all the 𝑘 jumps are all approximated to error ≤ 𝑘𝛿 , all fixed-point amplitude amplification consists of log(𝑘/𝛿) rounds, each calling the block-encoded (approximate) projectors. ■ 133 A simple lower bound on 𝛽 dependence In this section, we prove a simple lower-bound for the temperature dependence in the sense of implementing a reflection about the purified Gibbs state. Proposition 2.12.4 (Lower-bound on simulation time). A circuit implementing the reflection operator 𝑹 𝛽,𝑯 := 𝑰 − √ 𝝆 𝛽,𝑯 √ 𝝆 𝛽,𝑯 using Hamiltonian simulation for 𝑯 as a black-box must use Hamiltonian simulation time 𝑇 = Ω(𝛽). Proof. The idea is to argue that the reflection operator is sensitive to 𝛽 and the Hamiltonian 𝑯, so the Hamiltonian simulation time 𝑇 cannot be too short. First, we control the norm change of the reflection operator. Let 𝝆 𝛽,𝑯 =: 𝝆 and 𝝆 𝛽,𝑯′ =: 𝝆′, then √︁ √︁ √ √ ∥ 𝑹 𝛽,𝑯 − 𝑹 𝛽,𝑯′ ∥ = ∥| 𝝆⟩⟨ 𝝆| − | 𝝆′⟩⟨ 𝝆′ |∥ √︁ √︁  √ ⊥ √ √ √ ′ ≥ 𝝆 | 𝝆⟩⟨ 𝝆| − | 𝝆 ⟩⟨ 𝝆′ | 𝝆 √︁ √ ⊥ √ √︁ = ⟨ 𝝆| 𝝆′⟩⟨ 𝝆′ | 𝝆 ⟩ √︁ = 1 − 𝜃 2 · 𝜃, √︃ √ √ 2 where 𝜃 := 1 − ⟨ 𝝆| 𝝆′⟩ and √︁ E √ √ ⊥ 𝝆 ∈ 𝑆 𝑝𝑎𝑛{ 𝝆 , 𝝆′ } such that √ ⊥√ ⟨ 𝝆 | 𝝆⟩ = 0. Now, for infinitesimal 𝜖 → 0, let 𝑯 = |0⟩⟨0| + |1⟩⟨1| 𝑯′ = |0⟩⟨0| + (1 + 𝜖)|1⟩⟨1| such that √ 𝝆 ∝ |00⟩ + |11⟩ √︁ E 𝝆′ ∝ |00⟩ + (1 + 𝛽𝜖/2)|11⟩ + O (𝜖 2 ). Direct calculation gives 𝜃 = Ω(𝛽𝜖) for small 𝜖 → 0. To conclude the proof, suppose the block-box circuit uses only Hamiltonian simulation time 𝑇. Then, the resulting 134 circuits for 𝑯 and 𝑯′ can only differ by ∥𝑯 − 𝑯′ ∥𝑇 = 𝜖𝑇. Therefore, for small 𝜖 → 0, √︁ 𝑇𝜖 ≥ 1 − 𝜃 2 · 𝜃 = Ω(𝛽𝜖), proving the advertised result. ■ The above sensitivity argument similarly applies to algorithms preparing the Gibbs state using black-box Hamiltonian simulation. Proposition 2.12.5 (Lower-bound on simulation time). A circuit preparing the Gibbs state 𝝆 𝛽 ∝ e−𝛽𝑯 using Hamiltonian simulation for 𝑯 as a black-box must use Hamiltonian simulation time 𝑇 = Ω(𝛽). Proof. Again, consider 𝑯 and 𝑯′ as above and their Gibbs states 𝝆 ∝ |0⟩⟨0| + |1⟩⟨1| 𝝆′ ∝ |0⟩⟨0| + (1 + 𝛽𝜖)|1⟩⟨1| + O (𝜖 2 ). Then, for infinitesimal 𝜖 → 0, ∥ 𝝆 − 𝝆′ ∥ 1 ≥ Ω(𝛽𝜖), which implies 𝑇 = Ω(𝛽) as advertised. 2.13 ■ Appendix:Impossibility of boosted shift-invariant in-place phase estimation In this section, we include the proof that certain “boosted shift-invariant in place phase estimation” utilized in [167] is impossible. The impossibility result was developed in parallel with this work; we reproduce the main argument here with the permission of András Gilyén and Dávid Matolcsi until their manuscript becomes publicly available. We begin by reviewing the phase estimation assumptions made by [167]. First, they assume [167, Eqn.(11), Supplemental Information] the phase estimation map is shift-invariant in the sense that ∑︁ ∑︁ ∑︁ 𝜇¯ 𝜇¯ Φ := 𝑴𝜈¯ ⊗ | 𝜈⟩⟨ ¯ 𝜇| ¯ where 𝑴𝜈¯ := 𝛼(𝐸𝑖 , 𝜈¯ − 𝜇)|𝜓 ¯ 𝑖 ⟩⟨𝜓𝑖 | and 𝜈, ¯ 𝜇¯ ∈ {Z𝜔}. ¯ 𝜈¯ 𝜇¯ 𝑖 135 In [167], the shift-invariance was shown to hold for an unboosted phase estimation unitary [167, Eqn.(12), Supplemental Information], which has a slowly decaying tail when 𝐸𝑖 deviates substantially from 𝜈¯ − 𝜇. ¯ Second, to prove the correctness of the fixed point, they impose [167, Eqn.(10), Supplemental Information] that the profile 𝛼(𝐸𝑖 , 𝜈¯ − 𝜇) ¯ can be boosted:52 the only nonzero matrix elements are such that | 𝜈¯ − 𝜇¯ − 𝐸𝑖 | < 𝜔, ¯ where 𝜔¯ is the energy resolution for the phase estimation readout registers. Unfortunately, these two assumptions are not compatible with each other, as argued by the following. Proposition 2.13.1 (Impossibility for shift-invariant boosting). There exists no continuous family of “boosted shift-invariant in place phase estimation” unitaries. More precisely, for every constant 𝑘 for large enough 𝑁 there exists no profile 𝛼 : R × 2𝜋 𝑁 · {0, 1, 2, . . . , 𝑁 − 1} → C that simultaneously satisfies: Í 2𝜋 • (almost) unitarity: 𝑴𝐸 := 𝑖,𝑁−1 𝑗=0 𝛼(𝐸, 𝑁 (𝑖 − 𝑗 mod 𝑁))|𝑖⟩⟨ 𝑗 | is close to some unitary 𝑼𝐸 for all 𝐸 ∈ R : ∥ 𝑴𝐸 − 𝑼𝐸 ∥ ≤ 21 • boosting: 𝛼(𝐸, 𝜈¯ − 𝜇) ¯ = 0 if | 𝜈¯ − 𝜇¯ − 𝐸 mod 2𝜋| > 𝑘 2𝜋 𝑁 • continuity: 𝛼(𝐸, 𝜈) ¯ depends continuously on 𝐸 Proof. We prove the statement by contradiction. Let us assume that such a profile exists for 𝑁 ≫ 𝑘 3 . We will track how the profile changes as we increase the energy 2𝜋 from 𝐸 0 = 𝑘 2𝜋 𝑁 to 𝐸 1 = (3𝑘 + 1) 𝑁 . Í 2𝜋 𝑗 Now, consider the polynomial 𝑝 𝐸 (𝑧) = 4𝑘+1 𝑗=0 𝛼(𝐸, 𝑗 𝑁 )𝑧 (whose physical meaning will become clear). Due to boosting the polynomial 𝑝 𝐸0 (𝑧) has a degree at most 2𝑘, so it has at most 2𝑘 roots, and in particular, at most 2𝑘 roots are situated within the complex unit circle. Due to continuity, the polynomial 𝑝 𝐸 (𝑧) is continuously transformed to 𝑝 𝐸1 (𝑧) whose smallest nonzero coefficient comes with a power of 𝑧 at least 2𝑘 + 1 due to boosting. This implies that 0 is a root with multiplicity at least 2𝑘 + 1, and in particular, we have at least 2𝑘 + 1 roots within the unit 52This is explicitly stated in the last paragraph of page 13 [167, Supplemental Information]: “According to (10) we can replace the function 𝑓 (𝐸 𝑗 , 𝑘 − 𝑝) by its enhanced counterpart 𝛼𝐸 𝑗 (𝑘 − 𝑝), which acts as a binary amplitude for the two closest 𝑟-bit integers to the actual energy 𝐸 𝑗 .” 136 circle. Since the polynomial changes continuously, its multi-set of roots also changes continuously, which means that at some point, a root must enter the unit circle53 (here we acknowledge that some roots enter from infinity when the degree of the polynomial increases but that does not affect our argument — one can make this precise by tracking roots on the surface of the Riemann sphere). Thus, there is some energy 𝐸 ′ for which a complex unit number 𝑧 = e2𝜋𝑖𝜑 is a root of the corresponding polynomial 𝑝 𝐸 ′ (𝑧). We show that this implies that the plane wave with quasi-momentum 𝜑 is (almost) in the kernel of the shift-invariant matrix 𝑴𝐸 ′ : for every 𝑖 ≤ 𝑁 − 4𝑘 − 2 we have ⟨𝑖| 𝑴𝐸 ′ 𝑁−1 ∑︁ | 𝑗⟩𝑧− 𝑗 = 𝑗=0 𝑁−1 ∑︁ 𝛼(𝐸 ′, 2𝜋 (𝑖 − 𝑗 𝑁 mod 𝑁))𝑧− 𝑗 = 𝑧−𝑖 𝑝 𝐸 ′ (𝑧) = 0, 𝑗=0 implying ∥ 𝑴𝐸 ′ 𝑁−1 ∑︁ 𝑗=0 𝑁−1   ∑︁ | 𝑗⟩𝑧− 𝑗 ∥ 2 = O 𝑘 3 ≪ 𝑁 = ∥𝑼𝐸 | 𝑗⟩𝑧− 𝑗 ∥ 2 , 𝑗=0 a contradiction. 53The idea of tracking the roots of this polynomial is due to Dávid Matolcsi. ■ 𝛽2 ⟨𝑯 2 ⟩ 𝛽=0 3 √ ) 𝜖 𝜆 𝑔𝑎 𝑝 3/2 𝜆 𝑔𝑎 𝑝 + 𝛽 3/2 ) 𝜖𝜆 𝑔𝑎 𝑝 𝜆 𝑔𝑎 𝑝 Table 2.1: A comparison of existing thermal state preparation algorithms7 . certain measurement operator 𝑯 = 𝑯0 + 𝑽 for small 𝑽 and simple 𝑯0 Õ (𝑡 𝑚𝑖𝑥 𝛽) eO (𝛽∥𝑽 ∥) eO (𝛽∥𝑯∥) [90]:a perturbative approach QSVT [73]: directly implementing e−𝛽𝑯 - correlation length 𝐶 𝑑 𝑑 eÕ (𝐶 ln (𝛽𝑛/𝜖)) 𝑑 𝑑 Markov and clustering with length scale 𝐶 - rounding promise for 𝑯 perfect QPE eO (𝐶 ln (𝑛/𝜖)) 5 Gate complexity 𝑂˜ ( 𝛽2 ∥𝑯∥ Õ (𝑡𝑄𝑃𝐸 · √𝛽∥𝑯∥ ) 3 Õ (𝑡𝑄𝑃𝐸 · - Õ ( 𝛽𝑡 𝑚𝑖𝑥 𝜖 · 𝑡 𝑚𝑖𝑥 ) controllable, refreshable bath and ETH2 controllable, refreshable bath 3 𝛽𝑡 𝑚𝑖𝑥 𝜖 ) Poly(𝜖 −1 , 𝛽, 𝑛, 𝑡 𝑚𝑖𝑥 ) O( - 1 𝛽 𝑡 𝑚𝑖𝑥 1 Õ ( 𝛾𝑎𝑡𝑡 ) 𝜖7 3˜ shift-invariant, boosted QPE (provably impossible) assumptions Poly(𝜖 −1 , 𝛽, 𝑛)𝑡𝑄𝑃𝐸 · 𝑡 𝑚𝑖𝑥 Ham. sim. time [26]: patching via recovery maps QITE [132]: imaginary time evolution via local tomography [133]: drive transition by measurements Algorithms Quantum Metropolis [167]: a quantum version of rejection sampling via QPE [149]: simulate a Lindbladian with forbidden energy transitions [160]: system-bath evolution at weak-coupling [38]: system-bath evolution at weak-coupling Theorem 2.1.3: simulate a Lindbladian via operator FT Quantum2 Metropolis [186]: a coherent version of Quantum Metropolis via QPE [180]:a coherent version of Lindbladian via QPE Theorem 2.1.4: a coherent version of Lindbladian via operator FT not scalable nonconstructive, quasi-local recovery maps costly at low-temperature, strong correlation, or high accuracy the measurement must have “good overlap” with energy basis6 costly at low-temperature or nonperturbative regime - unknown error for unrounded Hamiltonians needs nondegenerate energies,4 eΩ(𝑛) runtime for QPE - large bath and large overheads overhead for randomized rounding; forbidding transitions impacts 𝑡˜𝑚𝑖𝑥 large bath; no guarantees without ETH. without the assumption, the QPE runtime is uncontrolled potential caveats 137 138 Chapter 3 AN EFFICIENT AND EXACTLY DETAILED-BALANCED QUANTUM GIBBS SAMPLER 3.1 Introduction One of the leading candidate applications of quantum computers [54] is to simulate quantum systems [68]. In particular, the preparation of thermal states or ground states for materials and molecules has received significant attention [14, 30, 37, 107]. Surprisingly, there has not been a consensus on the “go-to” quantum algorithm for this task due to a lack of provable guarantees or empirical evidence [108]. Recently, several Monte Carlo-style, nonunitary quantum algorithms have been proposed [40, 61, 149, 167, 180, 186]. While their efficacy has only been validated using small-scale numerics and under strong theoretical assumptions [38, 61, 160], there are reasons for optimism. On physical grounds, these algorithms resemble naturally occurring system-bath dynamics [134]; if a system rapidly cools in a refrigerator, the same plausibly applies to a “cooling algorithm” that emulates this process. Alternatively, from a computer science perspective, these algorithms are cousins of classical Markov chain Monte Carlo (MCMC) methods but with quantum mechanical effects and complications. This work sets out to complete this line of thought and construct an ideal quantum MCMC algorithm where the robustness, simplicity, and empirical success of the classical case may be transferrable. The cornerstone of classical Markov chain Monte Carlo methods is detailed balance (see, e.g., [110]): given a target state 𝜋, we impose a certain symmetry of the Markov chain 𝑴 𝑴𝑠 ′ 𝑠 𝜋 𝑠 = 𝜋 𝑠 ′ 𝑴𝑠 ′ 𝑠 for each configuration 𝑠, 𝑠′, ensuring stationarity 𝑴 [𝜋] = 𝜋. This simple recipe for the stationary state has been crucial in constructing and analyzing the Metropolis-Hastings algorithm and related Markov chains. Notably, detailed balance gives a conceptually simple picture of convergence via the spectral gap of 𝑴, a quantity amenable to numerical and analytic bounds. If the problem at hand has a local structure, detailed balance can often be imposed locally and efficiently, relegating the algorithm’s complexity to the mixing time, the time scale of convergence towards stationarity. While the mixing time may be challenging to 139 analyze, MCMC methods can often be employed heuristically. In particular, we are often interested in sampling the Gibbs distribution 𝜋 𝛽 ∝ 𝑒 −𝛽𝐻 of a certain energy functional 𝐻 at temperature 1/𝛽. Analogously, the central idea of quantum Gibbs sampling is to construct a detailed-balanced quantum process where the quantum Gibbs state is stationary. In this work, we focus on designing a Lindbladian L 𝛽 (the quantum analog of a continuous-time Markov chain generator) such that eL 𝛽 𝑡 [𝝆 𝛽 ] = 𝝆 𝛽 where 𝝆 𝛽 := e−𝛽𝑯 /Tr(e−𝛽𝑯 ), for any target quantum Hamiltonian 𝑯. As in the classical case, we can prepare samples of quantum Gibbs states if the Lindbladian evolution can be efficiently implemented and the state converges rapidly to the Gibbs state. The main issue with existing quantum Gibbs sampling algorithms is that quantum detailed balance (Figure 3.1) only holds approximately unless we can distinguish individual energy eigenstates exactly, which is generally intractable except for fastforwardable Hamiltonians (e.g., Hamiltonians with commuting terms). Consequently, we either lose accuracy guarantees for the stationary state or the efficiency for the individual steps of the Gibbs sampling algorithm, leading to significant aggregated complexity plaguing various constructions; see Ref. [40] for a comprehensive catalog. The algorithmic challenge in enforcing quantum detailed balance is the energy-time uncertainty principle rooted in metrology: for each energy estimate, the uncertainty scales inversely proportional to the Hamiltonian simulation time. Indeed, all existing quantum MCMC algorithms attempt to attain detailed balance via an “energy estimation” subroutine (quantum phase estimation [149, 167, 180] or operator Fourier Transform [40, Appendix A]). Consequently, this error propagates to the desired Gibbs state and impacts the implementation cost. To our knowledge, the best general lower-bound on the Hamiltonian simulation time is Ω(𝛽) per Gibbs sample [40, Proposition G.5]. This comes from a sensitivity argument that the Gibbs state is a smooth matrix function of 𝑯 with derivatives bounded by O (𝛽). This conceptual gap motivates our guiding question: Can we design an efficiently implementable yet exactly detailed-balanced quantum Gibbs sampler? If so, we may recover both the simplicity and versatility of classical MCMC algorithms. In this work, we answer this question in the affirmative by explicitly constructing an exactly detailed-balanced Lindbladian at a moderate cost: Õ (𝛽)-Hamiltonian 140 simulation cost per unit time of Linbladian evolution eL . (The unit-time evolution for a continuous-time generator should be regarded as “one step,” corresponding to one discrete Markov chain update.) Furthermore, for lattice Hamiltonians (with local jumps), our Lindbladian is (quasi-)local with locality scaling as Õ (𝛽). Thus, one step of the algorithm only needs to simulate localized Hamiltonian patches; this starkly contrasts with previous works, whose cost per unit time step generally scales with the system size due to simulating the global Hamiltonian. The mathematical and conceptual simplicity of our result immediately initiates a list of new directions, which we discuss in detail in section 3.4. The key revelation behind our construction is that quantum detailed balance can be enforced smoothly without ever knowing the energy. Indeed, the standard metrology lower bound ∼ Ω( 1𝜖 ) is not an obstruction because having access to a detailed-balanced Lindbladian (or the Gibbs state) does not give energy estimates. We have seen that Quantum Signal Processing [117] or Quantum Singular Value Transform (QSVT) [73] allows one to directly access smooth (polynomial) functions of Hamiltonians without a phase-estimation subroutine; the costs often scale linearly with the largest derivatives and only logarithmically with the precision. However, Lindbladians, as superoperators, are more restrictive to manipulate than Hermitian matrices. A key design ingredient is a carefully chosen coherent term in our Lindbladian L 𝛽 [𝝆] = −i[𝑩, 𝝆] + (“dissipative”), | {z } “coherent” which appears necessary to coherently and exactly cancel out certain unwanted errors from the dissipative part. As a by-product, purifying our Lindbladian yields a temperature-dependent family of “parent Hamiltonians” whose zero-eigenstate is a canonical purification of the Gibbs state.1 Similarly to how classical Markov chains can be “quantized” to prepare the purified stationary state, here we prepare the purified Gibbs state by following a prescribed adiabatic path (called quantum simulated annealing [164, 186]), drawing a surprisingly simple connection between thermal dissipation and adiabatic evolution. In particular, for lattice Hamiltonians, the parent Hamiltonian inherits the (quasi)locality, which curiously connects the purified Gibbs state to the ground state of (quasi-)local Hamiltonians. 1This purification coincides with the Thermal Field Double state featured in recent quantum gravity discussions. See, e.g., [122]. 141 E <latexit 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Classical Quantum Figure 3.1: (Left) For the classical Gibbs distribution, the detailed balance condition is a pairwise relation between heating (red) and cooling (blue) transition rates, depending on the energy difference 𝜈 of states. (Right) For the quantum Gibbs state, the detailed balance condition refers to pairs of matrix elements of the density operator (expanded in the energy basis), where each matrix element is described by a pair of energies (of the basis elements in the ket and bra respectively), and therefore the relation depends on both of the respective energy differences 𝜈1 and 𝜈2 . Main results Our main results, based on the algorithmic framework of [40, Section III], consider the following Lindbladian in the Schrodinger picture ª © ® ­ 1 ® ­ ˆ𝑎 𝛾(𝜔) ­ 𝑨 (𝜔)(·) 𝑨ˆ 𝑎 (𝜔) † − { 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔), ·}®d𝜔 ® ­| 2 {z } | −∞ ­ {z }® ∑︁ ∫ ∞ L 𝛽 [·] := −i[𝑩, ·] + | {z } 𝑎∈𝐴 “coherent” “transition” “decay” ¬ (3.1) « which is parameterized by the following terms (with convenient normalization conditions [40, Section I.B]): • The distinct jump operators 𝑨𝑎 “drive” the transitions, and can be chosen arbitrarily as long as their adjoints are included { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴} and ∑︁ 𝑨𝑎† 𝑨𝑎 ≤ 1. (3.2) 𝑎∈𝐴 For better mixing and ergodicity, the jumps should be “scrambling” and not commute with the Hamiltonian (e.g., breaking the symmetries of the Hamiltonian). For lattice Hamiltonians, the jump operators may be chosen simply to be the single-site Pauli operators, but global jumps could also be helpful in some cases as in the classical case (e.g., cluster updates). 142 • The Operator Fourier Transform (OFT)2 (section 3.2) weighted by a Gaussian filter with a tunable width ∼ 𝜎𝐸−1 ∫ ∞ 1 𝑎 ˆ 𝑨 (𝜔) := √ ei𝑯𝑡 𝑨𝑎 e−i𝑯𝑡 e−i𝜔𝑡 𝑓 (𝑡)d𝑡 (3.3) 2𝜋 −∞ √︃ √︁ 2 2 e−𝑡 /𝛽 1 −𝜎𝐸2 𝑡 2 where 𝑓 (𝑡) := e 𝜎𝐸 2/𝜋 = √︃ √︁ if 𝜎𝐸 = . 𝛽 𝛽 𝜋/2 ∫∞ In particular, the Gaussian is normalized −∞ | 𝑓 (𝑡)| 2 d𝑡 = 1. Naturally, the Heisenberg evolution ei𝑯𝑡 𝑨𝑎 e−i𝑯𝑡 diagnoses the energy difference 𝜔 before ∫∞ and after the jump. Integrating over time √1 −∞ (·)e−i𝜔𝑡 𝑓 (𝑡)d𝑡 yields the 2𝜋 operator Fourier Transform 𝑨ˆ 𝑎 (𝜔), which selects the transitions of 𝑨𝑎 that increase the energy by roughly ∼ 𝜔 ± O (𝜎𝐸 ). At first glance, the Gaussian filter seems to merely ensure good concentration for both the frequency and time domain, but it turns out to have a more intimate connection [133] to quantum detailed balance (see also section 3.8 for an alternative justification). • The transition weight 𝛾(𝜔) follows (yet another) Gaussian with a tunable variance 𝜎𝛾 > 0: ! 2𝜔 𝛾 (𝜔 + 𝜔 𝛾 ) 2 2 := with variance 𝜎 − 𝜎𝐸2 (3.4) 𝛾(𝜔) = exp − 𝛾 2 𝛽 2𝜎𝛾   (𝛽𝜔 + 1) 2 1 = exp − (3.5) if 𝜎𝐸 = 𝜎𝛾 = 𝜔 𝛾 = . 2 𝛽 The normalization is such that ∥𝛾(𝜔)∥ ∞ ≤ 1, and the maximum is attained at 𝜔 = −𝜔 𝛾 . • A coherent (i.e., nondissipative) term generated by a fine-tuned Hermitian matrix 𝑩. The expression depends on 𝜔 𝛾 , 𝜎𝐸 , 𝛽 in the general case (see Corollary 3.3.1), but it simplifies to ∫ ∞  ∑︁ ∫ ∞ −i𝛽𝑯𝑡 ′ i𝛽𝑯𝑡 ′ 𝑎† −2i𝛽𝑯𝑡 ′ 𝑎 i𝛽𝑯𝑡 ′ ′ i𝛽𝑯𝑡 𝑩 := 𝑏 1 (𝑡)e 𝑏 2 (𝑡 )e 𝑨 e 𝑨 e d𝑡 e d𝑡 𝑎∈𝐴 −∞ if 𝜔 𝛾 = 𝜎𝐸 = 𝜎𝛾 = −∞ 1 𝛽 (3.6) for some carefully chosen smooth and rapidly decaying functions 𝑏 1 , 𝑏 2 normalized by ∥𝑏 1 ∥ 1 , ∥𝑏 2 ∥ 1 ≤ 1. The coherent term 𝑩 may appear intimidating 2 Note the sign convention, which might differ from that of other works in the literature. 143 but plays a crucial role in ensuring quantum detailed balance for the Gibbs state 𝝆 𝛽 , defined as L †𝛽 [·] = √ √ √ √ 𝝆 𝛽 −1 L 𝛽 [ 𝝆 𝛽 · 𝝆 𝛽 ] 𝝆 𝛽 −1 for fixed 𝛽, 𝑯. (3.7) It implies the stationarity of Gibbs state exactly (see Definition 3.2.1). The Gaussian transition weight (3.4) is inspired by an observation of [133]:3 the functional form of Gaussians is naturally compatible with exact detailed balance4 if we make conscious choices of 𝜔 𝛾 , 𝜎𝐸 ! ! (𝜔 + 𝜔 𝛾 ) 2 2𝜔 𝛾 (−𝜔 + 𝜔 𝛾 ) 2 exp − = exp(− 2 · 𝜔) exp − . 2𝜎 2 𝜎 2𝜎 2 Compared with the usual step-function-like Metropolis weight min(1, e−𝛽𝜔 ), the Gaussian weight is more selective, only allowing energy transitions −𝜔 𝛾 ± O (𝜎𝛾 ); this narrower window could potentially freeze the dynamics, leading to long mixing time. Fortunately, quantum detailed balance 3.7 is preserved under linear combination of Lindbladians; hence, choosing a linear combination of 𝛾 covering a range of different width 𝜎𝛾 can remove the heavy restriction on energy transitions. Surprisingly, a suitable linear combination recovers Metropolis-like transition weights, which we focus on as the representative. To obtain the corresponding exactly detailed-balanced Lindbladian, the only change compared to (3.4)-(3.6) is the choice of transition weight    1 1 𝑀 (Metropolis-Style) 𝛾 (𝜔) := exp −𝛽 max 𝜔 + ,0 if 𝜎𝐸 = (3.8) 2𝛽 𝛽 with the corresponding coherent term 𝑩 𝑀 parameterized by another function 𝑏 2𝑀 (𝑡) (the function 𝑏 1𝑀 (𝑡) = 𝑏 1 (𝑡) remains the same as in (3.6)).5 Now, we present the first main result: the Gibbs state is an exact stationary state of the advertised Lindbladian (see section 3.2 for the proof). Although we have 3Their algorithm [133] seems qualitatively different from Monte Carlo style quantum algo- rithms [40, 149, 167, 180, 186] and closer to performing phase estimation on trial states; see the discussion in [133, Page 5]. 4We thank Jonathan Moussa for pointing us to his paper and raising the question of whether detailed balance can hold exactly in the precursor of this work [40]. 5The generalized function (distribution) 𝑏 𝑀 (𝑡) should be interpreted as the Cauchy principal value Í 2 lim 𝜂→0+ 1 (|𝑡| > 𝜂)𝑏 2𝑀 (𝑡). In case [𝑯, 𝑎∈ 𝐴 𝑨 𝑎† 𝑨 𝑎 ] ≠ 0 an additional correction term √1 𝛿(𝑡) 16 2 𝜋 should be added. 144 mainly focused on the Gibbs state, we can formally invoke Gibbs sampling for 𝛽𝑯 = log( 𝝆 𝑓 𝑖𝑥 ) for arbitrary target stationary state 𝝆 𝑓 𝑖𝑥 , albeit with potential overhead from implementing the matrix logarithm. Theorem 3.1.1 (Gibbs state is stationary). For any 𝛽 ≥ 0, the Lindbladian (3.1)-(3.3) with 𝜎𝐸 = 1𝛽 , Gaussian transition wieght (3.5), and the coherent term 𝑩 (3.6) satisfies detailed balance (3.7) exactly. Therefore, the Gibbs state is stationary L 𝛽 [𝝆 𝛽 ] = 0. The same applies for the Metropolis transition weight 𝛾 𝑀 (𝜔) (3.8) with the corresponding coherent term 𝑩 𝑀 . Furthermore, building on the algorithmic machinery developed in Ref. [40, Section III], the Lindbladian can be efficiently simulated at a moderate cost (see section 3.3 for the proof). Theorem 3.1.2 (Efficient implementation). Instantiate the Lindbladian parameters of Theorem 3.1.1 for either the Gaussian 𝛾(𝜔) or Metropolis 𝛾 𝑀 (𝜔) transition weight. Then, the Lindbladian evolution eL 𝛽 𝑡 for each 𝑡 ≥ 1 can be implemented efficiently in 𝜖-diamond distance with cost: Õ (𝑡 · 𝛽) (total Hamiltonian simulation time) Õ (1) (resettable ancilla) Õ (𝑡) (block-encodings for the jumps ∑︁ |𝑎⟩ ⊗ 𝑨𝑎 ) 𝑎∈𝐴 Õ (𝑡) (other two-qubit gates). The Õ (·) notation absorbs logarithmic dependencies on 𝑡, 𝛽, ∥𝑯∥, 𝑛, 1/𝜖, | 𝐴|. Here, our Lindbladian is normalized (3.2), (3.3), (3.4), (3.6) (or the Metropolis-like weight (3.8) with its coherent term 𝑩 𝑀 ) such that ∥L 𝛽 ∥ 1−1 = Õ (1). Therefore, evolving for unit time 𝑡 = 1 corresponds to a Õ (1)-strength update eL 𝛽 and only requires a characteristic Hamiltonian simulation time ∼ 𝛽. This is precisely 145 the cost for implementing both the operator Fourier Transform (3.3) and the coherent term (3.6) via Linear Combination of Unitaries. To be more careful, the Õ (·) notation also includes polylogarithmic factors due to discretization, truncation, and Hamiltonian simulation error, which is typical in quantum algorithms. To clarify, the idealized map L 𝛽 remains exactly detailed-balanced (the nice object to analyze), and the algorithmic implementation error can be made arbitrarily small given the desired runtime. We expect the total Hamiltonian simulation time to be the figure of merit for the algorithmic cost, among others. The jump operators 𝑨𝑎 can be as simple as Pauli operators, but we consider a black-box query model in case more complex jumps are needed for faster mixing. In the last line, the use of other two-qubit gates comes from easier-to-implement unitaries, including the Quantum Fourier Transform, state preparation unitary for the Gaussian filter | 𝑓 ⟩, controlled transition weight; see section 3.3 and [40, Section III.B]. Combining Theorem 3.1.1 and Theorem 3.1.2, we can prepare the Gibbs state by simulating the Lindblaidan until convergence, resulting in the cost
e 𝑡 𝑚𝑖𝑥 (L 𝛽 ) · 𝛽 . (total Hamiltonian simulation time per Gibbs sample) = O Formally, the mixing time 𝑡 𝑚𝑖𝑥 (L 𝛽 ) quantifies the shortest time scale that any two input states become indistinguishable (see Proposition 3.9.4). To obtain end-to-end gate complexities, we should also instantiate the Hamiltonian simulation cost, a subroutine whose complexity for various systems has been thoroughly studied. For example, on 𝐷-dimensional lattices with local jumps 𝑨𝑎 , we expect the actual cost to be (up to logarithmic error dependence) (gate complexity per unit evolution time) ∼ 𝛽 |{z} Ham. sim. time × (𝑣 𝐿𝑅 𝛽) 𝐷 , | {z } volume (spatially local Hamiltonians) where 𝑣 𝐿𝑅 is the Lieb-Robinson velocity and 𝑣 𝐿𝑅 𝛽 is roughly the radius of the Heisenberg evolution 𝑨𝑎 (·) at time ∼ 𝛽. Indeed, the Lindbladian is a sum over quasi-local Lindbladian operators (Figure 3.2) ∑︁ L𝛽 = L 𝛽𝑎 each centered at 𝑨𝑎 with radius Õ (𝑣 𝐿𝑅 𝛽). 𝑎∈𝐴 146 Aa⇤ <latexit 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Ha Figure 3.2: (Left) For lattice Hamiltonians, our Lindbladian is a sum of quasi-local terms L 𝛽𝑎 localized around each jump 𝑨𝑎 with radius Õ (𝛽). Indeed, detailed balance is really about the energy difference, which can be diagnosed by Fourier Transforming the Heisenberg evolution 𝑨𝑎 (𝑡) = ei𝑯𝑡 𝑨𝑎 e−i𝑯𝑡 . Due to the Lieb-Robinson bounds, the localized Lindbladian terms effectively only depend on the local Hamiltonian patch nearby (up to exponentially decaying tail). (Right) This locality persists after purification, where two copies of the system are glued together. In particular, the cost per unit Lindbladian evolution time is essentially independent of the system size (up to logarithmic dependencies), as we only need to simulate the Hamiltonian patch surrounding each jump 𝑨𝑎 .6 Purifying the Lindbladians We may purify the Lindbladian to prepare the purified Gibbs state [180] √ ∑︁ 1 𝝆 𝛽 := √︁ e−𝛽𝐸𝑖 /2 |𝜓𝑖 ⟩ ⊗ 𝜓𝑖∗ ; −𝛽𝑯 Tr[e ] 𝑖 (3.9) this particular purification is reminiscent of the quantum walk formalism for detailedbalanced classical Markov chains. The relevant “parent Hamiltonian” (playing the role of a “quantum walk” operator) is the discriminant associated with the Lindbladian; the expression may appear intimidating but resembles how classical detailed-balanced Markov chains are quantized H𝛽 := 𝝆 −1/4 L [𝝆 1/4 · 𝝆 1/4 ] 𝝆 −1/4 . The above should be regarded as the Lindbladian under similarity transformation. Further, to implement the quantum walk, the superoperator needs to be vectorized into an operator on duplicated Hilbert spaces (vectorization) 𝑛 𝑛 𝑛 𝑛 H𝛽 : B (C2 ) → B (C2 ) ≃ H 𝜷 ∈ B (C2 ⊗ C2 ). 6 In fact, we can further parallelize the Lindbladian evolution to improve the circuit depth; see section 3.7. 147 Indeed, the Gibbs state 𝝆 𝛽 , as the Lindbladian stationary state, corresponds to the √ purified Gibbs state 𝝆 𝛽 (3.9), as a zero-eigenvector of the discriminant. As a sanity check, the quantum detailed balance condition 3.7 naturally ensures that the operator is Hermitian H 𝜷 = H †𝜷 . Proposition 3.1.1 (Purifying Lindbladians). Instantiate the Lindbladian parameters of Theorem 3.1.1 for the Gaussian or Metropolis transition weight. Then, the corresponding discriminant H 𝜷 is Hermitian, frustration-free, and annihilates the purified Gibbs state. ∑︁ √ H𝜷 = H 𝒂𝜷 such that H 𝒂𝜷 𝝆 𝛽 = 0 for each 𝑎 ∈ 𝐴. 𝑎∈𝐴 In other words, preparing the purified Gibbs state boils down to the ground state problem (up to a negative sign) for a frustration-free parent Hamiltonian parameterized by 𝛽7; for lattice Hamiltonian with local jumps 𝑨𝑎 , the parent Hamiltonian inherits the quasi-locality of our Lindbladian, with individual terms of radius ∼ 𝑣 𝐿𝑅 𝛽. The algorithmic cost to implement the parent Hamiltonian is analogous to the Lindbladian case, where 𝑯 𝛽 roughly corresponds to a constant-time Lindbladian evolution. Theorem 3.1.3 (Block encodings for the discriminants). Instantiate the Lindbladian parameters of Theorem 3.1.1 for either the Gaussian or Metropolis transition weight. Then, the corresponding discriminant Õ 1(1) H 𝜷 can be block-encoded approximately in 𝜖-spectral norm using Õ (𝛽) (Hamiltonian simulation time) Õ (1) (resettable ancilla) Õ (1) (block-encodings for the jumps ∑︁ 𝑎∈𝐴 Õ (1) |𝑎⟩ ⊗ 𝑨𝑎 and its transposes ∑︁ |𝑎⟩ ⊗ ( 𝑨𝑎 )𝑇 ) 𝑎∈𝐴 (other two-qubit gates). The Õ (·) notation absorbs logarithmic dependencies on 𝑡, 𝛽, ∥𝑯∥, 𝑛, 1/𝜖, | 𝐴|. Like in Theorem 3.1.2, we have expanded many other unitaries in the last line; see section 3.3. Note that we had to downscale the discriminant to implement the 7 Strictly speaking, originating from a Lindbladian, here the parent Hamiltonian is negative semi-definite, and the purified Gibbs state is the top-eigenstate. Introducing a global negative sign will make it the ground state. 148 block encodings; see section 3.3 for the proof. For readers familiar with quantum walks, obtaining a block-encoding for 𝑰 + H 𝜷 (instead of H 𝜷 ) would be nicer as it leads to a quadratic speedup in terms of the spectral gap of H 𝜷 . However, we have not found such a direct block-encoding for our particular construction, leaving this an open problem. This is in contrast with our earlier approximate construction [40, 180], where such a direct block-encoding was possible, which might be advantageous in some cases. Roadmap The remaining body of text is organized thematically by the analysis and the algorithm. We begin with the analysis (section 3.2) regarding how we design our Lindbladian to satisfy quantum detailed balance exactly. Next, we give efficient algorithms (section 3.3) in terms of modularized block encodings to implement the advertised Lindbladian and its purification. In the appendix, we include independent expositions: a connection between the discriminant gap, mixing time, and the area law of entanglement (section 3.7) and an alternative heuristic derivation of our detailed balance Lindbladian in the time domain (section 3.8). 3.2 Analysis In this section, we execute the calculations circling the exact detailed balance condition. First, we review the operator Fourier Transform in the frequency domain. Second, we review the notion of detailed balance, including the stationary state and spectral theory of convergence. Third, we plug in the advertised functional forms and derive the required coherent term 𝑩 for achieving detailed balance. Operator Fourier Transform ˆ Our Lindbladian features the operator Fourier Transform 𝑨(𝜔) of a jump operator 𝑨 according to the Hamiltonian 𝑯 (dropping the jump label for this section). To analyze it, we need to consider the frequency domain representation instead of the time domain. Decompose the operator in the energy basis, and regroup in terms of the energy change (called the Bohr frequencies 𝜈 ∈ 𝐵(𝑯) := {𝐸𝑖 − 𝐸 𝑗 | 𝐸𝑖 , 𝐸 𝑗 ∈ Spec(𝑯)}) ∑︁ ∑︁ ∑︁ 𝑨 := 𝑷 𝐸2 𝑨𝑷 𝐸1 = 𝑷 𝐸2 𝑨𝑷 𝐸1 𝜈∈𝐵(𝑯) 𝐸 2 −𝐸 1 =𝜈 𝐸 1 ,𝐸 2 ∈spec(𝑯) =: ∑︁ 𝜈∈𝐵(𝑯) 𝑨𝜈 such that ( 𝑨𝜈 ) † = ( 𝑨† )−𝜈 , 149 where 𝑷 𝐸 denotes the orthogonal projector onto the eigensubspace of 𝑯 with exact energy 𝐸. This decomposition naturally solves the Heisenberg evolution ∑︁ ei𝑯𝑡 𝑨e−i𝑯𝑡 = 𝑨𝜈 ei𝜈𝑡 since [𝑯, 𝑨𝜈 ] = 𝜈 𝑨𝜈 . 𝜈∈𝐵 Indeed, the energy differences 𝜈 ∈ 𝐵 (shorthand of 𝐵(𝑯)) naturally arise from the commutator (as opposed to the absolute energies 𝐸 ∈ spec(𝑯)). At first glance, the reference to the exact energies seems unphysical as each of them individually requires a long (likely exponential) Hamiltonian simulation time to access algorithmically. Fortunately, all that we are manipulating are the smooth weights on these Bohr frequencies; indeed, the operator Fourier Transform can be conveniently expressed by ∫ ∞ ∑︁ 1 𝑨ˆ 𝑓 (𝜔) = √ ei𝑯𝑡 𝑨e−i𝑯𝑡 e−i𝜔𝑡 𝑓 (𝑡)d𝑡 = 𝑨𝜈 𝑓ˆ(𝜔 − 𝜈), 2𝜋 −∞ 𝜈∈𝐵 ∫ ∞ where 𝑓ˆ(𝜔) = √1 −∞ 𝑓 (𝑡)e−i𝜔𝑡 d𝑡 is the Fourier Transform of the filter function 2𝜋 𝑓 (𝑡). Our choice of 𝑓 (𝑡) is √︃ √︁ −𝜎𝐸2 𝑡 2 𝑓 (𝑡) := e 𝜎𝐸 2/𝜋 such that ! ∫ ∞ 2 1 𝜔 ˆ𝑓 (𝜔) = √︃ | 𝑓 (𝑡)| 2 d𝑡 = 1, and exp − 2 √ 4𝜎𝐸 −∞ 2𝜎𝐸 2𝜋 ˆ therefore 𝑨(𝜔) becomes simply 1 ˆ 𝑨(𝜔) = √︃ √ 2𝜎𝐸 2𝜋 ! (𝜔 − 𝜈) 2 exp − 𝑨𝜈 . 4𝜎𝐸2 𝜈∈𝐵 ∑︁ The above uses the Gaussian integrals, which will also be constantly recalled. Fact 3.2.1 (Gaussian integrals). For any 𝑏 ∈ C and 𝜎 > 0, we have that ∫ ∞ − ( 𝜔−𝑏) 2 √ 2𝜎 2 d𝜔 = 2𝜋𝜎. e −∞ We can think of the width 𝜎𝐸 as the uncertainty in energy, which scales inversely proportionally to the time width ∼ 𝜎𝐸−1 . Exact detailed balance from that of the transition part Our notion of detailed balance for Lindbladians is analogous to its classical cousin, ensuring a stationary state 𝝆. For the mathematical audience, we should mention that other forms of quantum detailed balance have also been studied (see section 3.9), but we will dominantly focus on the following as it appears to be especially nice. 150 Definition 3.2.1 (Kubo–Martin–Schwinger detailed balance condition). For a normalized, full-rank state 𝝆 ≻ 0, we say that an super-operator L satisfies 𝝆-detailed balance (or 𝝆-DB in short) if L † [·] = 𝝆 −1/2 L [𝝆 1/2 · 𝝆 1/2 ] 𝝆 −1/2 , or equivalently, whenever the associated discriminant is self-adjoint with respect to 𝝆, i.e., D ( 𝝆, L) := 𝝆 −1/4 L [𝝆 1/4 · 𝝆 1/4 ] 𝝆 −1/4 = 𝝆 1/4 L † [𝝆 −1/4 · 𝝆 −1/4 ] 𝝆 1/4 = D ( 𝝆, L) † . In the above, (L) † denotes the adjoint for superoperators with respect to trace (i.e., the Hilbert Schmidt inner product). One may interpret the conjugation with the state as a similarity transformation under which the Lindbladian becomes Hermitian (w.r.t. to the KMS inner product). The above generalizes classical detailed balance by considering super-operators and permitting the stationary distribution to be an operator. Proposition 3.2.1 (Fixed point). If a Lindbladian L is 𝝆-detailed-balanced, then L [𝝆] = 0. Recently, quantum approximate detailed balance has also been studied in the precursor of this work [40, Section II.A], discussing nonasymptotic error bounds relating mixing times to fixed point error. Exact detailed balance gives a much simpler conceptual picture. Still, we may again need to recall approximate detailed balance for non-fine-tuned Lindbladians (such as those from Nature) or amid intermediate steps of analysis (such as when truncating the radius of the quasi-local jumps). At first glance, the detailed balance condition is merely a linear equation that can be solved abstractly. However, the difficulty arises due to two additional constraints. • (Complelete Positivity.) Lindbladians have a particular quadratic dependence on the Lindblad operators to ensure complete positivity and trace preservation of e𝑡L for any 𝑡. • (Efficiency.) The Lindbladian (i.e., the block-encoding for the jumps and the Hamiltonian) must be efficiently implemented using a limited Hamiltonian simulation time. 151 The main challenge is to satisfy both constraints simultaneously. Indeed, Davies’ generator [56, 58] satisfies the first but not the second because it uses an infinitetime operator Fourier Transform; using QSVT, one might be able directly to implement the Boltzmann weight smoothly at moderate costs, but it may break the Lindbladian structure. Our approach begins by isolating the “transition” part of the Lindbladian (3.1) with abstract Lindblad operators 𝑳 𝑗 L [·] := −i[𝑩, ·] + 1 ∑︁ † 𝑳 𝑗 · 𝑳 †𝑗 − { 𝑳 𝑗 𝑳 𝑗 , ·}, 2 𝑗 𝑗 | {z } | {z } ∑︁ 𝑹:= T := where 𝑩 and 𝑹 are both Hermitian. This decomposition is helpful because conjugating with the stationary state preserves the form of the transition part ! ′ √ −1 ∑︁ ′ √ −1 ∑︁ √ √ 𝝆 𝑳 𝑗 ( 𝝆 · 𝝆) 𝑳 †𝑗 𝝆 = 𝑳 𝑗 (·) 𝑳 𝑗† . 𝑗 𝑗 However, the commutator 𝑩 and anti-commutator terms 𝑹 mix with each other under conjugation with Gibbs state. Based on the above observation, our recipe for constructing a detailed-balanced Lindbladian consists of three steps: 1. Guess a set of Lindblad operators 𝑳 𝑗 such that the transition part (which is Í ∫∞ T = 𝑎 −∞ 𝛾(𝜔) 𝑨ˆ 𝑎 (𝜔) · 𝑨ˆ 𝑎 (𝜔) † d𝜔 in our case) obeys detailed balance. 2. According to the transition part T , determine the decay part parameterized by 𝑹. This gives a purely dissipative Lindbladian. 3. According to the decay part 𝑹, tailor the commutator term 𝑩 to ensure detailed balance. Remarkably, such a 𝑩 always exists, can be found explicitly, and is essentially unique. Of course, whether the map is efficiently implementable is a separate question.8 To simplify the presentation, we introduce the following notation for conjugating any full-rank state 𝝆: Γ𝝆 [·] := 𝝆 1/2 (·) 𝝆 1/2 and Λ 𝝆 [·] := 𝝆 −1/2 (·) 𝝆 1/2 . Observe that for Hermitian operator 𝑿, we have the identities Γ𝝆 [ 𝑿] † = Γ𝝆 [ 𝑿] and Λ 𝝆 [ 𝑿] † = Λ−1 𝝆 [ 𝑿]. When the context is clear, we will omit subscript 𝝆. 8This is inspired by a related ongoing work [78] at an early stage. 152 The main calculation of this section is summarized as follows. For any 𝑹, we can give a general solution for the coherent term 𝑩; this calculation is possible because the coherent term is not constrained by the complete-positivity structure of Lindbladians and only needs to be Hermitian. Lemma 3.2.1 (Prescribing the coherent term). For any full-rank state 𝝆 and Hermitian operator 𝑹, there exists a unique Hermitian operator 𝑩 (up to adding any scalar multiples of the identity 𝑰) such that the super-operator 1 S [·] := −i[𝑩, ·] − {𝑹, ·} 2 satisfies 𝝆-DB.9 For a Gibbs state 𝝆 ∝ exp(−𝛽𝑯), we can express the solution decomposed according to the Bohr frequencies 𝜈 ∈ 𝐵 as   i ∑︁ 𝛽𝜈 𝑹𝜈 . 𝑩= tanh 2 𝜈∈𝐵 4 The above can be applied to a purely dissipative Lindbladian, where the transition part already satisfies 𝝆-DB. Corollary 3.2.1 (𝝆-DB Lindbladians). Suppose we have a purely-dissipative L 𝑑𝑖𝑠𝑠 Lindbladian such that the transition part satisfies 𝝆-DB for a full-rank state 𝝆 Γ𝝆−1 ◦ T ◦ Γ𝝆1 = T † , then we can accordingly prescribe 𝑩 such that −i[𝑩, ·] + L 𝑑𝑖𝑠𝑠 satisfies 𝝆-DB. Proof of Lemma 3.2.1. Let 𝑲 := 𝑩 − 2i 𝑹 and observe that 1 S [·] := −i[𝑩, ·] − {𝑹, ·} = −i𝑲 (·) + i(·)𝑲 † 2 1 S † [·] := i[𝑩, ·] − {𝑹, ·} = i𝑲 † (·) − i(·)𝑲 2 using that 𝑩 and 𝑹 are Hermitian. Then,   † −1 † −1 † S [·] − Γ ◦ S ◦ Γ[·] = i𝑲 (·) − i(·)𝑲 + iΓ 𝑲Γ[·] − Γ[·]𝑲     = i 𝑲 † + Λ[𝑲] (·) − i(·) 𝑲 + (Λ[𝑲]) † (using that Γ−1 𝑲Γ[·] = Λ[𝑲] (·)) =: i(𝑸(·) − (·)𝑸 † ), 9Actually, our proof shows an even stronger statement: for any Hermitian 𝑭 commuting with 𝝆 there is a unique 𝑩 (up to an additive term proportional to 𝑰) such that S † [·] − Γ −1 ◦S ◦ Γ[·] = −i[𝑭, ·]. The only change is that one should set 𝑩0 := − 12 𝑭. This relates to a more general notion of detailed balance that allows for a unitary drift Definition 3.9.1, see also [69, Section 5]. This might be useful for breaking degeneracies of the state 𝝆 (or the Hamiltonian 𝑯). 153 where we define 𝑸 := 𝑲 † + Λ[𝑲]. To ensure quantum detailed balance, we need the RHS to vanish 𝑸(·) − (·)𝑸 † = 0 ⇐⇒ 𝑸 = 0 + 𝜆𝑰 for 𝜆 ∈ R, that is, 𝑸 vanishes up to a real multiple of the identity 𝜆𝑰; for simplicity, the identity part can be dropped for now and added back. Since 𝝆 has full rank, we can assume without loss of generality that 𝝆 ∝ exp(−𝛽𝑯) for some 𝑯. Now we compute 𝑸 = 𝑲 † + Λ[𝑲] i = (1 + Λ) 𝑩 + (1 − Λ) 𝑹 2 ∑︁ i = (1 + e 𝛽𝜈/2 ) 𝑩𝜈 + (1 − e 𝛽𝜈/2 ) 𝑹𝜈 ., 2 𝜈∈𝐵 (using that Λ(𝑩𝜈 ) = e 𝛽𝜈/2 𝑩𝜈 ) where we denote 𝐵 := 𝐵(𝑯) the set of Bohr frequencies of 𝑯. Since the operators 𝑩𝜈 (and 𝑹𝜈 ) are linearly independent for different Bohr frequencies,   𝛽𝜈 i 𝑸=0 ⇐⇒ 𝑩𝜈 = tanh 𝑹𝜈 for each 𝜈 ∈ 𝐵. 2 4 2𝑥 ) (using that tanh(𝑥) = ee2𝑥 −1 +1 In particular, 𝑩0 = 0 since tanh(0) = 0. Finally, since 𝑹 is Hermitian and tanh(𝜔) is an odd function, we have that 𝑩 is Hermitian as well by   ∑︁ 𝛽𝜈 i ∑︁ † tanh − (𝑩𝜈 ) = ( 𝑹𝜈 ) † . (Using − tanh(𝑥) = tanh(−𝑥)) 2 4 𝜈∈𝐵 𝜈∈𝐵   i ∑︁ 𝛽𝜈 𝑹−𝜈 (Using that 𝑹 = 𝑹 † implies ( 𝑹𝜈 ) † = 𝑹−𝜈 ) = tanh − 2 𝜈∈𝐵 4   ∑︁ i ∑︁ 𝛽𝜈 = tanh 𝑹𝜈 = 𝑩𝜈 . (Change of variables 𝜈 → −𝜈) ■ 2 𝜈∈𝐵 4 𝜈∈𝐵 Adding the identity part 𝜆𝑰 to conclude the proof. Exact detailed balance for Gaussian weights and their linear combinations In this section, we carry out the abstract recipe for our advertised Lindbladian to prove the exact detailed balance condition (Definition 3.2.1). We begin with the transition part and then solve for the coherent term. 154 Exact detailed balance of the transition part First, we show that the Gaussian ansatz indeed leads to detailed balance for the “transition” part Γ𝝆−1 ◦ T ◦ Γ𝝆 = T † . It is instructive to rewrite the abstract equation above in terms of Bohr-frequencies. Let (·) be any input matrix, then the transition part of our Lindbaldian reads ∑︁ ∫ ∞ T = 𝛾(𝜔) 𝑨ˆ𝑎 (𝜔) (·) 𝑨ˆ𝑎 (𝜔) † d𝜔 𝑎∈𝐴 −∞ ∑︁ ∑︁ ∫ ∞ − ( 𝜔+𝜔2𝛾 ) 2 − ( 𝜔−𝜈21 ) 2 − ( 𝜔−𝜈22 ) 2 1 = e 2𝜎𝛾 e 4𝜎𝐸 e 4𝜎𝐸 𝑨𝜈𝑎1 (·)( 𝑨𝜈𝑎2 ) † d𝜔 √ 2𝜎𝐸 2𝜋 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 −∞ ∑︁ ∑︁ (𝜔 ,𝜎 ) =: 𝛼𝜈1 ,𝜈𝛾 2 𝛾 𝑨𝜈𝑎1 (·)( 𝑨𝜈𝑎2 ) † . (3.11) 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 Since the bilinear expression holds for any input (·), taking traces on both sides yields that ∑︁ ∫ ∞ ∑︁ ∑︁ (𝜔 ,𝜎 ) 𝛾(𝜔) 𝑨ˆ𝑎 (𝜔) † 𝑨ˆ𝑎 (𝜔)d𝜔 = 𝛼𝜈1 ,𝜈𝛾 2 𝛾 ( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 . 𝑎∈𝐴 −∞ 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 In terms of Bohr frequencies, the exact detailed balance condition is a certain symmetry of the coefficient matrix 𝛼 (𝜔 𝛾 ,𝜎𝛾 ) . Proposition 3.2.2 (Detailed balance in the Energy domain). Consider a superoperator parameterized by a Hamiltonian 𝑯, 𝛽, and a set of operators including its adjoints { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴}: ∑︁ ∑︁ T = 𝛼𝜈1 ,𝜈2 𝑨𝜈𝑎1 (·)( 𝑨𝜈𝑎2 ) † such that 𝛼𝜈1 ,𝜈2 = 𝛼−𝜈2 ,−𝜈1 e−𝛽(𝜈1 +𝜈2 )/2 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 for each 𝜈1 , 𝜈2 ∈ 𝐵 Then, Γ𝝆−1 ◦ T ◦ Γ𝝆1 = T † . Indeed, one recovers the classical detailed balance condition for inputs diagonal in the energy basis. However, the quantum detailed balance condition also constrains the amplitudes between off-diagonal matrix elements (Figure 3.1). 155 Proof. We can directly calculate √︃ √︃ ∑︁ ∑︁ √ 𝑎 √ 𝑎 † Γ𝝆−1 ◦ T ◦ Γ𝝆1 = 𝛼𝜈1 ,𝜈2 𝝆 −1 𝑨 𝝆 (·) 𝝆 ( 𝑨 ) 𝝆 −1 𝛽 𝛽 𝜈1 𝜈2 𝛽 𝛽 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 = ∑︁ ∑︁ = ∑︁ ∑︁ 𝛽 𝛽 𝛼𝜈1 ,𝜈2 e 2 𝜈1 𝑨𝜈𝑎1 (·)( 𝑨𝜈𝑎2 ) † e 2 𝜈2 (since 𝝆 𝛽 ∝ e−𝛽𝑯 ) 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 𝛼−𝜈2 ,−𝜈1 𝑨𝜈𝑎1 (·)( 𝑨𝜈𝑎2 ) † (since 𝛼𝜈1 ,𝜈2 e 𝛽 (𝜈1 +𝜈2 ) 2 = 𝛼−𝜈2 ,−𝜈1 ) 𝑎∈𝐴 𝜈 ,𝜈 ∈𝐵 = 2 ∑︁ 1∑︁ 𝛼−𝜈2 ,−𝜈1 (( 𝑨𝑎† )−𝜈1 ) † (·)( 𝑨𝑎† )−𝜈2 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 (since ( 𝑨𝜈 ) † = ( 𝑨† )−𝜈 ) = ∑︁ ∑︁ 𝑎 𝑎 𝛼−𝜈2 ,−𝜈1 ( 𝑨−𝜈 ) † (·) 𝑨−𝜈 1 2 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 (since { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴}) = ∑︁ ∑︁ 𝛼𝜈2 ,𝜈1 ( 𝑨𝜈𝑎1 ) † (·) 𝑨𝜈𝑎2 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 (since 𝐵 = spec(𝑯) − spec(𝑯) = −𝐵) = ∑︁ ∑︁ 𝛼𝜈1 ,𝜈2 ( 𝑨𝜈𝑎2 ) † (·) 𝑨𝜈𝑎1 = T † . (relabelling 𝜈1 ↔ 𝜈2 ) ■ 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 While the above representation explicitly addresses the energy basis, we note that positivity becomes obscured as the left and right energy labels 𝜈1 , 𝜈2 can differ. The positivity now becomes implicit in the coefficient matrix. Proposition 3.2.3 (Positive semi-definite). If the coefficients 𝛼𝜈1 ,𝜈2 as a matrix 𝜶 is positive-semi-definite 𝜶 ≥ 0,
∗ (and thus Hermitian 𝛼𝜈1 ,𝜈2 = 𝛼𝜈2 ,𝜈1 ), then   ∑︁ ∑︁ 1 𝑎 𝑎 † 𝑎 † 𝑎 𝛼𝜈1 ,𝜈2 𝑨𝜈1 (·)( 𝑨𝜈2 ) − {( 𝑨𝜈2 ) 𝑨𝜈1 , ·} 2 𝑎∈𝐴 𝜈 ,𝜈 ∈𝐵 1 gives a Lindbladian. 2 Positivity indeed holds for (3.11) and can be seen by the integral form of the coefficients. To conclude this section, it remains to verify that the coefficients arising from Gaussian indeed satisfy the symmetry. Lemma 3.2.2 (Exact “skew-symmetry” of coefficients). For each 𝜔 𝛾 , 𝜎𝛾 , the (𝜔 ,𝜎 ) coefficients 𝛼𝜈1 ,𝜈𝛾 2 𝛾 defined by (3.11) factorize ! ! (𝜈1 + 𝜈2 + 2𝜔 𝛾 ) 2 𝜎𝛾 (𝜈1 − 𝜈2 ) 2 (𝜔 𝛾 ,𝜎𝛾 ) 𝛼𝜈1 ,𝜈2 = √︃ · exp − · exp − ,(3.12) 2 + 𝜎2) 2 8(𝜎 8𝜎 2 2 𝛾 𝐸 𝐸 2 𝜎 +𝜎 𝐸 𝛾 156 and have a certain “skew-symmetry” under negation and transpose (𝜔 ,𝜎 ) (𝜔 ,𝜎 ) 𝛼𝜈1 ,𝜈𝛾 2 𝛾 = 𝛼−𝜈2𝛾,−𝜈𝛾1 e−𝛽(𝜈1 +𝜈2 )/2 2𝜔 𝛾 𝛽 := for 𝜎𝐸2 + 𝜎𝛾2 . (3.13) Proof. We directly calculate the Gaussian integrals in (3.11), preserving the quadratic nature of the exponents. ! ! ! ∫ ∞ 𝜔2𝛾 𝜈12 + 𝜈22 𝜔 𝛾 𝜈1 + 𝜈2 1 1 1 𝜔2 (𝜔 𝛾 ,𝜎𝛾 ) 𝛼𝜈1 ,𝜈2 = · + −𝜔· − − − d𝜔 exp − √ 2 𝜎𝛾2 𝜎𝐸2 𝜎𝛾2 2𝜎𝐸2 4𝜎𝐸2 2𝜎𝛾2 𝜎𝐸 8𝜋 −∞  ∫ ∞ © 𝜔+ ­ 1 = exp­­− √ 𝜎𝐸 8𝜋 −∞ ­ «  1 + 𝜎12 𝜎𝛾2 𝐸 2  −1 𝜔𝛾 1 +𝜈2 − 𝜈2𝜎 2 𝜎𝛾2 𝐸 1 + 𝜎12 𝜎𝛾2 𝐸 −1  2  + = √︂ · exp − 𝜎𝐸2 2 𝜎2 + 1 −1 2 !2 ª 𝜔2𝛾 ® 𝜈12 + 𝜈22 𝜈1 + 𝜈2 ®d𝜔 − − − 𝜎𝛾2 2𝜎𝐸2 4𝜎𝐸2 2𝜎𝛾2 ®® 𝜔𝛾 ¬ © 𝜔 𝛾 𝜈1 + 𝜈2 1 1 1  = · √︃ exp­­  − 2 1 1 2𝜎𝐸 1 1 𝜎 2𝜎𝐸2 𝛾 + 2 𝜎2 + 𝜎2 𝜎𝛾2 𝜎𝐸2 𝛾 « 𝐸 1 1 + 𝜎12 𝜎𝛾2 𝐸 (𝜈1 + 𝜈2 + 2𝜔 𝛾 ) 2 8(𝜎𝐸2 + 𝜎𝛾2 ) ! !2 − 𝜈12 + 𝜈22 (𝜈1 − 𝜈2 ) 2 · exp − 8𝜎𝐸2 4𝜎𝐸2 − 𝜔2𝛾 ª ® 2𝜎𝛾2 ® ¬ (by Fact 3.2.1) ! 𝛾 (since 2(𝜈12 +𝜈22 ) = (𝜈1 +𝜈2 ) 2 + (𝜈1 −𝜈2 ) 2 ) (𝜔 ,𝜎 ) = 𝛼−𝜈2𝛾,−𝜈𝛾1 e−𝛽(𝜈1 +𝜈2 )/2 . (due to our choice of 𝛽) ■ We see that we may tune the parameters to match a desirable exponent 𝛽. Remarkably, the width 𝜎𝐸 can be finite (i.e., do not scale linearly the precision 1𝜖 like in metrology) while retaining exact detailed balance; a reasonable choice is, e.g., 𝜔 𝛾 = 𝜎𝐸 = 𝜎𝛾 = 1 . 𝛽 This will imply that the algorithmic cost for implementing the Gaussian weighted operator Fourier Transform will only need to scale with 𝛽 (and polylogarithmically with the precision due to discretization and truncation error!) However, the Gaussian transition weight comes with the price of a narrower band of transitions peaked at 𝜔 𝛾 ± O (𝜎𝛾 ), which might result in a substantially increased mixing time compared to, e.g., Metropolis weight 𝛾(𝜔) = min(1, e−𝛽𝜔 ); we will revisit this issue by taking 157 linear combinations of Gaussians at section 3.2 since all our calculation are linear. For clarity, we first focus on the Gaussian weights. Why does Gaussian interplay so perfectly with quantum detailed balance? In section 3.8, we attempted to derive Gaussian from the first principle. Indeed, Gaussians are very natural if we impose several conditions on the function. Adding the unitary term Now that the “transition” part satisfies the detailed balance condition exactly, we proceed to complete the Lindbaldian by adding the “decay” part and the “coherent” part. The decay part is fixed by trace-preserving; Lemma 3.2.1 then uniquely prescribes the required coherent term, which we display as follows in the frequency domain. The explicit form will be useful for implementation. Corollary 3.2.2 (An exactly detailed-balanced Lindbladian with Gaussian filtering). The Lindbladian   ∑︁ ∑︁ ∑︁ 1 𝑎 † 𝑎 𝑎 𝑎 † L [·] := −i [𝑩𝜈 , ·] + 𝛼𝜈1 ,𝜈2 𝑨𝜈1 (·)( 𝑨𝜈2 ) − {( 𝑨𝜈2 ) 𝑨𝜈1 , ·} 2 𝜈∈𝐵 𝑎∈𝐴 𝜈 ,𝜈 ∈𝐵 1 2 corresponding to a self-adjoint set of jump operators { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴}, parametrized by coefficients 𝛼𝜈1 ,𝜈2 ∈ C satisfying 𝛼𝜈∗1 ,𝜈2 = 𝛼𝜈2 ,𝜈1 and 𝛼𝜈1 ,𝜈2 e 𝛽 (𝜈1 +𝜈2 ) 2 = 𝛼−𝜈2 ,−𝜈1 , and amended by the coherent terms ∑︁ ∑︁ tanh(−𝛽(𝜈1 − 𝜈2 )/4) 𝛼𝜈1 ,𝜈2 ( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 𝑩𝜈 := 2i 𝑎∈𝐴 𝜈1 −𝜈2 =𝜈 | {z } 𝜈 ,𝜈 ∈𝐵 1 2 (3.14) 𝑓ˆ(𝜈1 ,𝜈2 ):= satisfies 𝝆 𝛽 -detailed balance. Proof. Apply Lemma 3.2.1. Note that the operator ∑︁ 𝑷 𝐸 𝑘 𝑨𝑎† 𝑷 𝐸𝑖 𝑨𝑎 𝑷 𝐸 𝑗 ( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 = 𝐸 𝑖 −𝐸 𝑗 =𝜈1 ,𝐸 𝑖 −𝐸 𝑘 =𝜈2 must have the energy difference contained in the set of Bohr frequencies 𝐵 𝜈1 − 𝜈2 = 𝐸𝑖 − 𝐸 𝑗 − (𝐸𝑖 − 𝐸 𝑘 ) for some 𝐸𝑖 , 𝐸 𝑗 , 𝐸 𝑘 ∈ 𝑆 𝑝𝑒𝑐(𝑯) = 𝐸 𝑘 − 𝐸𝑖 ∈ 𝐵. ■ 158 The above corollary essentially leads to Theorem 3.1.1 but is written in the Bohrfrequency decomposition. Proof of Theorem 3.1.1. Combine Proposition 3.2.3, Lemma 3.2.2, Proposition 3.2.1, and Corollary 3.2.2 to conclude the proof. ■ Linear combination of Gaussians Can we go beyond Gaussians? As we discussed, the Gaussians have a narrower band of transitions; it would be desirable to lift this restriction to accelerate the mixing time. In this section, we give a family of filters by exploiting the freedom to tune the Gaussian parameters (𝜔 𝛾 , 𝜎𝐸 , 𝜎𝛾 ) and taking a linear combination of Gaussians. Corollary 3.2.3 (Linear combination of Gaussians). Fix 𝜎𝐸 and 𝑔 ∈ ℓ1 (R) and set 2 ∫∞ − ( 𝜔+𝑥 ) 2 4𝑥/𝛽−2𝜎 (𝑔) 𝐸 d𝑥.10 Then, analogous to (3.11), the coefficients as a 𝛾 (𝜔) := 𝛽 𝜎𝐸2 𝑔(𝑥)e 2 linear combination over integration variable 𝑥 ∫ ∞ √︃ (𝜔 𝛾 ,𝜎𝛾 ) (𝑔) (3.15) 𝛼𝜈1 ,𝜈2 := 𝛽 𝜎2 𝑔(𝑥)𝛼𝜈1 ,𝜈2 d𝑥 for (𝜔 𝛾 (𝑥), 𝜎𝛾 (𝑥)) = (𝑥, 2𝑥/𝛽 − 𝜎𝐸2 ), 𝐸 2 (𝑔)∗ (𝑔) (𝑔) (𝑔) satisfy the symmetries 𝛼𝜈1 ,𝜈2 = 𝛼𝜈2 ,𝜈1 and 𝛼𝜈1 ,𝜈2 = 𝛼−𝜈1 ,−𝜈2 e−𝛽(𝜈1 +𝜈2 )/2 for each 𝜈1 , 𝜈2 ∈ R. If 𝑔(𝑥) ≥ 0 for each 𝑥, then we also have that 𝜶 (𝒈) ≥ 0, however, this is not a necessary condition. (𝜔 ,𝜎 ) Proof. Recall the meaning of the superscripts 𝛼𝜈1 ,𝜈𝛾 2 𝛾 . The proof is merely the linearity of symmetries and that convex combination preserves the cone of positive semidefinite matrices. ■ − ( 𝜔+𝜔 𝛾 ) 2 2 2𝜎𝛾 To widen the band of transitions, a natural choice is to weigh each Gaussian e with its inverse ℓ1 -weight. Surprisingly, this leads to filters that resemble the Metropolis and Glauber weights; while other choices are plausible, we spell out the calculation for this as a natural representative. 10 Note that in principle we could also vary 𝜎 , but that would complicate both the analysis and the 𝐸 implementation due to the required adjustments to the parameters of the performed operator Fourier Transform. 159 Proposition 3.2.4 (Metropolis and Glauber-like filters). Setting 𝑔(𝜔 𝛾 ) = √ 1 2𝜋𝜎𝛾 = 1 √︃ yields 2𝜔 𝛾 2𝜋( 𝛽 −𝜎𝐸2 ) ∫ 𝛽 𝜎𝐸2 + 𝑠2 − ( 𝜔+𝑥 ) 2 4𝑥/𝛽−2𝜎 2 𝐸 e d𝑥 √︃ 2𝜋(2𝑥/𝛽 − 𝜎𝐸2 )   " ! ! ! !# 𝛽 𝜎2 𝛽 𝜎2 −𝛽 max 𝜔+ 2 𝐸 ,0 1 𝛽 𝜔+ 2 𝐸 𝛽𝜎𝐸2 𝛽𝜎𝐸2 𝑠 𝛽 𝑠 𝛽 =e erf + 𝜔 + · erf − 𝜔 + +1 +e −1 , 2 2 𝑠 2 2 𝑠 2 | {z } 𝛾𝜎(𝑠)𝐸 (𝜔) := 2 𝛽 𝛽 𝜎2 𝐸 2 ≤1 which, in the 𝑠 → ∞ limit, coincides with the Metropolis weight shifted by   𝛽 𝜎2 −𝛽 max 𝜔+ 2 𝐸 ,0 𝛾𝜎(∞) (𝜔) = e 𝐸 Restricting the above 𝑔(𝜔 𝛾 ) to the interval 𝜔 𝛾 ∈ smooth variant of (3.16)  (3.16) . 3𝛽𝜎𝐸2 2 ,∞ 𝛽𝜎𝐸2 2 :  results in the following   𝛽 𝜎2 −𝛽 max 𝜔+ 2 𝐸 ,0 𝛾˜ 𝜎(∞) (𝜔) = e · (3.17) 𝐸 !! !!# " 2 𝛽𝜎 𝛽 𝜔+ 2 𝐸 𝛽𝜎𝐸2 𝛽𝜎𝐸2 1 𝛽𝜎𝐸2 1 1 𝛽𝜎𝐸2 erfc , − 𝜔+ +e + 𝜔+ erfc 2 𝜎𝐸 2 2 𝜎𝐸 2 2 | {z } ≤1 which resembles the Glauber filter also shifted by 𝛽𝜎𝐸2 2 (Figure 3.3). Proof. We directly found the above by Mathematica. 3.3 ■ Algorithms This section presents efficient quantum algorithms for simulating the advertised Lindbladian and the associated parent Hamiltonian. The former mainly builds on black-box Lindbladian simulation algorithms [43, 49, 112], more precisely their improved variant described in [40, Theorem III.2] whose complexity boils down to constructing block encoding for the Lindblad operators (Definition 3.3.1); the latter merely requires block-encoding the parent Hamiltonian, which feeds into quantum 160 1 exp(−𝛽 max(𝜔, 0)) 𝛾 𝛽(∞) −1 (𝜔) in (3.16) (∞) 𝛾˜ 𝛽 −1 (𝜔) in (3.17) 𝛽𝜔 1/(1 + exp(𝛽𝜔)) 𝛾(𝜔) 1 2 −4 −3 −2 −1 1 2 3 4 Figure 3.3: A plot of the filter functions 𝛾(𝜔) for Metropolis, Glauber and our filters arising from Gaussian linear combination(3.16)-(3.17) (with 𝜎𝐸 = 1𝛽 ). simulated annealing (see [40, Appendix G] for a modern discussion) to prepare the purified Gibbs state. Thus, the main algorithmic contribution is to assemble the block encodings associated with our synthetic Lindbladian (3.1) and our parent Hamiltonians (Theorem 3.1.3). The frequency domain representation (section 3.2), which is natural in the context of analyzing quantum detailed balance, is less instructive for algorithmic implementation. Indeed, addressing the exact energy eigenstates (or the exact Bohr frequencies) is generally inefficient. Nevertheless, the algorithmic task becomes straightforward in the time domain representation. Indeed, our Lindbladian can be expressed in terms ∫ of weighted time integrals (·)d𝑡 of some rapidly decaying functions; a standard Linear-Combination-of-Unitary argument (under suitable discretization) leads to the algorithmic complexity in terms of controlled Hamiltonian simulation time. For the Lindbladian (3.1), block-encodings for the dissipative part are already constructed in [40, Section III.B]. Thus, it remains to construct the coherent term; for the parent Hamiltonian, we will need to construct block encodings from scratch, but the manipulations are analogous. In the following sections, we first present the timedomain expressions (section 3.3-3.3), which immediately yield the corresponding block encodings (section 3.3) and the overall complexities (section 3.3-3.3). Time-domain representation of our Lindbladians Applying a two-dimensional Fourier Transform for the coherent term (3.14) leads to the following time-domain representation where LCU techniques are naturally applicable. See section 3.5 for the calculations. Corollary 3.3.1 (Coherent term for the Gaussian case). For each 𝛽 > 0 and parameters 𝜎𝐸 = 𝜎𝛾 = 𝜔 𝛾 = 1𝛽 , the coherent term 𝑩 (3.14) corresponding to the 161  Gaussian weight 𝛾(𝜔) = exp − (𝛽𝜔+1) 2 𝑩 := ∑︁ ∫ ∞ 𝑎∈𝐴 −i𝛽𝑯𝑡  2 ∫ ∞ 𝑏 1 (𝑡)e −∞ can be written as ′ ′ ′ ′  𝑏 2 (𝑡 ) 𝑨 (𝛽𝑡 ) 𝑨 (−𝛽𝑡 )d𝑡 ei𝛽𝑯𝑡 d𝑡, (3.18) 𝑎† 𝑎 −∞ where √ 1 𝑏 1 (𝑡) := 2 𝜋e 8 ∥𝑏 1 ∥ 1 < 1 such that (3.19) 𝑏 2 (𝑡) := ∥𝑏 2 ∥ 1 < such that   1 2 ∗𝑡 sin(−𝑡) exp −2𝑡 cosh(2𝜋𝑡)  1 2𝜋 √︂   1 exp −4𝑡 2 − 2i𝑡 𝜋 1 . 16 (3.20) Indeed, we can verify that 𝑩 is Hermitian by ∫ ∞  ∑︁ ∫ ∞ † ∗ −i𝛽𝑯𝑡 ∗ ′ 𝑎† ′ 𝑎 ′ ′ i𝛽𝑯𝑡 𝑩 = 𝑏 1 (𝑡)e 𝑏 2 (𝑡 ) 𝑨 (−𝛽𝑡 ) 𝑨 (𝛽𝑡 )d𝑡 e d𝑡 −∞ −∞ ∑︁ ∫ ∞ ∫ ∞ 𝑎∈𝐴 = 𝑏 1 (𝑡)e −∞ −∞ ∑︁ ∫ ∞ ∫ ∞ 𝑎∈𝐴 = −i𝛽𝑯𝑡 𝑏 1 (𝑡)e−i𝛽𝑯𝑡 ′ ′ ′  𝑏 2 (−𝑡 ) 𝑨 (−𝛽𝑡 ) 𝑨 (𝛽𝑡 )d𝑡 ei𝛽𝑯𝑡 d𝑡 𝑎† 𝑎 (By 𝑏 ∗1 (𝑡) = 𝑏 1 (𝑡) and 𝑏 ∗2 (𝑡) = 𝑏 2 (−𝑡))  ′ 𝑎† ′ 𝑎 ′ ′ i𝛽𝑯𝑡 d𝑡 = 𝑩. 𝑏 2 (𝑡 ) 𝑨 (𝛽𝑡 ) 𝑨 (−𝛽𝑡 )d𝑡 e −∞ −∞ 𝑎∈𝐴 ′ (Change of variable 𝑡 ′ → −𝑡 ′) The explicit weights corresponding to Metropolis weights are slightly more cumbersome due to taming a mild (logarithmic) singularity. We can also verify that 𝑩 𝑀,𝜂 is 𝑀,𝜂∗ 𝑀,𝜂 Hermitian by 𝑏 2 (𝑡) = 𝑏 2 (−𝑡). Corollary 3.3.2 (Approximate coherent term for the Metropolis-like weight). If 𝜎𝐸 = 1𝛽 , then the coherent term 𝑩 𝑀 corresponding to the Metropolis-like weight    1 𝛾 𝑀 (𝜔) = exp −𝛽 max 𝜔 + 2𝛽 , 0 satisfies 𝑀 𝑩 −𝑩 𝑀,𝜂 ≤ ∑︁  𝑎† 𝑨 𝑨 𝑎∈𝐴 𝑎  𝜂𝛽∥𝑯∥  3 min √ , O (𝜂𝛽∥𝑯∥) , 2𝜋 (3.21) where 𝑩 𝑀,𝜂 := ∑︁ ∫ ∞ 𝑎∈𝐴 −∞ −i𝛽𝑯𝑡 ∫ ∞ 𝑏 1 (𝑡)e −∞ 𝑀,𝜂 𝑏 2 (𝑡 ′) 𝑨𝑎† (𝛽𝑡 ′) 𝑨𝑎 (−𝛽𝑡 ′)d𝑡 ′ +  1 𝑎† 𝑎 i𝛽𝑯𝑡 d𝑡, √ 𝑨 𝑨 e 16 2𝜋 (3.22) 162 with 𝑏 1 (𝑡) as in (3.19), and
1 exp −2𝑡 2 − i𝑡 + 1 (|𝑡| ≤ 𝜂)i(2𝑡 + i) √ 𝑡 (2𝑡 + i) 4 2𝜋 1 1 𝑀,𝜂 such that ∥𝑏 2 ∥ 1 < + √ ln(1/𝜂). (3.23) 5 2 2𝜋 Í  𝑎† 𝑎 Further, if 𝑎∈𝐴 𝑨 𝑨 , 𝑯 = 0, we can drop the second term in (3.22) after the ∫∞ integral in 𝑡 ′ since −∞ 𝑏 1 (𝑡) = 0. 𝑀,𝜂 𝑏 2 (𝑡) := See section 3.6 for the proof. After suitable truncation, this merely incurs a logarithmic overhead (∼ log(𝛽∥𝑯∥/𝜖)) to the algorithmic cost for an 𝜖-approximation due to subnormalization 𝑀,1 𝜂 . ∥𝑏 2 ∥1 Time-domain representation of our parent Hamiltonians Recall that the central mathematical object for coherent Gibbs sampling is the discriminant (i.e., the Lindbladian under a similarity transformation) H ( 𝝆, L) := 𝝆 −1/4 L [𝝆 1/4 · 𝝆 1/4 ] 𝝆 −1/4 . This amounts to a mild calculation that transfers the heavy lifting done already in the Lindbladidan context. One adaption for the coherent algorithm is that the fundamental object is not a superoperator but an operator on a doubled Hilbert space. Formally, we define the vectorization of a super-operator by ∑︁ ∑︁ C [·] = 𝛼 𝑗 𝑨 𝑗 [·] 𝑩 𝑗 → C = 𝛼 𝑗 𝑨 𝑗 ⊗ 𝑩𝑇𝑗 , (vectorization) 𝑗 𝑗 where 𝑩𝑇𝑗 denotes the transpose of the matrix 𝑩 𝑗 in the computational basis |𝑖⟩. We use curly fonts C for super-operators and bold fonts C for the vectorized superoperators (which is, a matrix). For a matrix 𝝆, let us denote its vectorized version by | 𝝆⟩ := (𝐼 ⊗ 𝑇 −1 ) 𝝆 where 𝑇 |𝑖⟩ = ⟨𝑖| for each |𝑖⟩. In the time domain, our parent Hamiltonian takes the following form (see section 3.5 for the calculations): ∑︁ ∫ ∞ ∫ ∞ H𝛽 = ℎ− (𝑡− )ℎ+ (𝑡+ ) · 𝑨𝑎 (𝑡+ − 𝑡− ) ⊗ 𝑨𝑎 (−𝑡− − 𝑡+ )𝑇 d𝑡+ d𝑡− 𝑎∈𝐴 −∞ −∞ | 1 + (𝑵 ⊗ 𝑰 + 𝑰 ⊗ 𝑵 ∗ ) . 2 | {z } decay and coherent part {z transition part } 163 Compared with the Lindbladian case, the first term is essentially the transition part under a similarity transformation, and the second term combines the decay and coherent part. Here, we do not care about complete positivity or trace-preserving but merely Hermiticity. The transition part The transition parts for both Gaussian and Metropolis weights are as follows. Corollary 3.3.3  (The2 transition part for Gaussian weights). For the Gaussian weight 𝛾(𝜔) = exp − (𝛽𝜔+1) with 𝜔 𝛾 = 𝜎𝐸 = 𝜎𝛾 = 1/𝛽, the discriminant H 𝛽 is described 2 in the time domain by ℎ+ (𝑡) = 1 −1/4 4𝑡 2 e exp(− 2 ) 𝛽 𝛽 ℎ− (𝑡) = and 1 2𝑡 2 exp(− 2 ) 𝜋𝛽 𝛽 such that ∥ℎ− ∥ 1 , ∥ℎ+ ∥ 1 ≤ 1. (3.24) Corollary 3.3.4 (The transition part for Metropolis weights). For 𝜎𝐸 = 1𝛽 , the    1 Metropolis-like weight 𝛾 𝑀 (𝜔) = exp −𝛽 max 𝜔 + 2𝛽 , 0 yields H 𝛽 described in the time domain by the same ℎ− (𝑡) as in Eq. (3.24) and by 2 2 e−1/8 e−2𝑡 /𝛽 ℎ+ (𝑡) = √ 𝛽 4 2𝜋( 𝑡 22 + 1 ) such that ∥ℎ+ (𝑡)∥ 1 ≤ 1. 16 𝛽 The 𝑵-term The time-domain presentation of the 𝑵 term is as follows. Corollary 3.3.5 (𝑵 term   for Gaussian weights). For each 𝛽, the Gaussian weight (𝛽𝜔+1) 2 𝛾(𝜔) = exp − 2 with 𝜎𝐸 = 𝜎𝛾 = 𝜔 𝛾 = 1𝛽 corresponds to the discriminant where ∫ ∞  ∑︁ ∫ ∞ −i𝛽𝑯𝑡 ′ 𝑎† ′ 𝑎 ′ ′ i𝛽𝑯𝑡 𝑵= 𝑛1 (𝑡)e 𝑛2 (𝑡 ) 𝑨 (𝛽𝑡 ) 𝑨 (−𝛽𝑡 )d𝑡 e d𝑡, 𝑎∈𝐴 −∞ −∞ where    1 √ 1 2 𝑛1 (𝑡) := · 2 𝜋 ∗𝑡 exp −2𝑡 4 cosh(2𝜋𝑡) 1 𝑛2 (𝑡) := 4 · 2 · 2𝜋 √︂ with 𝑏 2 as in (3.20).   1 2 exp −4𝑡 − 2i𝑡 = 8 · 𝑏 2 (𝑡) 𝜋 such that 𝜋 ∥𝑛1 ∥ 1 = √ < 1 4 2 (3.25) such that ∥𝑛2 ∥ 1 < 1 2 164 Corollary 3.3.6 (𝑵 term for Metropolis weights). If 𝜎𝐸 = 1𝛽 , then the Metropolis-like    1 𝑀 weight 𝛾 (𝜔) = exp −𝛽 max 𝜔 + 2𝛽 , 0 corresponds to the discriminant where 𝑵 𝑀 satisfies   ∑︁ 𝜂𝛽∥𝑯∥  𝑀 𝑀,𝜂 𝑎† 𝑎 3 𝑵 −𝑵 ≤ 𝑨 𝑨 min √ , O (𝜂𝛽∥𝑯∥) , 2𝜋 𝑎∈𝐴 where 𝑵 𝑀,𝜂 := ∑︁ ∫ ∞ 𝑎∈𝐴 −i𝛽𝑯𝑡 ∫ ∞ 𝑛1 (𝑡)e −∞ −∞ 𝑀,𝜂 with 𝑛1 (𝑡) as in (3.25), and 𝑛2 𝑀,𝜂 𝑛2 (𝑡 ′) 𝑨𝑎† (𝛽𝑡 ′) 𝑨𝑎 (−𝛽𝑡 ′)d𝑡 ′ + 𝑀,𝜂 = 𝑏2  1 𝑎† 𝑎 i𝛽𝑯𝑡 d𝑡, √ 𝑨 𝑨 e 16 2𝜋 as in (3.23). In the above cases, we can verify that 𝑵 is Hermitian by 𝑛∗1 (𝑡) = 𝑛1 (𝑡) and 𝑛∗2 (𝑡) = 𝑛2 (−𝑡). For 𝑵 𝑀 , we also use that ( 𝑨𝑎† 𝑨𝑎 ) † = 𝑨𝑎† 𝑨𝑎 . Block-encodings Our simulation algorithm extensively uses block encodings that are largely borrowed from [40]. This section aims to instantiate them to state the theorems appropriately, and the curious reader may refer to [40, Section III.B]. Definition 3.3.1 (Block-encoding for Lindbladian). Given a purely irreversible Lindbladian  ∑︁  1 † 1 † † L [𝝆] := 𝑳 𝑗 𝝆𝑳 𝑗 − 𝑳 𝑗 𝑳 𝑗 𝝆 − 𝝆𝑳 𝑗 𝑳 𝑗 , 2 2 𝑗 ∈𝐽 we say that a unitary 𝑼 is a block-encoding for Lindblad operators {𝑳 𝑗 } 𝑗 ∈𝐽 if 11 ∑︁ | 𝑗⟩ ⊗ 𝑳 𝑗 for 𝑏 ≤ 𝑐 ∈ Z+ . ( 0𝑏 ⊗ 𝑰) · 𝑼 · (|0𝑐 ⟩ ⊗ 𝑰) = 𝑗 ∈𝐽 • Block-encoding 𝑽 𝑗𝑢𝑚 𝑝 of the jump operators 𝑨𝑎 in the form of Definition 3.3.1: ∑︁ |𝑎⟩ ⊗ 𝑨𝑎 . ( 0𝑏 ⊗ 𝑰𝑎 ⊗ 𝑰 𝑠𝑦𝑠 )] · 𝑽 𝑗𝑢𝑚 𝑝 · (|0𝑐 ⟩ ⊗ 𝑰 𝑠𝑦𝑠 ) = 𝑎∈𝐴 To implement the discriminant, we also assume access to a block-encoding Í 𝑽 𝑗𝑢𝑚 𝑝𝑇 for the partial transpose 𝑎∈𝐴 |𝑎⟩ ⊗ ( 𝑨𝑎 )𝑇 . 11 In the first register, we could use any orthonormal basis. Sticking to computational basis elements | 𝑗⟩ is just for ease of presentation. Intuitively, one can think about 𝑏 as the number of ancilla qubits used for implementing the Lindblad operators 𝑳 𝑗 , while typically 𝑎 − 𝑏 ≈ log |𝐽 |. 165 • Quantum Fourier Transform 1 ∑︁ −i𝜔¯ 𝑡¯ | 𝜔⟩. 𝑸𝑭𝑻𝑁 : | 𝑡¯⟩ → √ e ¯ 𝑁 𝜔∈𝑆 ¯ 𝜔 0 We use “bar” to denote variables taking discrete values. In particular, the Fourier frequencies 𝜔¯ and times 𝑡¯ are integer multiples of 𝜔0 and 𝑡0 respectively such that   2𝜋 ⌈𝑁⌋ 𝜔0 𝑡 0 = , and 𝑆 := − ⌈(𝑁 − 1)/2⌉, . . . , −1, 0, 1, . . . , ⌊(𝑁 − 1)/2⌋ , 𝑁 and 𝑆𝜔⌈𝑁⌋ := 𝜔0 · 𝑆 ⌈𝑁⌋ , 0 𝑆𝑡⌈𝑁⌋ := 𝑡0 · 𝑆 ⌈𝑁⌋ . 0 • Controlled Hamiltonian simulation ∑︁ | 𝑡¯⟩⟨𝑡¯| ⊗ e±i𝑡¯𝑯 . 𝑡¯∈𝑆𝑡0 • State preparation oracles for the Fourier Transform weights, acting on the frequency register 𝑷𝒓𝒆 𝒑 𝒇 : 0̄ → | 𝑓 ⟩. • Controlled filter for the Boltzmann factors acting on the frequency register and the Boltzmann weight register ∑︁ 𝑾 := 𝒀1−𝛾(𝜔) ¯ 𝜔| ¯ where 0 ≤ 𝛾( 𝜔) ¯ ≤ 1 and 𝛾( 𝜔) ¯ = 𝛾(−𝜔)e ¯ −𝛽𝜔¯ . ¯ ⊗ | 𝜔⟩⟨ 𝜔∈𝑆 ¯ 𝜔0 • Single qubit Pauli-Y rotations √ 𝒀𝜃 := √ ! 1−𝜃 − 𝜃 √ √ . 𝜃 1−𝜃 • Reflection on 𝑏-qubits 𝑹𝑏 := 2 0𝑏 0𝑏 − 𝑰 𝑏 . To feed into the black-box Lindbladian simulation algorithm [40, Theorem III.2], we need block-encodings for the dissipative part and the coherent term; for the coherent Gibbs sampler, we need block-encoding for the discriminant H𝛽 , which we obtain by adding the transition part and the 𝑵 part. 166 𝒀1/2 |0⟩ 𝒀1/2 ⟨0| 𝑹𝑏+1 0𝑏 0𝑏 0𝑐−𝑏 e−i𝑯𝑡¯ 𝝆 0̄ 𝑽 †𝑗 𝑝 𝑽𝑗 𝑝 ei𝑯𝑡¯ e−i𝑯𝑡¯ 0𝑐−𝑏 ei𝑯𝑡¯ 𝑷𝒓𝒆 𝒑′ 𝑷𝒓𝒆 𝒑 † 0̄ Figure 3.4: Circuit for block-encoding (3.3). The gate 𝒑𝒓𝒆 𝒑 is a shorthand for 𝑷𝒓𝒆 𝒑 √| 𝑓 | and 𝒑𝒓𝒆 𝒑′ for 𝑷𝒓𝒆 𝒑 𝑓 /√| 𝑓 | . + + + Proposition 3.3.1 (Block-encoding for the coherent term). Suppose ∥ 𝑓− ∥ 1 , ∥ 𝑓+ ∥ 1 ≤ 1. Then, there is a block-encoding for ∑︁ ∑︁ © ∑︁ ª 𝑓− ( 𝑡¯− )e−i𝑯𝑡¯− ­ 𝑓+ ( 𝑡¯+ ) 𝑨𝑎† ( 𝑡¯+ ) 𝑨𝑎 (−𝑡¯+ ) ®ei𝑯𝑡 − 𝑎∈𝐴 𝑡¯− ∈𝑆𝑡0 «𝑡¯+ ∈𝑆𝑡0 ¬ using constant calls to controlled Hamiltonian simulation, 𝑽 𝑗𝑢𝑚 𝑝 , 𝒑𝒓𝒆 𝒑 √| 𝑓 | , 𝒑𝒓𝒆 𝒑 𝑓 /√| 𝑓 | , + + + 𝒑𝒓𝒆 𝒑 √ , 𝒑𝒓𝒆 𝒑 √ and their adjoints. | 𝑓+ | 𝑓+ / | 𝑓+ | The identical statement applies to the 𝑵 term by replacing 𝑓± → 𝑛± . Proof. It suffices to construct ∑︁ 𝑓+ ( 𝑡¯+ ) 𝑡¯+ ∈𝑆𝑡0 ∑︁ 𝑨𝑎† ( 𝑡¯+ ) 𝑨𝑎 (−𝑡¯+ ) 𝑎∈𝐴 and then apply the operator Fourier Transform. When there is only one jump (| 𝐴| = 1), this merely uses iterations of LCU and controlled Hamiltonian simulation. When a block-encoding gives the set of jumps, see Figure 3.4. To see that this yields the desired expression, observe that 𝒀1/2 |0⟩ = |+⟩ † |0⟩ = |−⟩. and 𝒀1/2 Thus, the expression remains the same if we drop the −𝑰 𝑏+1 term in 𝑹𝑏+1 . ■ Proposition 3.3.2 (Bilinear). Suppose ∥ℎ− ∥ 1 = ∥ℎ+ ∥ 1 = 1. Then, there is a block-encoding for ∑︁ ∑︁ ∑︁ ℎ− ( 𝑡¯− )ℎ+ ( 𝑡¯+ ) · 𝑨𝑎 ( 𝑡¯+ − 𝑡¯− ) ⊗ 𝑨𝑎 (−𝑡¯− − 𝑡¯+ )𝑇 𝑡¯− ∈𝑆𝑡0 𝑡¯+ ∈𝑆𝑡0 𝑎∈𝐴 167 using constant calls to controlled Hamiltonian simulation, 𝑽 𝑗𝑢𝑚 𝑝 , 𝑽 𝑗𝑢𝑚 𝑝𝑇 , 𝒑𝒓𝒆 𝒑 √|ℎ | , 𝒑𝒓𝒆 𝒑 ℎ /√|ℎ | , + + + 𝒑𝒓𝒆 𝒑 √|ℎ | , 𝒑𝒓𝒆 𝒑 ℎ /√|ℎ | and their adjoints. + + + Proof. The proof is a modification of Proposition 3.3.1 by replacing 𝑽 †𝑗𝑢𝑚 𝑝 with 𝑽 𝑇𝑗𝑢𝑚 𝑝 and duplicating the system register. ■ Proving Theorem 3.1.2: complexity for Lindbladian simulation We now combine the block-encoding to give the overall cost of Gibbs sampling. We can implement the controlled time-evolution up to a truncation time as long as the profiles 𝑓+ , 𝑓− are well-concentrated and smooth in the time domain (and that the Fourier Transforms are well-defined), which is the case since the frequency profiles 𝑓ˆ+ , 𝑓ˆ− are smooth and concentrated Corollary 3.3.1-Proposition 3.6.1. Also, the following results all require discretization of the time integrals, which is fortunately handled by [40, Appendix C]; this sets the required size of the Fourier Transform register, which uses polylogarithmically many qubits. Proof of Theorem 3.1.2. Use the black-box Lindbladian simulation algorithm [40, Theorem III.2] for block-encoding for the coherent term (Proposition 3.3.1) and the dissipative part [40, Section III.B.1]. For the Metropolis weight, a logarithmic overhead is incurred for taming the mild singularity in the 𝑏 2𝑀 ; 𝑩 is especially subnormalized by ∥𝑏 2𝑀 ∥ −1 1 = 1/O (log(𝛽∥𝐻 ∥/𝜖)) to fit an approximation of 𝑩 into a unitary block encoding. ■ Proving Theorem 3.1.3: complexity for the discriminant We may now construct the advertised block encoding for the discriminant. The 𝑵 term is analogous to the coherent term (Proposition 3.3.1). Proof of Theorem 3.1.3. Add the block encodings for the transition part (from Proposition 3.3.2), 𝑵 ⊗ 𝑰, and 𝑰 ⊗ 𝑵 (from Proposition 3.3.1), up to mild subnormalization. Again, the Metropolis case incurs an additional logarithmic factor. ■ Note that we do not implement 𝑰 + H 𝛽 (as in [40, Proposition III.5]) but rather H 𝛽 itself; implementing the former would allow us to obtain a quadratic speedup on the discriminant gap, which we currently do not have. 168 3.4 Discussion We have constructed the quantum analog of the classical Monte Carlo algorithms with desirable features. We highlight potential future directions as listed. • Quantum simulation applications. A key factor in industrial quantum simulation applications [30, 37, 107] is effective quantum algorithms for lowenergy states. Our algorithm can be employed for any Hamiltonian without substantial variational parameters or a case-by-case trial state or adiabatic path. While the mixing time can vary widely, the fact that physically relevant states (molecules or materials) exist in Nature suggests a reasonable mixing time in practice. Regarding practical gate complexities, the locality of our algorithm for lattice Hamiltonian may be favorable as we merely need to simulate a Õ (𝛽)-radius Hamiltonian patch localized around each jump 𝑨𝑎 . • Locality and complexity of quantum Gibbs state. Our algorithm opens new angles on the locality and complexity of Gibbs states (such as the decay of correlation, quantum conditional mutual information, recovery channels, and quantum belief propagation). In particular, the combination of localized jumps and exact detailed balance enables the rigorous study of convergence [31, 96] for noncommuting lattice Hamiltonians. Rapid mixing also directly implies the circuit complexity of the purification (Appendix 3.7) through the Lindbladian gap, giving a dynamic perspective on the area law of entanglement [84]. • New open-system physics. Just as quantum computing lacks a go-to Monte Carlo algorithm, open system physics lacks a simple, universal Lindbladian that succinctly captures open system thermodynamics. Our algorithm qualifies due to its elegant properties. For example, our Lindbladian enables a precise definition of dynamical thermal phase transitions in terms of mixing time that may contrast with static thermal phase transitions. Related concepts include metastable states, the energy landscape, quantum spin glass, and self-correcting quantum memories, whose precise formulation for noncommuting Hamiltonian has also been lacking. • A new algorithmic subroutine. Classical MCMC algorithms have been widely employed to solve other problems beyond physical simulation, and we may expect the same for our algorithm. A natural example is optimization problems (e.g., constraint satisfaction problems and modern optimization problems), whether applying to classical Hamiltonian (in a setting similar 169 to Quantum Approximate Optimization Algorithms (QAOA) [66]) or quantum Hamiltonians. Another application is Quantum Semidefinite Program Solvers [11, 27], where Quantum Gibbs state preparation is routinely invoked. • Comparision with existing algorithms. With a new algorithm at hand, we expect fruitful comparison with existing (quantum or classical) algorithms such as the adiabatic algorithm [64], phase estimation with trial states, tensor network, Quantum Monte Carlo, etc. In particular, understanding the distinction from classical algorithms could either inspire better classical algorithms or expose potential sources of quantum advantage in quantum simulation (e.g., the sign problem or difficulty in contracting PEPS). • Numerical studies. As the complement to theory, the explicit form of our Lindbladian also enables direct numerical studies regarding the above notions, e.g., the scaling of mixing time for thermal state or ground states, dynamic phase transitions, and noncommuting quantum memories, and the interplay with tensor networks. To conclude, given the celebrated theoretical and empirical triumph of Markov chain Monte Carlo methods and its successors over the past 70 years, we argue that this work should serve similar roles in quantum computing. Especially given the current skepticism on the practical applicability of quantum computers, our new algorithms bring hope to the community by initiating a new wave of directions covering theory, experiment, numerics, and application. Acknowledgments We thank Jonathan Moussa for raising the question of whether exact detailed balance is possible and pointing us to his related work [133] after the precursor of this work [40] became public; at that time, we thought that exact detailed balance is incompatible with the energy-time uncertainty principle. We also thank Aaron J. Friedman, Jinkang Guo, Oliver Hart, and Andrew Lucas for collaborations on related topics and Alvaro Alhambra, Anurag Anshu, Simon Apers, Mario Berta, Fernando Brandao, Angela Capel, Garnet Chan, ChatGPT-4, Alex Dalzell, Zhiyan Ding, Steve Flammia, Hsin-Yuan (Robert) Huang, Lin Lin, Yunchao Liu, Sam McArdle, Mehdi Soleimanifar, Frederik Nathan, Umesh Vazirani, and Tong Yu for helpful discussions. CFC was supported through an internship of the AWS Center for Quantum Computing. AG acknowledges funding from the AWS Center for Quantum Computing. 170 3.5 Appendix:Deriving time-domain represantations Our calculation for detailed balance has focused on the frequency domain. This appendix applies Fourier transforms to obtain the time-domain representation. The arguments are conceptually straightforward but require some bookkeeping. For both the Lindbladians and the parent Hamiltonians, we will often encounter a two-dimensional sum over Bohr frequencies. Since there are two energy labels, we employ a two-dimensional Fourier Transform. For any function of frequencies 𝑓ˆ(𝜈1 , 𝜈2 ), the time-domain representation of the bilinear expression gives ∑︁ ∑︁ ∑︁ 1 ∫ ∞ ∫ ∞ ˆ𝑓 (𝜈1 , 𝜈2 ) 𝑨𝜈𝑎 ⊗ ( 𝑨𝜈𝑎 ) † = 𝑓 (𝑡1 , 𝑡2 ) 𝑨𝑎 (𝑡 1 ) ⊗ 𝑨𝑎† (−𝑡2 )d𝑡1 d𝑡2 , 1 2 2𝜋 −∞ −∞ 𝑎∈𝐴 𝜈 ,𝜈 ∈𝐵 𝑎∈𝐴 1 2 where we introduced the two-dimensional Fourier Transform ∫ ∞∫ ∞ 1 𝑓 (𝑡 1 , 𝑡2 ) := 𝑓ˆ(𝜈1 , 𝜈2 )ei𝜈1 𝑡1 ei𝜈2 𝑡2 d𝜈1 d𝜈2 . 2𝜋 −∞ −∞ Fortunately, for our usage, the Fourier Transform decouples into two iterations of one-dimensional Fourier Transforms, significantly simplifying the presentation and implementation. Corollary 3.5.1 (Factorized time-domain functions). If the function factorizes in the energy domain such that 1 ˆ 𝑓 (𝜈1 , 𝜈2 ) = 𝑓ˆ+ (𝜈1 + 𝜈2 ) · 𝑓ˆ− (𝜈1 − 𝜈2 ), 2𝜋 then ∑︁ ∑︁ 𝑓ˆ(𝜈1 , 𝜈2 ) 𝑨𝜈𝑎1 ⊗ ( 𝑨𝜈𝑎2 ) † 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 ∑︁ ∫ ∞ ∫ ∞ = −∞ 𝑎∈𝐴 𝑓− (𝑡− ) 𝑓+ (𝑡+ ) 𝑨𝑎 (−𝑡− − 𝑡+ ) ⊗ 𝑨𝑎† (𝑡+ − 𝑡− )d𝑡+ d𝑡 − −∞ and ∑︁ ∑︁ 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 ∑︁ ∫ ∞ = 𝑎∈𝐴 −∞ 𝑓ˆ(𝜈1 , 𝜈2 )( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 −i𝑯𝑡 − ∫ ∞  𝑓+ (𝑡+ ) 𝑨 (𝑡+ ) 𝑨 (−𝑡+ )d𝑡+ ei𝑯𝑡 − d𝑡− , 𝑎† 𝑓− (𝑡− )e 𝑎 −∞ where the function 𝑓± are inverse Fourier Transforms of 𝑓ˆ± . (3.26) 171 Crucially, the RHS can be implemented via Linear Combination of Unitaries by discretizing the integral. Proof. Since the expression is linear in the sum over jumps 𝑎 ∈ 𝐴, it suffices to prove for any operator 𝑨, dropping the jump labels 𝑎. ∑︁ 2𝜋 𝑓ˆ+ (𝜈1 + 𝜈2 ) · 𝑓ˆ− (𝜈1 − 𝜈2 ) 𝑨𝜈1 ⊗ ( 𝑨𝜈2 ) † 𝜈1 ,𝜈2 ∈𝐵 = ∑︁  † − 2𝜋 𝑓ˆ+ (𝜈+ ) · 𝑓ˆ− (𝜈− ) 𝑨 𝜈+ +𝜈− ⊗ 𝑨 𝜈+ −𝜈 2 2 𝜈1 ,𝜈2 ∈𝐵 = ∑︁ ∫ ∞ 𝜈1 ,𝜈2 ∈𝐵 = 𝑓+ (𝑡+ )𝑒 −i𝜈+ 𝑡+ d𝑡+ −∞ −∞ ∑︁ ∫ ∞ ∫ ∞ 𝜈1 ,𝜈2 ∈𝐵 −∞ ∫ ∞ −∞ (Let 𝜈+ := 𝜈1 + 𝜈2 and𝜈− := 𝜈1 − 𝜈2 )  † − 𝑓− (𝑡− )𝑒 −i𝜈 − 𝑡 − d𝑡− 𝑨 𝜈+ +𝜈− ⊗ 𝑨 𝜈+ −𝜈 2 2 (Fourier Transform) † − 𝑓+ (𝑡+ ) 𝑓− (𝑡− ) 𝑨 𝜈+ +𝜈− (−𝑡+ − 𝑡− ) ⊗ 𝑨 𝜈+ −𝜈 (𝑡 + − 𝑡 − )d𝑡+ d𝑡− 2  2 (Since ( 𝑨𝜈 ) † = ( 𝑨† )−𝜈 ) ∫ ∞∫ ∞ = −∞ 𝑓+ (𝑡 + ) 𝑓− (𝑡− ) 𝑨(−𝑡+ − 𝑡− ) ⊗ 𝑨† (𝑡+ − 𝑡− )d𝑡+ d𝑡 − . −∞ The fourth equality uses that e−i𝜈+ 𝑡+ e−i𝜈 − 𝑡 − = exp( i(𝜈+ − 𝜈− )(𝑡+ − 𝑡− ) i(𝜈+ + 𝜈− )(−𝑡+ − 𝑡− ) ) · exp( ) 2 2 to conclude the proof. ■ Now, we plug in the appropriate functions to arrive at the time-domain functions. Our Lindbladians We evaluate the Fourier transform for the coherent term 𝑩. The expression looks intimidating, but all that matters for the algorithmic complexity is that they decay rapidly (in the time domain). Corollary 3.5.2 (Explicit time-domain functions). In the time domain, the coherent term 𝑩 in (3.14) corresponding to coefficients constructed in (3.15) reads   ∫ ∞ 4𝑡 2 𝑥 𝑓+ (𝑡) = 𝛽 𝜎2 𝑔(𝑥)𝜎𝛾 (𝑥) exp − − 2i𝑡𝑥 d𝑥, (3.27) 𝐸 𝛽 2 𝑓− (𝑡) =   2 2ª 𝜎𝐸 𝛽2 𝜎𝐸2 ©­ 1   ∗𝑡 sin −𝛽𝜎𝐸2 𝑡 e−2𝜎𝐸 𝑡 ®®, e 8 ­ 𝜋𝛽 cosh 2𝜋𝑡 𝛽 « ¬ (3.28) 172 depending on parameters 𝜎𝐸 , 𝑔(𝑥), 𝛽.12 Recall that the convolution of two functions over variable 𝑡 is defined by ∫ ∞ ( 𝑓 ∗𝑡 𝑔)(𝑡) := 𝑓 (𝑠)𝑔(𝑡 − 𝑠)d𝑠. −∞ Nicely, the product structure persists under a convex combination of Gaussians as only 𝑓+ (𝑡) depends on 𝑔(𝑥). Otherwise, we could have had to consider a linear combination of function products, which is messier to implement. Proof. First, we confirm that the energy domain function indeed has a product 1 ˆ 𝑓 (𝜈1 , 𝜈2 ) = 𝑓ˆ+ (𝜈1 + 𝜈2 ) · 𝑓ˆ− (𝜈1 − 𝜈2 ) due to (3.12), (3.15) and (3.14) for structure 2𝜋   ∫ ∞ 2 𝑔(𝑥)𝜎𝛾 (𝑥) (𝜈 + 2𝑥) exp − 𝑓ˆ+ (𝜈) = 𝛽 𝜎2 √︃ d𝑥, and (3.29) 𝐸 16𝑥/𝛽 2 2 𝜎𝐸2 + 𝜎𝛾2 (𝑥) ! ! 2 2 tanh(−𝛽𝜈/4) 𝜈 1 1 sinh(−𝛽𝜈/4) 𝜈 1 exp − 2 = · exp − 2 . 𝑓ˆ− (𝜈) = 2𝜋 2i 2𝜋 cosh(−𝛽𝜈/4) 2i 8𝜎𝐸 8𝜎𝐸 Since we work with well-concentrated integrable functions, the Fourier Transforms exist, and we can compute them as follows. We begin with the Gaussian integral associated with 𝑓+ √︄     ∫ ∞ 1 (𝜈 + 2𝑥) 2 i𝜈𝑡 2𝑥 4𝑡 2 𝑥 exp − e d𝜈 = 2 exp − − 2i𝑡𝑥 . √ 16𝑥/𝛽 𝛽 𝛽 2𝜋 −∞ √︃ Thus, using (3.29) and the definition of 𝜎𝛾 (𝑥) = 2𝑥/𝛽 − 𝜎𝐸2 , we get (3.27). In order √ to compute 𝑓− (𝑡), we use the convolution theorem F −1 (F ( 𝑓 ) · F (𝑔)) = 𝑓 ∗ 𝑔/ 2𝜋. Individually, we have ! ∫ ∞   𝛽2 𝜎2 2 2 1 sinh(−𝛽𝜈/4) 𝜈 2 i𝜈𝑡 𝐸 2 8 exp − 2 e d𝜈 = 𝜎𝐸 e sin −𝛽𝜎𝐸 𝑡 e−2𝜎𝐸 𝑡(3.30) √ 2i 8𝜎𝐸 2𝜋 −∞ ∫ ∞ 1 1 2  . ei𝜈𝑡 d𝜈 = √ √ 2𝜋 −∞ 2𝜋 cosh(−𝛽𝜈/4) 2𝜋𝛽 cosh 2𝜋𝑡 𝛽 1 12 Note that the function 𝑏 (𝑡) seems to have width ∼ 𝛽𝜎 due to the convolution by 1 𝐸 cosh(4 𝜋𝑡/(𝛽 𝜎𝐸 ) ) , meaning that the integral in 𝑡 seems to require Hamiltonian evolution times up to ∼ 𝛽. In fact, the numerics show a 1/poly decay until about ∼ 𝛽𝜎
𝐸 (after which the exponential decay starts), suggesting that about min 𝜎𝐸−1 Poly(1/𝜖), 𝛽 log(1/𝜖) Hamiltonian simulation time is required in order to achieve 𝜖 precision for the block-encoding of the coherent term. Thus, it might be difficult to obtain exact detailed balance below √︁ Ω(𝛽) Hamiltonian evolution times. On the contrary, the function 𝑏 2 (𝑡 ′ ) has a width only about 1/ 𝛽𝜔 𝛾 , implying that the corresponding other integral in 𝑡 ′ can be √︁ well-approximated by only using Hamiltonian evolution time ∼ 𝛽/𝜔 𝛾 . 173 Take the convolution to conclude the calculation. ■ The above explicit form allows us to compute the explicit form of the function 𝑓+ (𝑡) corresponding to our two main settings. We begin with the Gaussian case. Corollary 3.3.1 (Coherent term for the Gaussian case). For each 𝛽 > 0 and parameters 𝜎𝐸 = 𝜎𝛾 = 𝜔 𝛾 = 1𝛽 , the coherent term 𝑩 (3.14) corresponding to the   2 Gaussian weight 𝛾(𝜔) = exp − (𝛽𝜔+1) can be written as 2 ∫ ∞  ∑︁ ∫ ∞ −i𝛽𝑯𝑡 ′ 𝑎† ′ 𝑎 ′ ′ i𝛽𝑯𝑡 𝑩 := 𝑏 1 (𝑡)e 𝑏 2 (𝑡 ) 𝑨 (𝛽𝑡 ) 𝑨 (−𝛽𝑡 )d𝑡 e d𝑡, (3.18) 𝑎∈𝐴 −∞ −∞ where √ 1 𝑏 1 (𝑡) := 2 𝜋e 8 such that such that    1 2 ∗𝑡 sin(−𝑡) exp −2𝑡 cosh(2𝜋𝑡) ∥𝑏 1 ∥ 1 < 1 1 𝑏 2 (𝑡) := 2𝜋 1 . ∥𝑏 2 ∥ 1 < 16 (3.19) √︂   1 2 exp −4𝑡 − 2i𝑡 𝜋 (3.20) Indeed, both functions of time are rapidly decaying and have bounded ℓ1 -norm (as required for LCU implementation). Proof. Setting 𝑔(𝑥) = 𝛿(𝑥 − 𝜔 𝛾 ) in Corollary 3.5.1 yields the desired Gaussian weight and 𝑓+ (𝑡) in (3.27) becomes √︄ !   ∫ ∞ 2𝜔 2 4𝑡 2𝜔 𝛾 4𝑡 𝑥 𝛾 𝑓+ (𝑡) = 𝛿(𝑥 − 𝜔 𝛾 )𝜎𝛾 exp − − 2i𝑡𝑥 𝑑𝑥 = − 𝜎𝐸2 exp − − 2i𝑡𝜔 𝛾 . 𝛽 𝛽 𝛽 −∞ √ √ Setting 𝑏 1 (𝑡) := 2𝜋 𝜋 𝑓− (𝑡/𝜎𝐸 ) and 𝑏 2 (𝑡 ′) := 𝑓+ (𝛽𝑡 ′)/(2𝜋 𝜋) and applying a change of variables in the integral (3.26) yields the desired result (3.18). Note that the convolution ∗𝑡 implicitly is an integral over 𝑡, so we should not forget to rescale d𝑡 there. Lastly, we bound the ℓ1 -norm of the functions by Hölder’s inequality √︂ 𝜋 2 −1 2 ∥𝑏 1 ∥ 1 ≤ (1 + 𝑡 ) 2 (1 + 𝑡 )𝑏 1 2 = (1 + 𝑡 2 )𝑏 1 2 ≤ 1, 2 ∫∞ ∫∞ 2 1 𝜋 2 using individual bounds −∞ (1+2𝑡 2 ) 2 d𝑡 = 2 and −∞ (1 + 𝑡 )𝑏 1 (𝑡) d𝑡 < 0.625. The −1/4 1 norm ∥𝑏 2 ∥ 1 = 𝑒 4𝜋 < 16 is a Gaussian integral (Fact 3.2.1). ■ 174 The explicit weights corresponding to Metropolis weights are slightly more cumbersome due to taming a logarithmic singularity; see section 3.6. Our parent Hamiltonians Based on the Lindbladian, we explicitly evaluate the discriminant in the frequency domain. Proposition 3.5.1 (Symmerized discriminant). In the setting of Corollary 3.2.2, the discriminant corresponding to the 𝝆 𝛽 -DB Lindbladian reads ∑︁ ∑︁ 1 H𝛽 = ℎ 𝜈1 ,𝜈2 𝑨𝜈𝑎1 (·)( 𝑨𝜈𝑎2 ) † + (𝑵(·) + (·) 𝑵) 2 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 ∑︁ ∑︁ 1 or H 𝛽 = ℎ 𝜈1 ,𝜈2 𝑨𝜈𝑎1 ⊗ ( 𝑨𝜈𝑎2 ) ∗ + (𝑵 ⊗ 𝑰 + 𝑰 ⊗ 𝑵 ∗ ) 2 {z } | 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 | {z } transition part where ℎ 𝜈1 ,𝜈2 := e 𝛽(𝜈1 +𝜈2 )/4 𝛼 𝑵 := − coherent and decay part 𝜈1 ,𝜈2 = ℎ−𝜈2 ,−𝜈1 and 𝛼𝜈1 ,𝜈2 ( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 = 𝑵 † . cosh(𝛽(𝜈 − 𝜈 )/4) 1 2 𝑎∈𝐴 𝜈 ,𝜈 ∈𝐵 ∑︁ ∑︁ 1 2 Proof. We calculate H𝛽 = 𝝆 −1/4 L [𝝆 1/4 · 𝝆 1/4 ] 𝝆 −1/4 ∑︁ ∑︁ ∑︁ 𝛽𝜈 𝛽𝜈 = e 𝛽(𝜈1 +𝜈2 )/4 𝛼𝜈1 ,𝜈2 𝑨𝜈𝑎1 (·)( 𝑨𝜈𝑎2 ) † − i (e 4 𝑩𝜈 (·) − e− 4 (·) 𝑩𝜈 ) 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 𝜈∈𝐵  𝛽 (𝜈1 −𝜈2 ) 𝛽 (𝜈1 −𝜈2 ) 1 ∑︁ ∑︁  − 𝛼𝜈1 ,𝜈2 e 4 ( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 (·) + 𝛼𝜈1 ,𝜈2 e− 4 (·)( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 2 𝑎∈𝐴 𝜈 ,𝜈 ∈𝐵 1 2 ∑︁ ∑︁ 1 1 = ℎ 𝜈1 ,𝜈2 𝑨𝜈𝑎1 (·)( 𝑨𝜈𝑎2 ) † + 𝑵(·) + (·) 𝑵 † , 2 2 𝑎∈𝐴 𝜈 ,𝜈 ∈𝐵 1 2 where 𝑵 = −2i ∑︁ 𝜈∈𝐵 e 𝛽𝜈/4 𝑩𝜈 − ∑︁ ∑︁ 𝛼𝜈1 ,𝜈2 e 𝛽 (𝜈1 −𝜈2 ) 4 ( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 . 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 We further simplify 𝑵 by expressing 𝑩 as a linear combination of ( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 as in (3.14).  ∑︁ ∑︁  𝑵= exp(𝛽(𝜈1 − 𝜈2 )/4) tanh(𝛽(𝜈1 − 𝜈2 )/4) − exp(𝛽(𝜈1 − 𝜈2 )/4) 𝛼𝜈1 ,𝜈2 ( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 =− 𝛼𝜈1 ,𝜈2 ( 𝑨𝜈𝑎2 ) † 𝑨𝜈𝑎1 cosh(𝛽(𝜈 − 𝜈 )/4) 1 2 𝑎∈𝐴 𝜈 ,𝜈 ∈𝐵 ∑︁ ∑︁ 1 † =𝑵 , −1 (By e𝑥 (tanh(𝑥) − 1) = cosh(𝑥) ) 2 (Invariance under 𝜈1 ↔ 𝜈2 ) 175 as advertised. ■ The transition part For the transition part, we quickly obtain the time-domain representation using a two-dimensional Fourier Transform as a corollary of Corollary 3.5.1. Corollary 3.5.3 (Time integrals). Suppose ℎ 𝜈1 ,𝜈2 = 2𝜋 · ℎˆ + (𝜈+ ) · ℎˆ − (𝜈− ), then ∑︁ ∑︁ ℎ 𝜈1 ,𝜈2 𝑨𝜈𝑎1 ⊗ ( 𝑨𝜈𝑎2 ) ∗ 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 ∑︁ ∫ ∞ ∫ ∞ = 𝑎∈𝐴 −∞ ℎ− (𝑡− )ℎ+ (𝑡+ ) · 𝑨𝑎 (𝑡 + − 𝑡 − ) ⊗ 𝑨𝑎 (−𝑡− − 𝑡+ )𝑇 d𝑡+ d𝑡− , −∞ where ℎ± (𝑡) are Fourier Transforms of ℎˆ ± (𝜈). Now, we can evaluate the Fourier transforms explicitly. Proposition 3.5.2 (Linear combination for ℎ). For 𝛼𝜈1 ,𝜈2 defined in (3.15) and each 𝜎𝐸 , 𝑔(𝑥), we have that ℎ 𝜈1 ,𝜈2 = 2𝜋 ℎˆ + (𝜈+ ) · ℎˆ − (𝜈− ) for the discriminant (Proposition 3.5.1) where   ∫ ∞ 𝑔(𝑥)𝜎𝛾 (𝑥) 𝛽𝜈 2 𝛽𝑥 ˆℎ+ (𝜈) = exp − − d𝑥 √︃ 𝛽 𝜎2 𝐸 16𝑥 4 2 2 2 2 𝜎𝐸 + 𝜎𝛾 (𝑥) ! 2 𝜈 1 exp − 2 , and ℎˆ − (𝜈) := 2𝜋 8𝜎𝐸 with the corresponding time-domain functions   ∫ ∞ 4𝑡 2 𝑥 𝛽𝑥 ℎ+ (𝑡) = 𝛽 𝜎2 𝑔(𝑥)𝜎𝛾 (𝑥) exp − − d𝑥 𝐸 𝛽 4 2 𝜎𝐸 and ℎ− (𝑡) = exp(−2𝜎𝐸2 𝑡 2 ). 𝜋 (3.31) Proof. For ℎˆ + (𝜈) term, the cross term in the exponential e−𝛽𝜈/4 is precisely cancelled by ℎ 𝜈1 ,𝜈2 := e 𝛽(𝜈1 +𝜈2 )/4 𝛼𝜈1 ,𝜈2 ; the ℎˆ − (𝜈) term remains the same. To obtain the time-domain functions, we simply carry out the Gaussian integral √︄     ∫ ∞ 𝛽𝜈 2 𝛽𝑥 i𝜈𝑡 4𝑡 2 𝑥 𝛽𝑥 1 2𝑥 exp − − e d𝜈 = 2 exp − − √ 16𝑥 4 𝛽 𝛽 4 2𝜋 −∞ and use that 𝜎𝐸2 + 𝜎𝛾2 (𝑥) = 𝛽/2𝑥. ■ 176 Proposition 3.5.3 (The Gaussian case). In the setting of Proposition 3.5.2 and for 𝛼𝜈1 ,𝜈2 defined in (3.13), we have ! ! 2 𝜎𝛾 𝜈 2 + (2𝜔 𝛾 ) 2 𝜈 1 ℎˆ + (𝜈) := √︃ exp − 2 . and ℎˆ − (𝜈) := · exp − 2 + 𝜎2) 2𝜋 8(𝜎 8𝜎𝐸 2 2 𝛾 𝐸 2 𝜎𝐸 + 𝜎𝛾 Corollary 3.3.3  (The2 transition part for Gaussian weights). For the Gaussian weight 𝛾(𝜔) = exp − (𝛽𝜔+1) with 𝜔 𝛾 = 𝜎𝐸 = 𝜎𝛾 = 1/𝛽, the discriminant H 𝛽 is described 2 in the time domain by ℎ+ (𝑡) = 1 −1/4 4𝑡 2 e exp(− 2 ) 𝛽 𝛽 ℎ− (𝑡) = and 1 2𝑡 2 exp(− 2 ) 𝜋𝛽 𝛽 such that ∥ℎ− ∥ 1 , ∥ℎ+ ∥ 1 ≤ 1. (3.24) Corollary 3.3.4 (The transition part for Metropolis weights). For 𝜎𝐸 = 1𝛽 , the    1 Metropolis-like weight 𝛾 𝑀 (𝜔) = exp −𝛽 max 𝜔 + 2𝛽 , 0 yields H 𝛽 described in the time domain by the same ℎ− (𝑡) as in Eq. (3.24) and by 2 2 e−1/8 e−2𝑡 /𝛽 ℎ+ (𝑡) = √ 𝛽 4 2𝜋( 𝑡 22 + 1 ) 𝛽 ∥ℎ+ (𝑡)∥ 1 ≤ 1. such that 16 1 on the interval 2𝜋𝜎𝛾 (𝑥) Proof. Setting 𝑔(𝑥) = √  𝛽𝜎𝐸2 2 , ∞ , yields in (3.31)     ∫ ∞ 4𝑡 2 𝑥 𝛽𝑥 1 4𝑡 2 𝑥 𝛽𝑥 ℎ+ (𝑡) = 𝛽 𝜎2 𝑔(𝑥)𝜎𝛾 (𝑥) exp − − d𝑥 = √ exp − − d𝑥 𝛽 𝜎2 𝐸 𝐸 𝛽 4 𝛽 4 2𝜋 2 2 ∫ ∞  2 2 2 2 1 e−2𝜎𝐸 𝑡 −𝛽 𝜎𝐸 /8 = . √ 𝛽 4 2𝜋( 𝑡 22 + 1 ) 𝛽 16 Set 𝜎𝐸 = 1/𝛽 to conclude the calculation. To obtain the ℓ1 -norm bound, integrate 2 √ √︁ −1/8 ∫ ∞ e√ e −2𝑥 d𝑥 = 𝜋2 erfc(1/ 8) < 0.78. ■ −∞ 𝑥 2 +1/16 4 2𝜋 The 𝑵-term We instantiate the time-domain presentation of the 𝑵 term (adapted from the calculation for the coherent term 𝑩 (Corollary 3.5.2)). Corollary 3.5.4 (Explicit time-domain functions). In the time domain, the 𝑵 term corresponding to coefficients constructed in (3.15) can be written as ∫ ∞  ∑︁ ∫ ∞ −i𝑯𝑡 − 𝑎† 𝑎 𝑵= 𝑓− (𝑡− )e 𝑓+ (𝑡 + ) 𝑨 (𝑡 + ) 𝑨 (−𝑡+ )d𝑡+ ei𝑯𝑡 − d𝑡− 𝑎∈𝐴 −∞ −∞ 177 for 𝑓+ (𝑡) as in (3.27) and 𝑓− (𝑡) = 2 2ª 2𝜎𝐸 ©­ 1   ∗𝑡 e−2𝜎𝐸 𝑡 ®®, ­ 𝜋𝛽 cosh 2𝜋𝑡 𝛽 « ¬ depending on parameters 𝜎𝛾 , 𝜎𝐸 , 𝑔(𝑥), 𝛽. Proof. Follow the proof of Corollary 3.5.2, but drop the sinh(−𝛽𝜈/4)/(2i) term in (3.30). The convolution then gives ! ∫ ∞ 2 2 𝜈 2 i𝜈𝑡 1 exp − 2 e d𝜈 = 2𝜎𝐸 e−2𝜎𝐸 𝑡 . √ 8𝜎𝐸 2𝜋 −∞ ■ We instantiate similar results with minor modifications without replicating the proofs. Corollary 3.3.5 (𝑵 term   for Gaussian weights). For each 𝛽, the Gaussian weight (𝛽𝜔+1) 2 𝛾(𝜔) = exp − 2 with 𝜎𝐸 = 𝜎𝛾 = 𝜔 𝛾 = 1𝛽 corresponds to the discriminant where ∫ ∞  ∑︁ ∫ ∞ −i𝛽𝑯𝑡 ′ 𝑎† ′ 𝑎 ′ ′ i𝛽𝑯𝑡 𝑵= 𝑛1 (𝑡)e 𝑛2 (𝑡 ) 𝑨 (𝛽𝑡 ) 𝑨 (−𝛽𝑡 )d𝑡 e d𝑡, 𝑎∈𝐴 −∞ −∞ where    1 √ 1 2 ∗𝑡 exp −2𝑡 𝑛1 (𝑡) := · 2 𝜋 4 cosh(2𝜋𝑡) 1 𝑛2 (𝑡) := 4 · 2 · 2𝜋 √︂   1 exp −4𝑡 2 − 2i𝑡 = 8 · 𝑏 2 (𝑡) 𝜋 such that 𝜋 ∥𝑛1 ∥ 1 = √ < 1 4 2 (3.25) such that ∥𝑛2 ∥ 1 < 1 2 with 𝑏 2 as in (3.20). 1 Proof. As both functions are positive cosh(2𝜋𝑡) , exp(−2𝑡 2 ) > 0, we have that     √ ∫∞ ∫∞ ∫∞ 𝜋 1 ∥𝑛1 (𝑡)∥ 1 = −∞ 𝑛1 (𝑡)d𝑡 = 2 −∞ cosh(2𝜋𝑡) d𝑡 · −∞ exp(−2𝑡 2 )d𝑡 , where the second equality follows from the fact that the integral of a convolution is the product of the integrals of the convolved functions due to Fubini’s theorem. The first integral √︁ evaluates to 1/2 and the second evaluates to 𝜋/2 due to Fact 3.2.1, therefore 178 √ ∥𝑛1 (𝑡)∥ 1 = 𝜋/ 32. The second bound follows from the analogous bound on 𝑏 2 in Corollary 3.3.1. The factor of 4 and 14 is redistributed to ensure both 𝑛1 and 𝑛2 are normalized. Note the overall extra factor of 2 in Corollary 3.5.4 compared with Corollary 3.5.2. ■ Corollary 3.3.6 (𝑵 term for Metropolis weights). If 𝜎𝐸 = 1𝛽 , then the Metropolis-like    1 weight 𝛾 𝑀 (𝜔) = exp −𝛽 max 𝜔 + 2𝛽 , 0 corresponds to the discriminant where 𝑵 𝑀 satisfies   ∑︁ 𝜂𝛽∥𝑯∥  𝑀 𝑀,𝜂 𝑎† 𝑎 3 𝑵 −𝑵 ≤ 𝑨 𝑨 min √ , O (𝜂𝛽∥𝑯∥) , 2𝜋 𝑎∈𝐴 where 𝑵 𝑀,𝜂 := ∑︁ ∫ ∞ 𝑎∈𝐴 −i𝛽𝑯𝑡 ∫ ∞ 𝑛1 (𝑡)e −∞ −∞ 𝑀,𝜂 with 𝑛1 (𝑡) as in (3.25), and 𝑛2 3.6 𝑀,𝜂 𝑛2 (𝑡 ′) 𝑨𝑎† (𝛽𝑡 ′) 𝑨𝑎 (−𝛽𝑡 ′)d𝑡 ′ + 𝑀,𝜂 = 𝑏2  1 𝑎† 𝑎 i𝛽𝑯𝑡 d𝑡, √ 𝑨 𝑨 e 16 2𝜋 as in (3.23). Appendix:Calculating the coherent term for the Metropolis-like weights In this section, we dedicate to calculating the weights corresponding to the Metropolislike weight in order to arrive at an expression that enables efficient implementation via LCU. Proposition 3.6.1 (Metropolis-like weights). Setting 𝑔(𝑥) = √ 1 on the interval 2𝜋𝜎𝛾 (𝑥)  2 2 𝛽𝜎𝐸 𝑠 2 , 𝛽 , yields in (3.27) 𝑓+(𝑠) (𝑡) = 2 −4 𝑡 2 𝑠2 −2i 𝛽𝑡 𝑠2 −2𝜎𝐸2 𝑡 2 −i𝛽𝜎𝐸2 𝑡 1e 𝛽 −e √ 𝑡 𝑡 2 2𝜋 𝛽 (2 𝛽 + i) 𝛽 . (3.32) In particular, in the 𝑠 → ∞ limit, the second term vanishes, but we can only interpret the result as a distribution √︂ 2 2 2 1 e−2𝜎𝐸 𝑡 −i𝛽𝜎𝐸 𝑡 𝜋 (∞) 𝑓+ (𝑡) = lim 1 (|𝑡| ≥ 𝜂) √ 𝑡 𝑡 𝛿(𝑡), + 𝜂→0+ 𝛽 2 2𝜋 (2 + i) 8 𝛽 𝛽 which again decays rapidly; we would need to pay attention to the 𝑡 ∼ 0 regime due to the delta function 𝛿(𝑡) and 1/𝑡 dependence. 179 Proof. We substitute 𝑔(𝑥) = √ 𝑓+(𝑠) (𝑡) := 2𝜔 1 in (3.27), which due to 𝜎𝐸2 + 𝜎𝛾2 = 𝛽 𝛾 gives 2𝜋𝜎𝛾 (𝑥)    ∫ 𝑠2 𝛽 4𝑡 2 𝑥 1 4𝑡 2 𝑥 𝜎𝛾 (𝑥) exp − − 2i𝑡𝑥 d𝑥 = √ exp − − 2i𝑡𝑥 d𝑥 √ 𝛽 𝜎2 𝛽 𝜎2 𝐸 𝐸 𝛽 𝛽 2𝜋𝜎 (𝑥) 2𝜋 𝛾 2 2 ∫ ∞  1 −2𝜎𝐸2 𝑡 2 −i𝛽𝜎𝐸2 𝑡 = 2 −4 𝑡 2 𝑠2 −2i 𝛽𝑡 𝑠2 −e √ 𝑡 𝑡 2 2𝜋 𝛽 (2 𝛽 + i) 1e 𝛽 𝛽 . In order to properly understand the 𝑠 → ∞ limit we also need to analyze (3.29):     ∫ ∞ ∫ 𝑠2 √︂ 2 2 𝛽 𝑔(𝑥)𝜎𝛾 (𝜈 + 2𝑥) 𝛽 (𝜈 + 2𝑥) (𝑠) exp − 𝑓ˆ+ (𝜈) = 𝛽 𝜎2 √︃ d𝑥 = 𝛽 𝜎2 exp − d𝑥 𝐸 𝐸 16𝑥/𝛽 16𝜋𝑥 16𝑥/𝛽 2 2 2 𝜎𝐸2 + 𝜎𝛾2 (3.33)    𝛽𝜈 1 − 𝛽𝜈 𝜈 𝛽𝜎𝐸 𝛽𝜎𝐸 𝑠 𝛽𝜈 𝜈 𝑠 𝛽𝜈 − = e 2 erf √ − √ − erf √ + √ + e 2 erf − + erf + , 4 2 4𝑠 2 4𝑠 8𝜎𝐸 8 8𝜎𝐸 8       which in the 𝑠 → ∞ limit becomes      𝜈 𝛽𝜎𝐸 𝜈 𝐸 − 𝛽𝜈 ˆ𝑓+(∞) (𝜈) = 1 erfc 𝛽𝜎 2 erfc √ − √ +e . √ +√ 4 8 8𝜎𝐸 8 8𝜎𝐸 (3.34) (∞) In order to compute 𝑓+(∞) (𝜈), let us consider the function 𝑓ˆ+,0 (𝜈) := 𝑓ˆ+(∞) (𝜈) − 1−sgn(𝜈) , where 4 sgn(𝜈) =   0  if 𝜈=0   sgn(𝜈) = 𝜈/|𝜈|  if 𝜈≠0 . (∞) Since 𝑓ˆ+,0 (𝜈) ∈ ℓ1 , its inverse Fourier Transform exists, and it can be computed as e (∞) 𝑓+,0 (𝑡) = 1𝛽 −2𝜎 2 𝑡 2 −i𝛽 𝜎 2 𝑡 𝐸 𝐸 +2i 𝑡 −1 𝛽 √ 𝑡 𝑡 2 2𝜋 𝛽 (2 𝛽 +i) after a change of variables by utilizing the (complex) (∞) Laplace transform of the error function erf [146]. Moreover, since 𝑓ˆ+,0 (𝜈) is smooth, apart from 𝜈 = 0 where its value is the mean of its left and right side limits, standard results in the theory of Fourier integrals (see, e.g., [189, Chapter 18.3.1.d]) imply 1 (∞) 𝑓ˆ+,0 (𝜈) = lim √ 𝜂→0+ 2𝜋 ∫ 1 𝜂 − 𝜂1 1 𝜂 −2𝜎𝐸2 𝑡 2 −i𝛽𝜎𝐸2 𝑡 1e 2𝜋 − 𝜂1 𝛽 1 (∞) 𝑓+,0 (𝜈)𝑒 −i𝜈𝑡 d𝑡 = lim √ 𝜂→0+ ∫ + 2i 𝛽𝑡 − 1 √ 2 2𝜋 𝛽𝑡 (2 𝛽𝑡 + i) d𝑡. 180 On the other hand, it is a standard calculation that 1 1 sgn(𝜈) = lim 𝜂→0+ 8𝜋 4 ∫ ∞ 𝜂 sin(𝜈𝑡) 1 d𝑡 = lim √ 𝜂→0+ 2𝜋 𝑡 ∫ −𝜂 √ − 𝜂1 i 8𝜋𝑡 e−i𝜈𝑡 d𝑡 + ∫ 𝜂 1 𝜂 i e−i𝜈𝑡 d𝑡 . √ 8𝜋𝑡 (∞) Since 𝑓+,0 (𝜈) is bounded in the neighborhood of 0 we can combine the above two expression, yielding 1 sgn(𝜈) (∞) (𝜈) − = 𝑓ˆ+,0 𝑓ˆ+(∞) (𝜈) − 4   ∫4 ∞ i 1 (∞) = lim √ 1 (|𝑡| ∈ (𝜂, 1/𝜂)) 𝑓+,0 (𝜈) − √ 𝑒 −i𝜈𝑡 d𝑡 𝜂→0+ 2𝜋 −∞ 8𝜋𝑡 ∫ ∞ −2𝜎𝐸2 𝑡 2 −i𝛽𝜎𝐸2 𝑡 1 1 e = lim √ 1 (|𝑡| ∈ (𝜂, 1/𝜂)) √ 𝑡 𝑡 e−i𝜈𝑡 d𝑡 𝜂→0+ 2𝜋 −∞ 𝛽 2 2𝜋 (2 + i) 𝛽 𝛽 ∫ ∞ −2𝜎𝐸2 𝑡 2 −i𝛽𝜎𝐸2 𝑡 1 1 e = lim √ e−i𝜈𝑡 d𝑡, 1 (|𝑡| ≥ 𝜂) √ 𝑡 𝑡 𝜂→0+ 2𝜋 −∞ 𝛽 2 2𝜋 (2 + i) 𝛽 𝛽 where the last equality holds because of the rapid decay of the function for large 𝑡 values. We can conclude the proof by observing that the Fourier Transform of √︁ 𝜋 1 ■ 8 𝛿(𝑡) is 4 . In the 𝑠 = ∞ case, the diverging ℓ1 norm of 𝑓+(∞) (𝑡) raises the question: how big could the norm 𝑩 𝑀 be? In the following we spell out the explicit form of 𝑩 by removing the singularity at 𝑡 = 0, which shows that in fact 𝑩 𝑀 ≤ O (log(𝛽∥𝑯∥)), because ∥ 𝑓− ∥ 1 = O (1). Although the exact formula becomes cumbersome, it enables us to find a very precise approximation by a much simpler formula in Corollary 3.3.2. Proposition 3.6.2 (Exact form of Metropolis coherent term). The coherent term 𝑩 𝑀 corresponding to the quasi-Metropolis weight 𝛾𝜎(∞) (𝜔) in (3.16) can be written as 𝐸 ∑︁ ∫ ∞ 𝑀 𝑩 = 𝑓− (𝑡− )e−i𝑯𝑡 − 𝑶 𝑎𝑀 ei𝑯𝑡 − d𝑡− , 𝑎∈𝐴 −∞ ! 181 where 𝑓− (𝑡− ) is defined in (3.28), and for arbitrary 𝜃 > 0 we have   𝑡 −2𝜎𝐸2 𝑡 2 −i𝛽𝜎𝐸2 𝑡 ∫ ∞ e + 1 (|𝑡| ≤ 𝜃)i 2 + i © 𝛽 sinc(2𝑯𝑡) ª® 𝑎 i𝑯𝑡 −2i𝑯𝑡 𝑶 𝑎𝑀 = ei𝑯𝑡 𝑨𝑎† ­­ e − 1 (|𝑡| ≤ 𝜃)𝑯 √ √ ® 𝑨 e d𝑡 −∞ 8𝜋𝑡 (2 𝛽𝑡 + i) 2𝜋 « ¬ 1 +√ 8𝜋 ∫ 𝜃 √︂ + cos(𝑯𝑡) 𝑨𝑎† cos(2𝑯𝑡) 𝑨𝑎 𝑯 sinc(𝑯𝑡) + sinc(𝑯𝑡)𝑯 𝑨𝑎† cos(2𝑯𝑡) 𝑨𝑎 cos(𝑯𝑡)d𝑡 −𝜃 𝜋 𝑎† 𝑎 𝑨 𝑨 . 8 Proof. By Corollary 3.5.1-Proposition 3.6.1 we know that ∫ ∞ 𝑶 𝑎𝑀 = lim 𝜂→0+ 2 2 2 e−2𝜎𝐸 𝑡 −i𝛽𝜎𝐸 𝑡 + 1 (|𝑡| ≥ 𝜂) √ −∞ 8𝜋𝑡 (2 𝛽𝑡 + i) √︂ ! 𝜋 𝛿(𝑡) 𝑨𝑎† (𝑡) 𝑨𝑎 (−𝑡)𝑑𝑡. 8 We decompose the above integral in order to remove its singularity at 𝑡 = 0. We start with the decomposition   𝑡 −2𝜎𝐸2 𝑡 2 −i𝛽𝜎𝐸2 𝑡 2 2 2 e + 1 (|𝑡| ≤ 𝜃)i 2 𝛽 + i e−2𝜎𝐸 𝑡 −i𝛽𝜎𝐸 𝑡 i = − 1 (|𝑡| ≤ 𝜃) √ , √ √ 𝑡 𝑡 8𝜋𝑡 (2 𝛽 + i) 8𝜋𝑡 (2 𝛽 + i) 8𝜋𝑡 implying that 2 2 2 e−2𝜎𝐸 𝑡 −i𝛽𝜎𝐸 𝑡 𝑎† 𝑨 (𝑡) 𝑨𝑎 (−𝑡) √ 𝑡 8𝜋𝑡 (2 𝛽 + i) ! 2 2 2 e−2𝜎𝐸 𝑡 −i𝛽𝜎𝐸 𝑡 −2i𝑯𝑡 𝑎 i𝑯𝑡 =e 𝑨 · √ e 𝑨 e 8𝜋𝑡 (2 𝛽𝑡 + i)   𝑡 −2𝜎𝐸2 𝑡 2 −i𝛽𝜎𝐸2 𝑡 e + 1 (|𝑡| ≤ 𝜃)i 2 + i © 𝛽 i ª® −2i𝑯𝑡 𝑎 i𝑯𝑡 = ei𝑯𝑡 𝑨𝑎† · ­­ − 1 (|𝑡| ≤ 𝜃) 𝑨 e , √ √ ®e 8𝜋𝑡 (2 𝛽𝑡 + i) 8𝜋𝑡 « ¬ i𝑯𝑡 𝑎† where √ i i 1 cos(2𝑯𝑡) sinc(2𝑯𝑡) e−2i𝑯𝑡 = √ cos(2𝑯𝑡) + √ sin(2𝑯𝑡) = i √ +𝑯 √ . 8𝜋𝑡 8𝜋𝑡 8𝜋𝑡 | {z } 8𝜋𝑡 2𝜋 2𝑯𝑡 sinc(2𝑯𝑡) 182 Finally, observe that due to parity reasons, we have ∫ ∞ i cos(2𝑯𝑡) 𝑎 i𝑯𝑡 1 (|𝑡| ≥ 𝜂)ei𝑯𝑡 𝑨𝑎† 1 (|𝑡| ≤ 𝜃) √ 𝑨 e d𝑡 𝑡 −∞ 8𝜋   ∫ ∞ −1 𝑎† cos(2𝑯𝑡) 𝑎 𝑎† cos(2𝑯𝑡) 𝑎 =√ 1 (𝜂 ≤ |𝑡| ≤ 𝜃) cos(𝑯𝑡) 𝑨 𝑨 sin(𝑯𝑡) + sin(𝑯𝑡) 𝑨 𝑨 cos(𝑯𝑡) d𝑡 𝑡 𝑡 8𝜋 −∞ ∫ ∞ −1 =√ 1 (𝜂 ≤ |𝑡| ≤ 𝜃) −∞ 8𝜋   𝑎† 𝑎 𝑎† 𝑎 cos(𝑯𝑡) 𝑨 cos(2𝑯𝑡) 𝑨 𝑯 sinc(𝑯𝑡) + sinc(𝑯𝑡)𝑯 𝑨 cos(2𝑯𝑡) 𝑨 cos(𝑯𝑡) d𝑡. Since in the above provided decomposition, every (matrix) function is bounded in the neighborhood of 0, we can obtain the 𝜂 → 0+ limit by simply removing the indicator 1 (|𝑡| ≥ 𝜂). ∫ ∞ 2 2 2 e−2𝜎𝐸 𝑡 −i𝛽𝜎𝐸 𝑡 𝑎† 𝑨 (𝑡) 𝑨𝑎 (−𝑡)𝑑𝑡 lim 1 (|𝑡| ≥ 𝜂) √ 𝑡 𝜂→0+ −∞ 8𝜋𝑡 (2 𝛽 + i)   𝑡 −2𝜎𝐸2 𝑡 2 −i𝛽𝜎𝐸2 𝑡 ∫ ∞ + 1 (|𝑡| ≤ 𝜃)i 2 𝛽 + i ©e sinc(2𝑯𝑡) ª® 𝑎 i𝑯𝑡 −2i𝑯𝑡 = ei𝑯𝑡 𝑨𝑎† · ­­ e − 1 (|𝑡| ≤ 𝜃)𝑯 √ √ ® 𝑨 e d𝑡 −∞ 8𝜋𝑡 (2 𝛽𝑡 + i) 2𝜋 « ¬ 1 +√ 8𝜋 ∫ 𝜃 cos(𝑯𝑡) 𝑨𝑎† cos(2𝑯𝑡) 𝑨𝑎 𝑯 sinc(𝑯𝑡) + sinc(𝑯𝑡)𝑯 𝑨𝑎† cos(2𝑯𝑡) 𝑨𝑎 cos(𝑯𝑡)d𝑡. −𝜃 ∫ ∞ √︁ √︁ We conclude by noting that −∞ 𝜋8 𝛿(𝑡) 𝑨𝑎† (𝑡) 𝑨𝑎 (−𝑡)𝑑𝑡 = 𝜋8 𝑨𝑎† 𝑨𝑎 . ■ Note that this exact formula could be directly and efficiently implemented using QSVT. However, the gains are minimal, as it would only reduce subnormalization from O (log(𝛽∥𝐻 ∥/𝜖)) (this being the ℓ1 norm required to achieve 𝜖 precision in Corollary 3.3.2) to O (log(𝛽∥𝐻 ∥)) (the ℓ1 norm of the weight function corresponding to the natural choice 𝜃 = 1/𝛽∥𝑯∥ in Proposition 3.6.2). More interestingly, this exact formula seems also to hold in the infinite-dimensional case, giving rise to exact detailed balance in the infinite-dimensional version as well. We leave it for future work to verify that the construction and its analysis can indeed be generalized to infinite dimensional systems. 183 Moreover, the above result actually also implies that the Lindbladian corresponding to (3.32) also well-approximates its 𝑠 → ∞ limit, i.e., the above Metropolis-like Lindbladian. The argument goes as follows: first, “round” the Hamiltonian 𝑯 to discretize its spectrum at some finite resolution ≪ 1/𝛽. Due to the form of (3.32), the resulting perturbation of the Lindbladian is bounded. Then, take the 𝑠 → ∞ limit. Since (3.33) is exponentially close to its limit (3.34) and there is a limited number of Bohr frequencies due to rounding, the resulting perturbation of the Lindbladian is once again bounded. Finally, undo the rounding, which again causes a bounded perturbation due to the exact form of Metropolis coherent term (Proposition 3.6.2). Carefully executing these bounds in the 𝜎𝐸 = 𝛽 case should show that for 𝑠 = Θ((𝛽∥𝐻 ∥ + log(1/𝜖) + 1) 2 ) the resulting Lindbladian is 𝜖-close to its 𝑠 = ∞ limit. Now we show how to approximate the exact Metropolis Lindbladian in a different efficient way. Corollary 3.3.2 (Approximate coherent term for the Metropolis-like weight). If 𝜎𝐸 = 1𝛽 , then the coherent term 𝑩 𝑀 corresponding to the Metropolis-like weight    1 𝑀 𝛾 (𝜔) = exp −𝛽 max 𝜔 + 2𝛽 , 0 satisfies 𝑀 𝑩 −𝑩 𝑀,𝜂 ≤ ∑︁  𝑎† 𝑨 𝑨 𝑎∈𝐴 𝑎  𝜂𝛽∥𝑯∥  3 min √ , O (𝜂𝛽∥𝑯∥) , 2𝜋 (3.21) where 𝑩 𝑀,𝜂 := ∑︁ ∫ ∞ 𝑎∈𝐴 −i𝛽𝑯𝑡 ∫ ∞ 𝑏 1 (𝑡)e −∞ −∞ 𝑀,𝜂 𝑏 2 (𝑡 ′) 𝑨𝑎† (𝛽𝑡 ′) 𝑨𝑎 (−𝛽𝑡 ′)d𝑡 ′ +  1 𝑎† 𝑎 i𝛽𝑯𝑡 d𝑡, √ 𝑨 𝑨 e 16 2𝜋 (3.22) with 𝑏 1 (𝑡) as in (3.19), and
1 exp −2𝑡 2 − i𝑡 + 1 (|𝑡| ≤ 𝜂)i(2𝑡 + i) √ 𝑡 (2𝑡 + i) 4 2𝜋 1 1 𝑀,𝜂 such that ∥𝑏 2 ∥ 1 < + √ ln(1/𝜂). (3.23) 5 2 2𝜋 Í  𝑎† 𝑎 Further, if 𝑎∈𝐴 𝑨 𝑨 , 𝑯 = 0, we can drop the second term in (3.22) after the ∫∞ integral in 𝑡 ′ since −∞ 𝑏 1 (𝑡) = 0. 𝑀,𝜂 𝑏 2 (𝑡) := Proof. First, let us establish the norm bound (3.23). Observe that 𝑀,𝜂 𝑏2 1 i (𝑡) = 𝑏 2𝑀,1 (𝑡) − √ 1 (𝜂 < |𝑡| ≤ 1) . 𝑡 4 2𝜋 184 Using the above identity and applying a triangle inequality and then Hölder’s inequality gives 1 i ∥ 1 ≤ (1 + 4𝑡 2 ) −1 2 (1 + 4𝑡 2 )𝑏 2𝑀,1 + √ 1 (𝜂 < |𝑡| ≤ 1) . 2 4 2𝜋 𝑡 1 ∫1 ∫∞ ∫ ∞ exp(−2𝑡 2 ) (2𝑡−i) 2 1 𝜋 To obtain (3.23), use that 𝜂 1𝑡 d𝑡 = ln(1/𝜂), −∞ (1+4𝑡 d𝑡 = 2 ) 2 d𝑡 = 4 , 1 𝑡 ∫ 2 2 2 √ 1 exp ( −2𝑡 −i𝑡 ) (2𝑡−i)+i ( 1+4𝑡 ) e−4 − 𝜋 erfc(2), and a direct computation of −1 < 16. 𝑡 𝑀,𝜂 ∥𝑏 2 Next, setting 𝜃 = 𝜂𝛽 in Proposition 3.6.2 and scaling the variables 𝑡, 𝑡 ′ by a factor of 𝛽 reveals that ∫ 1 ∑︁ ∞ 𝑀 𝑀,𝜂 𝑏 1 (𝑡)e−i𝛽𝑯𝑡 𝑸 𝑎 ei𝛽𝑯𝑡 d𝑡, (3.35) 𝑩 −𝑩 = √ 2 2 2𝜋 𝑎∈𝐴 −∞ where 1 𝑸𝑎 = 2 ∫ 𝜂 cos(𝛽𝑯𝑡 ′) 𝑨𝑎† cos(2𝛽𝑯𝑡 ′) 𝑨𝑎 𝛽𝑯 sinc(𝛽𝑯𝑡 ′) −𝜂 + sinc(𝛽𝑯𝑡 ′) 𝛽𝑯 𝑨𝑎† cos(2𝛽𝑯𝑡 ′) 𝑨𝑎 cos(𝛽𝑯𝑡 ′)d𝑡 ′ ∫ 𝜂
′ ′ − ei𝛽𝑯𝑡 𝑨𝑎† 𝛽𝑯 sinc(2𝛽𝑯𝑡 ′) 𝑨𝑎 ei𝛽𝑯𝑡 d𝑡 ′ . −𝜂 ′ We can decompose in the second term ei𝛽𝑯𝑡 = cos(𝛽𝑯𝑡 ′) + i sin(𝛽𝑯𝑡 ′). Due to parity reasons, we can see that the second term can be replaced by ∫ 𝜂

− cos(𝛽𝑯𝑡 ′) 𝑨𝑎† 𝛽𝑯 sinc(2𝛽𝑯𝑡 ′) 𝑨𝑎 cos(𝛽𝑯𝑡 ′) − sin(𝛽𝑯𝑡 ′) 𝑨𝑎† 𝛽𝑯 sinc(2𝛽𝑯𝑡 ′) 𝑨𝑎 sin(𝛽𝑯𝑡 ′)d𝑡 ′ . −𝜂 Now, let us define 𝜇𝑎 ( 𝑿) := cos( 𝑿) 𝑨𝑎† cos(2𝑿) 𝑨𝑎 𝑿 sinc( 𝑿) + sinc( 𝑿) 𝑿 𝑨𝑎† cos(2𝑿) 𝑨𝑎 cos( 𝑿)

− 2 cos( 𝑿) 𝑨𝑎† 𝑿 sinc(2𝑿) 𝑨𝑎 cos( 𝑿) + 2 sin( 𝑿) 𝑨𝑎† 𝑿 sinc(2𝑿) 𝑨𝑎 sin( 𝑿), so that 𝜇𝑎 (𝛽𝑯𝑡 ′) ′ d𝑡 . 2𝑡 ′ −𝜂 ∫ 𝜂 𝑸𝑎 = If 𝑿 is Hermitian, by the triangle inequality, we can see that ∥𝜇𝑎 ( 𝑿) ∥ ≤ 6∥ 𝑨𝑎 ∥ 2 ∥ 𝑿 ∥. Moreover, from its definition, we can see that 𝜇𝑎 ( 𝑿) is an odd analytic entire 185 function, and is 0 at 𝑿 = 0. From this it directly follows that  its derivative  𝜇𝑎 ( 𝑿) = O ∥ 𝑨𝑎 ∥ 2 ∥ 𝑿 ∥ 3 for every Hermitian 𝑿. Due to the triangle inequality,  ∫ 𝜂 ∫ 𝜂  ∥𝜇𝑎 (𝛽𝑯𝑡 ′) ∥ ′ 6  𝑎 2 ′ 2 ∥𝑸 𝑎 ∥ ≤ ∥ 𝑨 ∥ ∥ 𝛽𝑯∥ min , O ∥ 𝛽𝑯𝑡 ∥ d𝑡 ′ d𝑡 ≤ ′ 2|𝑡 | 2 −𝜂 −𝜂    𝑎 2 3 = ∥ 𝑨 ∥ min 6𝜂𝛽∥𝑯∥, O (𝜂𝛽∥𝑯∥) . Finally, we can prove (3.21) for individual jumps by combining the above bound and the inequality ∥𝑏 1 ∥ 1 ≤ 1 within (3.35). To obtain the bound in terms of the Í norm-of-sum ∥ 𝑎 𝑨𝑎† 𝑨𝑎 ∥, note that for any matrices 𝑿, 𝒀, 𝒁, we have ∑︁ ∑︁ ∑︁ ∥ 𝑿 𝑨𝑎†𝒀 𝑨𝑎 𝒁∥ ≤ ∥ 𝑿 ∥ · ∥ 𝑨𝑎† ⊗ ⟨𝑎|∥ · ∥𝒀 ⊗ 𝑰∥ · ∥ 𝑨𝑎 ⊗ |𝑎⟩∥ · ∥ 𝒁∥ 𝑎 𝑎 = ∥ 𝑿 ∥∥𝒀 ∥∥ 𝒁∥ · ∥ 𝑎 ∑︁ 𝑎† 𝑎 𝑨 𝑨 ∥. 𝑎 And rewrite 𝜇𝑎 ( 𝑿) as a linear combination of terms each bounded by O (∥ by canceling out the linear O ( 𝑿) terms. 3.7 Í 𝑎 𝑨 𝑎† 𝑨𝑎 ∥∥ 𝑿 ∥ 3 ) ■ Appendix:The discriminant gap and area law The fact that the discriminant H 𝛽 can be regarded as frustration-free, spatially local Hamiltonian (Proposition 3.1.1) has direct implications on the “locality” of its zero-eigenstate, the purified Gibbs state. In particular, existing analytic techniques for studying gapped phases immediately lead to explicit low-depth circuits assuming a large discriminant gap for H 𝛽 . Elegantly, lowering the temperature can now be interpreted as the Nature-given adiabatic paths parameterized by the inverse temperature 𝛽 (known as quantum simulated annealing). The particularly helpful tool is quasi-adiabatic evolution [85], a version of adiabatic evolution that exploits spatial locality. Lemma 3.7.1 (Quasi-adiabatic evolution [15, Proposition 2.4]). Consider a oneparameter family of Hamiltonians 𝑯(𝑠) for each 𝑠 ∈ [0, 1] with minimal gap Δ. Then, the family of ground-state projectors can be generated by a time-dependent Hamiltonian 𝑷′ (𝑠) = i[𝑾 (𝑠), 𝑷(𝑠)] where ∫ ∞ ∫ 𝑡 𝑾 (𝑠) := d𝑡 𝑤(𝑡) 𝑑𝑢 ei𝑢𝑯(𝑠) 𝑯′ (𝑠)e−i𝑢𝑯(𝑠) −∞ 0 for each 𝑠 ∈ [0, 1] 186 for any weight function 𝑤(𝑡) satisfying ∫ ∞ 𝑤(𝑡)d𝑡 = 1 and 𝑤(𝜔) ˜ =0 if |𝜔| ≥ |Δ|. −∞ For geometrically local Hamiltonian on 𝑑-dimention lattice with single-site jumps ∥ 𝑨𝑎 ∥ = 1 (note that the natural normalization here is to make the strength extensive Í ∥ 𝑎∈𝐴 𝑨𝑎† 𝑨𝑎 ∥ ∝ 𝑛), we can plug the discriminant into the above by 𝑯(𝑠) → H 𝑠𝛽 and Δ → min 𝜆 𝑔𝑎 𝑝 ( H 𝑠𝛽 ). 𝑠 Strictly speaking, we are interested in the top eigenvector instead of the ground state due to its Lindbladian origin (Lindbladian spectrum is always nonpositive), but this is also handled by quasi-adiabatic continuation. We obtain a unitary circuit preparing the purified Gibbs state driven by the following time-dependent Hamiltonian: ∫ 𝑡 ∑︁ ∫ ∞ ′ 𝑾 (𝑠) = d𝑡 𝑤(𝑡) 𝑑𝑢 ei𝑢H 𝑠𝛽 H 𝑎𝑠𝛽 e−i𝑢H 𝑠𝛽 . (3.36) 0 𝑎 |−∞ {z } =:𝑾 𝑎 (𝑠) Algorithmically, the above should be considered an alternative to modernized adiabatic algorithms based on amplitude amplification [40, Appedix G]. The Heisenberg evolution is particularly powerful in the case of local Hamiltonians as it manifestly gives a quasi-local unitary circuit. Based on existing tools, we quickly instantiate several notations of locality. First, the unitary preserves spatially local operators. Proposition 3.7.1 (Quasi-local unitary [15]). Consider the one-parameter family of unitaries generated by the time-dependent Hermitian matrix 𝑾 (𝑠) as in (3.36) 𝑽 ′ (𝑠) = i𝑾 (𝑠)𝑽 (𝑠) for each 𝑠 ∈ [0, 1]. Then, the unitary satisfies a Lieb-Robinson bound with almost exponential decay   𝑑 † ∥ [𝑽 (𝑠)𝑶 1𝑽 (𝑠), 𝑶 2 ] ∥ ≤ O exp(− ) where 𝑑 = 𝑑 (𝑶 1 , 𝑶 2 ). Poly𝐷 (𝛽) ln(𝑑 2 ) Second, the unitary generates limited entanglement. Since the unitary is generated by (quasi)-local operators, integrating the entangling rate from 0 → 𝛽 13 gives an area law of entanglement [171] for purified Gibbs state. 13The infinite temperature Gibbs state is the maximally entangled state. It is a tensor product of bell pairs across the two copies and has trivially entanglement across regions. 187 Corollary 3.7.1 (Thermal area-law from entangling rates (similar argument to [171])). The purified Gibbs state satisfies an entanglement area-law (the entanglement entropy for any region 𝑅) ≤ Poly𝐷 (𝛽) · |𝜕𝑅|. Δ The above thermal area-law is weaker than existing bounds14 and should be thought of as a sanity check. However, what’s new is the explicitly parameterized low-depth unitary circuit (which is not a consequence of the area law). The unitary 𝑽 (1) corresponding to (3.36) for a 𝐷-dimensional lattice Hamiltonian can be decomposed into a depth Poly𝐷 (𝛽) · Poly log(𝑛/𝜖) Δ circuit of 2-qubit gates up to 𝜖-error in the operator norm using the HHKL algorithm [80]. The discriminant gap Δ is expected to be independent of the system size in the rapid mixing regime (under the normalization ∥ 𝑨𝑎 ∥ = 1). To obtain gapped ground states, we could scale the inverse temperature with the Hamiltonian gap Δ(𝑯)(not to confuse with the discriminant gap Δ)
𝛽 = O Δ(𝑯) log(𝑛/𝜖) . Of course, there might be phase transitions at low temperatures where the discriminant gap may close. The logarithmic dependence on the system size marks a technical difference from the area law setting. Nevertheless, the explicit circuit appears novel and could potentially lead to new numerical ansatz for low-energy states. The Lindbladian case Of course, we may apply the same line of thought to Lindbladians, assuming the quasi-local Lindbladian mixes in logarithmic time. However, there are subtle differences. The spectral gap of the parent Hamiltonian is qualitatively the same as the mixing time of our Lindbladian (which has a vanishing anti-Hermitian part) ln(2) ln(2∥ 𝝆 −1/2 ∥) ≤ 𝑡 𝑚𝑖𝑥 (L) ≤ . 𝜆 𝑔𝑎 𝑝 (H ) 𝜆 𝑔𝑎 𝑝 (H ) The conversion overhead ln(2∥ 𝝆 −1/2 ∥) = O (𝛽∥𝑯∥) could scale with the system size in general. However, to drop this O (𝛽∥𝑯∥) factor (i.e., proving rapid mixing 14 Ref. [104] states that the mutual information between regions satisfies 𝐼 (𝑅 : 𝑅 ′ ) ≤ O (𝛽2/3 )|𝜕𝑅| in any 𝐷-dimension lattice. 188 such that the mixing time scaling logarithmically with the system size) amounts to proving a more stringent quantum log-Sobelov inequality, which has been highly nontrivial [31]. Even if we assume rapid mixing, to get a low-depth circuit for the Gibbs state, we still need an efficient algorithm to implement the Lindbladian evolution that parallelizes the Lindblad operators. However, the HHKL algorithm [80] achieving this parallelization for local Hamiltonians made critical use of reversing time evolutions, which does not obviously apply to the dissipative setting. As a remedy, one may consider trotterizing into quasi-local brickwork nonunitary circuits, giving a discrete-time Quantum Markov chain15 Ö ∑︁ eL𝜃 → N := eL𝑔 𝜃 where L = L𝑔 𝑔 such that each 𝑔 L𝑔 is a sum of quasi-local, nearly commuting block. Í That is, we regroup the Lindbladian L = 𝑎∈𝐴 L 𝑎 into quasi-local blocks to exploit the fact that nearly disjoint Lindbladians can be efficiently implemented in parallel (after spatial truncation). However, the Trotter error from first-order product formulas is extensive; thus, the discrete-time channel would not approximate the continuoustime evolution in general, and the mixing time analysis might require quantitatively different formalism. Nevertheless, the Gibbs state remains stationary N [𝝆 𝛽 ] = 𝝆 𝛽 , and could potentially share similar mixing behavior as the continuous case in practice. 3.8 Appendix:Why Gaussians? Our direct calculation confirms the correctness of our ansatz. But why does the filter 𝑓 (𝑡) need to be Gaussian (3.3)? In this section, we try to derive the Gaussians from scratch, which can be viewed as an alternative view of detailed balance in the time domain. While we try to make our arguments precise, we stop at physicists’ level of rigor and do not attempt to extract a mathematical theorem. Assume that the filter function is real on the real axis 𝑓 (𝑡) = 𝑓 (𝑡) ∗ for each 𝑡 ∈ R and complex analytic, independent of 𝛽, and that the jump operator 𝑨 = 𝑨† is Hermitian (Let us focus on a single jump and drop the jump labels 𝑎 ∈ 𝐴.). The goal is to solve for the viable choices of 𝑓 (𝑡). 15The observation that discretization can be helpful despite deviating from the continuum was made in [61]. 189 We calculate the associated “transition” part (3.1) in the time domain ∫ ∞ † † ˆ ˆ T [·] = 𝛾(𝜔) 𝑨(𝜔) (·) 𝑨(𝜔)d𝜔 −∞ ∫ ′ = 𝛾(𝜔)ei𝜔(𝑡−𝑡 ) 𝑓 (𝑡) 𝑓 (𝑡 ′) 𝑨(𝑡)(·) 𝑨(𝑡 ′)d𝜔d𝑡d𝑡 ′ (using 𝑓 (𝑡) = 𝑓 (𝑡) ∗ ) ∫ 1 = √ 𝑐(𝑡 − 𝑡 ′) 𝑓 (𝑡) 𝑓 (𝑡 ′) 𝑨(𝑡)(·) 𝑨(𝑡 ′)d𝑡d𝑡 ′ . 2𝜋 The third line uses the inverse Fourier Transform ∫ ∞ 1 𝑐(𝑡) = √ ei𝜔𝑡 𝛾(𝜔)d𝜔. 2𝜋 −∞ We may interpret 𝑐(𝑡) as a certain correlation function (hence the notation). Now, note that conjugating the Gibbs state yields ∫ ∞ ˆ Λ( 𝑨(𝜔)) = e−i𝜔𝑡 𝑓 (𝑡)ei𝑯(𝑡−i𝛽/2) 𝑨e−i𝑯(𝑡−i𝛽/2) d𝑡 ∫−∞ = e−i𝑠𝜔 e 𝛽𝜔/2 𝑨(𝑠) 𝑓 (𝑠 + i𝛽/2)d𝑠 (setting 𝑠 := 𝑡 − i𝛽/2). Then, we get that Γ−1 ◦ T ◦ Γ[·] ∫ ∞ −1 ˆ ˆ = 𝛾(𝜔)Λ( 𝑨(𝜔))(·)Λ ( 𝑨(𝜔) † )d𝜔 −∞ ∫ ∞∫ ∫ ′ 𝛾(𝜔)e−i𝜔(𝑠−𝑠 +i𝛽) 𝑓 (𝑠 + i𝛽/2) 𝑓 (𝑠′ − i𝛽/2) 𝑨(𝑠)(·) 𝑨(𝑠′)d𝑠d𝑠′d𝜔 = −∞ (setting 𝑠′ := 𝑡 ′ + i𝛽) 1 =√ 2𝜋 ∫ ∫ 1 =√ 2𝜋 ∫ ∫ 𝑐(𝑠′ − 𝑠 − i𝛽) 𝑓 (𝑠 + i𝛽/2) 𝑓 (𝑠′ − i𝛽/2) 𝑨(𝑠)(·) 𝑨(𝑠′)d𝑠d𝑠′ (continuing 𝑐(𝑧), 𝑧 ∈ C) 𝑐(𝑡 ′ − 𝑡 − i𝛽) 𝑓 (𝑡 + i𝛽/2) 𝑓 (𝑡 ′ − i𝛽/2) 𝑨(𝑡)(·) 𝑨(𝑡 ′)d𝑡d𝑡 ′ . (shifting integration) The third line uses that ( 𝑨(𝑧)) † = 𝑨(𝑧∗ ) for Heisenberg evolution at complex times. The last line shifts the integrals by 𝑠 → 𝑡 and 𝑠′ → 𝑡 ′ (assuming the absence of poles across the strip). Comparing the coefficient of product integrals over 𝑨(𝑡) · 𝑨(𝑡 ′) for each 𝑡, 𝑡 ′, the condition 𝑐(𝑡 − 𝑡 ′) 𝑓 (𝑡) 𝑓 (𝑡 ′) = 𝑐(𝑡 ′ − 𝑡 − i𝛽) 𝑓 (𝑡 + i𝛽/2) 𝑓 (𝑡 ′ − i𝛽/2) for each 𝑡, 𝑡 ′ ∈ R ensures T † = Γ−1 ◦ T ◦ Γ. 190 This condition can be rearranged as (whenever the denominators are non-zero) 𝑐(𝑡 − 𝑡 ′) 𝑓 (𝑡 + i𝛽/2) 𝑓 (𝑡 ′ − i𝛽/2) = 𝑐(𝑡 ′ − 𝑡 − i𝛽) 𝑓 (𝑡) 𝑓 (𝑡 ′) for each 𝑡, 𝑡 ′ ∈ R. (3.40) Solving the functional equation We proceed to solve (3.40) for a fixed 𝛽. Consider a change of variable 𝑡 ↔ 𝑡 ′ 𝑐(𝑡 ′ − 𝑡) 𝑓 (𝑡 ′ + i𝛽/2) 𝑓 (𝑡 − i𝛽/2) = . 𝑐(𝑡 − 𝑡 ′ − i𝛽) 𝑓 (𝑡 ′) 𝑓 (𝑡) (3.41) Then, divide the two equations and observe that the LHS depends only on the time difference 𝑡 − 𝑡 ′ (3.40)/(3.41) : 𝑓1 (𝑡 − 𝑡 ′) = 𝑓 (𝑡 + 𝑖𝛽/2) 𝑓 (𝑡 ′ − i𝛽/2) · := 𝑔(𝑡)/𝑔(𝑡 ′). 𝑓 (𝑡 − i𝛽/2) 𝑓 (𝑡 ′ + i𝛽/2) We must have that 𝑔(𝑡) = 𝑎e𝑏𝑡 for constants 𝑎, 𝑏 independent of 𝑡. Thus, 𝑎e𝑏𝑡 𝑓 (𝑡 − i𝛽/2) = 𝑓 (𝑡 + i𝛽/2) =⇒ 𝑎e𝑏𝑡 e−𝑖𝛽/2·𝜕𝑡 𝑓 (𝑡) = e𝑖𝛽/2·𝜕𝑡 𝑓 (𝑡) =⇒ e𝑎 +𝑏 𝑡−𝑖𝛽·𝜕𝑡 𝑓 (𝑡) = 𝑓 (𝑡) ′ ′ (analyticity) (since [𝑡, 𝜕𝑡 ] ∝ 1). The last equality regards 𝑡 and 𝜕𝑡 as linear operators acting on functions and use the matrix-exponential fact that [ 𝑿, 𝒀] commutes with 𝑿, 𝒀 implies e 𝑿 e𝒀 = e 𝑿+𝒀+[ 𝑿,𝒀]/2 . We can get rid of the exponential by regarding it as an eigenproblem e 𝑿 |𝑣⟩ = |𝑣⟩ implies 𝑿 |𝑣⟩ = i2𝜋Z · |𝑣⟩. This amounts to solving the differential equation (𝑎 1 𝜕𝑡 + 𝑎 2 𝑡 + 𝑎 3 ) 𝑓 (𝑡) = 0 which has the general solution being Gaussians (the constants may depend on the fixed 𝛽.). Plugging back into (3.40), we can solve for 𝑐(𝑡), which would not be unique as it allows for linear combinations. 191 As a sanity check, we re-derive the constraints between the Gaussian parameters and the temperature in the time picture. Let 2 2 𝑐(𝑥) = 𝑒 −𝑖𝑎𝑥 e−𝑥 /𝛿 , a𝑛𝑑 2 2 𝑓 (𝑡) = 𝑒 −𝑡 /𝜅 , where 𝑎, 𝛿, 𝜅 are all real parameters. Then, letting 𝑡 − 𝑡 ′ := 𝑥 we get:     𝑐(𝑡 − 𝑡 ′) 𝑓 (𝑡 + i𝛽/2) 𝑓 (𝑡 ′ − i𝛽/2) log = log 𝑐(𝑡 ′ − 𝑡 − i𝛽) 𝑓 (𝑡) 𝑓 (𝑡 ′) " # " # 2 2 2 2 ′ 2 2 𝑒 −𝑖𝑎𝑥 e−𝑥 /𝛿 𝑒 −(𝑡+i𝛽/2) /𝜅 e−(𝑡 −i𝛽/2) /𝜅 =⇒ log = log , 2 2 2 2 ′2 2 𝑒𝑖𝑎(𝑥+i𝛽) e−(𝑥+i𝛽) /𝛿 𝑒 −𝑡 /𝜅 −𝑡 /𝜅 𝛽 −𝑖𝑥 𝛽 𝛽2 𝛽 + 2. it suffices if 2𝑖𝑥( 2 − 𝑎) + 𝑎𝛽 − 2 = 𝛿 𝛿 𝜅2 2𝜅 Equating the imaginary and real parts leads to two linearly dependent equations with the same solution: 1 1 𝑎 = 𝛽( 2 + 2 ). 𝛿 2𝜅 −2 2 −2 Then identifying 𝑎 ≡ 𝜔 𝛾 , 𝛾 = 𝜎𝛾 /2, and 𝜅 = 𝜎𝐸2 leads to the same relationship between 𝜔 𝛾 , 𝜎𝛾2 , 𝜎𝐸2 and 𝛽 as in (3.4). 3.9 Appendix:Other notions of detailed balance In addition to our KMS detailed balance condition (Definition 3.2.1), other quantum detailed balance has also been studied (see, e.g., [8, 32, 69]). In this section, we will discuss two variants of quantum detailed balance, and only the first one seems to work. Detailed balance with unitary drift We constructed a Lindbladian satisfying the KMS detailed balance condition. However, much of our results remain to hold even if we add a suitable Hamiltonian term, which might enable alternative constructions. Definition 3.9.1 (KMS-detailed balance with unitary drift [69, Section 5]). We say that the Lindbladian L satisfies 𝝆-detailed balance with unitary drift (in short 𝝆-DBU) with respect to a full-rank state 𝝆 if there exists a Hermitian operator 𝑸 such that √ −1 √ √ √ −1 (3.45) L † [·] − 𝝆 L [ 𝝆 · 𝝆] 𝝆 = i[𝑸, ·]. Or, in terms of discriminants, A ( 𝝆, L) := D ( 𝝆, L) − D ( 𝝆, L) † = −i𝝆 1/4 [𝑸, 𝝆 −1/4 (·) 𝝆 −1/4 ] 𝝆 1/4 . 2 192 That is, we relax the KMS detailed balance condition (Definition 3.2.1) by allowing the RHS to be any commutator term i[𝑸, ·] 16; this enlarges the family of possible Lindbladian for the stationary state 𝝆. Proposition 3.9.1 (Fixed point). If a Lindbladian L is 𝝆-detailed-balanced with unitary drift, then L [𝝆] = 0. √ √ √ √ Proof. Evaluate the superoperator (3.45) for the identity 𝑰 and conclude 𝝆 −1 L [ 𝝆𝑰 𝝆] 𝝆 −1 = L † [𝑰] = 0. ■ Intuitively, we can certainly add any Hermitian 𝑸 that commutes with 𝝆 without changing the stationary state. In fact, this is the only possibility17. Proposition 3.9.2 (Structure of 𝑸 [69, Lemma 28]). In the setting of Definition 3.9.1, 𝑸 must commute with 𝝆. Indeed, we can solve for 𝑩 and 𝑸 by modifying the argument for Corollary 3.2.1 𝑸 = 𝑸† ⇕ (Λ − Λ−1 )(𝑩) = i (2I − (Λ + Λ−1 ))( 𝑹) 2 ⇕ ∑︁ (e 𝛽𝜈 2 −e −𝛽𝜈 2 ) 𝑩𝜈 = 𝜈∈𝐵 𝛽𝜈 𝛽𝜈 i ∑︁ (2 − e− 2 − e 2 ) 𝑹𝜈 2 𝜈∈𝐵 ⇕ ∑︁ 𝜈∈𝐵\{0}   𝛽𝜈 i ∑︁ 𝑩𝜈 = tanh 𝑹𝜈 . 2 4 𝛽𝜈 −𝛽𝜈 (e 2 − e 2 ≠ 0 ⇐⇒ 𝜈 ≠ 0) 𝜈∈𝐵\{0} Thus, allowing the term 𝑸 ≠ 0 merely amounts to dropping constraints on the 𝑩0 component (which is exactly the set of operators that commute with the Hamiltonian). This relaxed version of detailed balance comes with the conceptual price of making the Lindbladian “non-self-adjoint” under similarity transformation. Due to the antiself-adjoint component A ( 𝝆, L) † , the spectral theory of convergence (at first glance) 16 In an email exchange with Jonathan Moussa, he mentioned analogous unitary effect for discrete quantum channels. 17We thank Jonathan Moussa for pointing us to [69, Lemma 28]. This further simplifies the derivation and the presentation, clarifying that our construction actually has 𝑸 = 0 and a vanishing anti-Hermitian component. 193 seems to break since the right eigenvectors are not orthogonal in the Hilbert-Schmidt norm. Fortunately, the following observation comes to the rescue. Proposition 3.9.3. If a Lindbladian L is 𝝆-detailed-balanced with unitary drift, then √ √ A ( 𝝆, L) [ 𝝆] = H ( 𝝆, L) [ 𝝆] = 0. In other words, the specific form of detailed balance implies that the anti-self-adjoint √ √ component preserves the eigenvector 𝝆; the particular eigenvector 𝝆, which we care about, is orthogonal to other eigenvectors. Therefore, the second eigenvalue of H corresponds to the contraction of the Hilbert-Schmidt norm of the other eigenvector, controlling the mixing time. Proposition 3.9.4 (Mixing time from spectral gap (adapted from [40, Proposition II.2])). If a Lindbladian L satisfies 𝝆-DBU, then 𝑡 𝑚𝑖𝑥 (L) ≤ ln(2∥ 𝝆 −1/2 ∥) 𝜆 𝑔𝑎 𝑝 (H ) where H := D ( 𝝆, L) † + D ( 𝝆, L) , 2 and the mixing time 𝑡 𝑚𝑖𝑥 is the smallest time for which eL𝑡 𝑚𝑖 𝑥 [𝝆1 − 𝝆2 ] 1 ≤ 1 ∥ 𝝆1 − 𝝆2 ∥ 1 2 for any states 𝝆1 , 𝝆2 . In other words, adding a coherent term only seems to help with the convergence. Proof. Write 𝑹 = 𝝆1 − 𝝆2 , then eL𝑡 [𝑹] 1 = 𝝆 1/4 eD𝑡 [𝝆 −1/4 𝑹𝝆 −1/4 ] 𝝆 1/4 ≤ 𝝆 1/4 ≤e ≤e · eD𝑡 [𝝆 −1/4 𝑹𝝆 −1/4 ] 4 −𝜆 𝑔𝑎 𝑝 (H )𝑡 −𝜆 𝑔𝑎 𝑝 (H )𝑡 𝝆 −1/4 ∥𝝆 𝑹𝝆 1 2 · 𝝆 1/4 4 −1/4 2 −1/4 2 ∥ ∥ 𝑹∥ 2 ≤ e−𝜆𝑔𝑎 𝑝 (H )𝑡 ∥ 𝝆 −1/4 ∥ 2 ∥ 𝑹∥ 1 † = e−𝜆𝑔𝑎 𝑝 (L )𝑡 ∥ 𝝆 −1/2 ∥∥ 𝑹∥ 1 . The first inequality uses Hölder’s inequality. The second inequality uses the √ orthogonality to the leading eigenvector such that Tr[ 𝝆 · 𝝆 −1/4 𝑹𝝆 −1/4 ] = Tr[𝑹] = 0. Take the logarithm to conclude the proof. ■ 194 However, if we wish to obtain a parent Hamiltonian corresponding to this Lindbladian, the anti-Hermitian component forces us to consider the symmetrized discriminant D + D† H𝜷 = . 2 Unfortunately, the presence of the anti-Hermitian component breaks the qualitative analogy between mixing times and discriminant gaps; fast mixing can hold despite a small symmetrized discriminant gap, marking a limitation of the parent Hamiltonian approach. Incorporating a coherent term, the general conversion bound is as follows [40]; unfortunately, some bounds are only meaningful if the anti-Hermitian part A is small enough. Proposition 3.9.5 (Spectral gap from mixing time [40, Proposition E.5]). For any Lindbladian L, let −𝜆 Re(𝑔𝑎 𝑝) (L) be the second largest real part in its spectrum (counted by algebraic multiplicity), then 𝜆 𝑔𝑎 𝑝 (H ) + 2∥A ∥ 2−2 ≥ ∥A ∥ 2−2 − 𝜆 2 (H ) ≥ 𝜆 Re(𝑔𝑎 𝑝) (L) ≥ ln(2) . (3.46) 𝑡 𝑚𝑖𝑥 (L) Moreover, if 𝜆 Re(𝑔𝑎 𝑝) (L) ≥ 2∥A ∥ 2−2 , then there is unique eigenvalue 𝜆 1 (H ) ≥ −∥A ∥ 2−2 and 𝜆 𝑔𝑎 𝑝 (H ) + 2∥A ∥ 2−2 ≥ ∥A ∥ 2−2 − 𝜆 2 (H ) ≥ 𝜆 Re(𝑔𝑎 𝑝) (L). (3.47) Corollary 3.9.1 (Spectral gap from mixing time). If a Lindbladian L satisfies ln(2) 𝝆-DBU and 2𝑡 𝑚𝑖 ≥ ∥A ∥ 2−2 , then18 𝑥 (L) 𝜆 𝑔𝑎 𝑝 (H ) ≥ ln(2) . 2𝑡 𝑚𝑖𝑥 (L) ln(2) Proof. Observe that 2𝑡 𝑚𝑖 ≥ ∥A ∥ 2−2 implies 𝜆 Re(𝑔𝑎 𝑝) (L) ≥ 2∥A ∥ 2−2 due to 𝑥 (L) (3.46), which in turn by (2.79) also implies that 𝜆1 (H ) = 0 (since 0 is an eigenvalue due to Proposition 3.9.3). Thus combining (3.46)-(3.47) yields ∥A ∥ 2−2 + 𝜆 gap (H ) ≥ 𝜆Re(𝑔𝑎 𝑝) (L) ≥ ln(2) . 𝑡 𝑚𝑖𝑥 (L) ■ 18The bound cannot be qualitatively strengthened without additional structural understanding or assumptions, e.g., consider 𝑯 := 𝒁 − (1 + 𝜖) 𝑰, 𝑨 := i𝑿; the eigenvalues of 𝑯 + 𝑨 are −1 − 𝜖, while 𝜆 1 (𝑯) = −𝜖, ∥ 𝑨∥ = 1, and the relaxation (mixing) time is O (1). 195 The proof of the above proceeds by showing that 𝜆 Re(𝑔𝑎 𝑝) (L) ≥ 𝑡 𝑚𝑖ln(2) , however 𝑥 (L) without any prior knowledge on ∥A ∥ 2−2 it does not seem to be possible to lower bound 𝜆 𝑔𝑎 𝑝 (H ) in general. Indeed, if the Lindbladian L satisfies 𝝆-detailed balance with unitary drift, then L and D ( 𝝆, L) = H + A are related by a similarity transform and √ are thus co-spectral. Due to Proposition 3.9.3 we know that 𝑉 := Span( 𝝆) ⊥ is an invariant subspace of both H and A, and thus also of L. Since H|𝑉 ⪯ −𝜆 𝑔𝑎 𝑝 (H )I † and A |𝑉 is anti-Hermitian we get that every eigenvalue of L |𝑉 has real part at most −𝜆 𝑔𝑎 𝑝 (H ), thus 𝜆 Re(𝑔𝑎 𝑝) (L) ≥ 𝜆 𝑔𝑎 𝑝 (H ), which is an inequality in the “wrong” direction. 𝑠-detailed balance Our KMS detailed balance condition (Definition 3.9.2) is a special case (𝑠 = 1/2) of a larger family of 𝑠-detailed balance condition. Definition 3.9.2 (𝑠-detailed balance condition). For a normalized, full-rank state 𝝆 ≻ 0 and an scalar 0 ≤ 𝑠 ≤ 1, we say that an super-operator L satisfies (𝑠, 𝝆)-detailed balance (or 𝑠-DB in short) if L † [·] = 𝝆 𝑠−1 L [𝝆 1−𝑠 · 𝝆 𝑠 ] 𝝆 −𝑠 . Since 𝝆 is an operator, different choices of 0 ≤ 𝑠 ≤ 1 yield different detailed balance conditions. Nevertheless, they all prescribe the same fixed point. Proposition 3.9.6 (Fixed point). If a Lindbladian L is (𝑠, 𝝆)-detailed-balanced, then L [𝝆] = 0. The case 𝑠 = 0 corresponds to the so-called Gelfand-Naimark-Segal (GNS) inner product. One naturally wonders if our constructions also apply here. Unfortunately, the case 𝑠 = 1/2 appears to be a special point in the interval 0 ≤ 𝑠 ≤ 1. Let us revisit the energy domain representation of detailed balance (Proposition 3.2.2) for the transition part. Consider a super-operator parameterized by a Hamiltonian 𝑯, 𝛽, and a set of operators including its adjoints { 𝑨𝑎 : 𝑎 ∈ 𝐴} = { 𝑨𝑎† : 𝑎 ∈ 𝐴}: ∑︁ ∑︁ T = 𝛼𝜈1 ,𝜈2 𝑨𝜈𝑎1 (·)( 𝑨𝜈𝑎2 ) † . 𝑎∈𝐴 𝜈1 ,𝜈2 ∈𝐵 Then, the 𝑠-detailed balance condition T † [·] = 𝝆 𝑠−1 T [𝝆 1−𝑠 · 𝝆 𝑠 ] 𝝆 −𝑠 𝛼𝜈1 ,𝜈2 = 𝛼−𝜈2 ,−𝜈1 exp(−𝛽(1 − 𝑠)𝜈1 − 𝛽𝑠𝜈2 ) for each demands that 𝜈1 , 𝜈2 ∈ 𝐵. 196 However, this appears to be a strong condition; if we apply a change of variable (𝜈1 , 𝜈2 ) → (−𝜈2 , −𝜈1 ), 𝛼−𝜈2 ,−𝜈1 = 𝛼𝜈1 ,𝜈2 exp(𝛽(1 − 𝑠)𝜈2 + 𝛽𝑠𝜈1 ) for each 𝜈1 , 𝜈2 ∈ 𝐵. Multiply the two to see that (see [32, Lemma 2.5] for an abstract argument) 𝛼𝜈1 ,𝜈2 𝛼−𝜈2 ,−𝜈1 = 𝛼𝜈1 ,𝜈2 𝛼−𝜈2 ,−𝜈1 exp(𝛽(1 − 2𝑠)(𝜈2 − 𝜈1 )) for each 𝜈1 , 𝜈2 ∈ 𝐵. Therefore, if 𝑠 ≠ 21 and 𝛽 ≠ 0, we must have that 𝛼𝜈1 ,𝜈2 = 0 if 𝜈1 ≠ 𝜈2 (3.48) which contradicts our construction, and currently, we do not know how to algorithmically ensure (3.48). The only existing Lindbladian we knew that satifies (3.48) is the Davies’ generator [56], which requires resolving the level spacing using a (exponentially) long Hamiltonian simulation time. 197 Chapter 4 COOLING IS UNIVERSAL FOR QUANTUM COMPUTATION 4.1 Introduction Finding ground states and other low-energy states of quantum many-body systems is a central problem in physics, materials science, and chemistry. To address this problem, many powerful computational methods, such as density functional theory (DFT) [89, 103], quantum Monte Carlo (QMC) [17, 34, 155], variational optimization with tensor network ansatzes [82, 93, 145, 156, 173, 177, 178] or neural network ansatzes [33, 59, 88], and data-driven machine learning approaches [71, 91, 111, 147], have been developed. These methods, which can be implemented using classical computers, work well for many physically relevant problem instances but fail badly in other cases. Quantum computers may be able to find ground states efficiently in cases that are too hard to solve classically, but conclusively identifying such cases is challenging. Even when known classical methods falter, one cannot easily rule out that alternative classical methods will have much better performance. Furthermore, finding ground states of local Hamiltonians is known to be QMA-hard [98, 100], and therefore is expected to be intractable even for quantum computers for some physical systems. To establish the efficacy of existing quantum algorithms for the ground-state problem, assumptions are required that might not be justified [108], such as the presence of a trial state with sufficient ground state overlap [70, 114] or a parameterized adiabatic path whose spectral gap remains open [65]. If finding ground states is hard for quantum computers, we expect that the task is hard for Nature as well. In searching for physics problems where quantum computers could have an advantage over classical computers, we should follow Nature’s guidance. With that in mind, we observe that when a quantum system is placed in a low-temperature thermal bath [28, 55, 115, 135], it seeks a local minimum of the energy which may be far from the global minimum. For some physical systems, such as spin glasses [23, 63, 99, 136], finding a ground state is indeed known to be hard; such systems, when cooled, almost always find a local minimum instead of the ground state. In these cases, the ground state of the Hamiltonian is physically irrelevant because it is unlikely to be observed in experiments. 198 Low energy Thermal perturbation Ground State Suboptimal local minimum (c) Thermal perturbation E lan ne ds rgy ca pe (b) E lan nerg y ds ca pe (a) High energy Ground State Local unitary perturbation Energy landscape Ground State Figure 4.1: (a) Energy landscape with multiple local minima. Some Hamiltonians, such as magnets in a small external magnetic field, have multiple local minima. One is the ground state while the other local minima are suboptimal. (b) Energy landscape with one local minimum. For some local Hamiltonians, such as a family of BQP-hard Hamiltonians, the energy landscape over the entire 𝑛-qubit state space has a nice bowl shape, and the only local minimum is the global minimum. However, for QMA-hard Hamiltonians, the energy landscape necessarily contains many suboptimal local minima. (c) Energy landscape under local unitary perturbations. For any local Hamiltonian 𝑯, there are always doubly exponentially many local minima within the 𝑛-qubit state space that stems from a large barren plateau. Local unitary perturbations are reversible, while thermal perturbations are irreversible. This perspective encourages us to formulate and study a physically motivated alternative to the ground-state problem. We ask: How tractable is the problem of finding local minima of the energy in quantum systems using classical and quantum computers? We address this question by proving that a machine that cools physical systems to local minima is a universal quantum computer; that is, by finding local minima one can solve any problem that is efficiently solvable by a quantum computer, and furthermore a quantum computer can always find local minima efficiently for any physical system. Hence, under the widely accepted assumption that quantum computing is more powerful than classical computing, finding local minima is both classically hard and quantumly easy. This result raises the hope that exploring local minima in quantum systems arising from physics, chemistry, and materials science may offer new opportunities for solving classically intractable and physically relevant problems using quantum computers. 4.2 Results We now present our main results concerning the tractability of finding local minima in quantum systems. A collection of notational conventions and some background 199 on quantum thermodynamics can be found in Appendix 4.5. Based on the standard definition in mathematical optimization [2, 5, 25, 94, 144], we consider a local minimum in a quantum system governed by Hamiltonian 𝑯 to be a quantum state such that the expectation value of 𝑯 does not decrease under any small perturbation applied to the state; see a brief review of mathematical optimization in Appendix 4.6. Let P𝜶 be a perturbation parameterized by a small vector 𝜶 that maps quantum states to quantum states. An 𝜖-approximate local minimum of an 𝑛-qubit Hamiltonian 𝑯 under perturbation P is a state 𝝆 with an energy tr(𝑯𝝆) that is an approximate minimum under perturbations, i.e., tr(𝑯𝝆) ≤ tr(𝑯P𝜶 ( 𝝆)) + 𝜖 ∥𝜶∥ for all small enough 𝜶. The formal definition is given in Appendix 4.6. The local minima of 𝑯 form a subset of the entire quantum state space, which contains the global minima, the ground states of 𝑯. See Fig. 4.1 for energy landscapes with different numbers of local minima. We say an algorithm A has solved the problem of finding local minima under perturbation P if given any 𝑛-qubit Hamiltonian 𝑯, written as a sum of few-qubit Hermitian operators, and any few-qubit observable 𝑶, the algorithm A can output a real value tr(𝑶 𝝆) corresponding to any approximate local minimum 𝝆 of 𝑯 under perturbations P up to a small error.1 Inspired by the physical process that Nature uses to cool quantum systems, we define local minima based on thermal perturbations rooted in open system thermodynamics [28, 39, 55, 115, 135]. Thermal perturbations are fundamentally different from local unitary perturbations, which are short-time unitary evolution governed by a sum of few-body Hermitian operators, as may arise in an adaptive variational quantum eigensolver (VQE) [35, 77, 143]. These two types of perturbations yield distinct energy landscapes as illustrated in Fig. 4.1, leading to vastly different computational complexity for the task of finding a local minimum. Indeed, we argue in Section 4.2 that for local unitary perturbations, in contrast to thermal perturbations, finding a local minimum is classically easy. Local minima under thermal perturbations In this section, we consider local minima under thermal perturbations induced by a heat bath, formally defined in Appendix 4.6. When the coupling between an 𝑛-qubit system and a thermal bath is weak, and the bath is memoryless, the complicated joint 1 Since there could be multiple local minima and we consider finding one instance to be sufficient, this problem is closer to a relational problem than to a decision problem. 200 system-bath Hamiltonian dynamics reduces to a Markovian Lindbladian evolution of the system alone, 𝝆(𝑡) = eL𝑡 [𝝆]. Remarkably, this continuous time generator L can be defined by merely the system Hamiltonian 𝑯, the jump operators 𝑨𝑎 through which the bath interacts with the system, and thermodynamic quantities of the bath: inverse temperature 𝛽 and a characteristic time-scale 𝜏. See Appendix 4.5 for an introduction and Appendix 4.10 for an in-depth discussion. Under these assumptions, we may effectively consider a thermal perturbation of 𝑛-qubit state 𝝆 to be ! 𝑚 ∑︁ 𝛽,𝜏,𝑯 (thermal perturbation): 𝝆 → exp 𝛼𝑎 L 𝑎 ( 𝝆), (4.1) 𝑎=1 𝛽,𝜏,𝑯 where L𝑎 is the thermal Lindbladian associated with each jump operator 𝑨𝑎 acting Í on a few qubits, 𝑚 = poly(𝑛) is the number of jump operators, and 𝜶 = 𝑎 𝛼𝑎 𝒆ˆ 𝑎 ∈ R𝑚 ≥0 is a nonnegative vector close to zero. Here, the vector is nonnegative because thermodynamic processes are generally irreversible. The irreversibility in thermal perturbations is crucial to ensure that there are fewer than doubly-exponentially many local minima in the energy landscape; see the discussion in Appendix 4.6. We define a local minimum under thermal perturbations to be a state 𝝆 with the minimum energy tr(𝑯𝝆) under thermal perturbations given in Eq. (4.1). Given the definition, we next study how tractable is the problem of finding a local minimum under thermal perturbations. Our complexity-theoretic results show that finding local minima under thermal perturbations is both quantumly easy (Section 4.2) and classically hard (Section 4.2), the latter under the well-accepted assumption that not all quantum circuits can be efficiently simulated on classical computers (BPP ≠ BQP). Finding local minima is easy for quantum computers If our notion of local minima properly captures how a quantum system behaves in a cold environment, we expect finding local minima to be quantumly easy. Indeed, in the following theorem, we prove that a quantum computer can always efficiently find a local minimum of 𝑯 under thermal perturbations starting from any initial state. Theorem 4.2.1 (Quantumly easy to find local minima under thermal perturbations; informal). The problem of finding an 𝜖-approximate local minimum of an 𝑛-qubit local Hamiltonian 𝑯 under thermal perturbations with inverse temperature 𝛽 and time scale 𝜏 can be solved in poly(𝑛, 1/𝜖, 𝛽, 𝜏) quantum computational time. 201 The formal statement is given in Theorem 4.9.1 and is proven in Appendix 4.11. To establish the theorem, we propose a quantum thermal gradient descent algorithm that cools down a quantum system by performing gradient descent based on quantum thermodynamics. This result builds on the recent algorithmic developments for quantum Markov Chain Monte Carlo methods [41, 61, 149, 167]. Finding local minima is hard for classical computers While local minima are easier to find than ground states for quantum computers, how hard is it to find them? As our second main result, we address this question by considering a class of geometrically local Hamiltonians {𝑯𝐶 } on a 2D lattice, such that the ground state of 𝑯𝐶 encodes the outcome of a polynomial-size quantum circuit 𝐶 using a modified Kitaev’s circuit-to-Hamiltonian construction [3, 100, 142]. To understand the computational hardness, we provide the following characterization of the energy landscape. Theorem 4.2.2 (No suboptimal local minimum in BQP-hard Hamiltonians; informal). For any quantum circuit 𝐶 with size |𝐶 |, all approximate local minima of the geometrically local 2D Hamiltonian 𝑯𝐶 under thermal perturbations with inverse temperature 𝛽 = Poly(|𝐶 |) and time scale 𝜏 = Poly(|𝐶 |) are close to the ground state. This theorem is the most technically involved contribution of this work. The formal statement is given in Theorem 4.9.2 and is proven in Appendix 4.14. Conceptually, the landscape of these 2D Hamiltonians has a nice bowl shape, like in convex optimization [25]. Therefore, performing thermal gradient descent (Theorem 4.2.1) allows us to prepare the ground state starting from an arbitrary initial state. To prove the theorem, we develop a set of techniques for establishing that a Hamiltonian 𝑯 has no suboptimal local minima, i.e., all local minima of 𝑯 are global minima. Using this energy landscape characterization, we can show that finding a local minimum under thermal perturbations is classically intractable, assuming quantum computation is more powerful than classical computation. Suppose that a classical computer can efficiently find any local minima under thermal perturbations. Then by Theorem 4.2.2, the classical computer can efficiently find the ground state of 𝑯𝐶 and thus simulate any efficient quantum circuit 𝐶, which is widely believed to be impossible. See Theorem 4.9.3 for a formal statement and the proof. 202 Theorem 4.2.3 (Classically hard to find local minima under thermal perturbations; informal). Assume the widely believed conjecture that BPP ≠ BQP. The problem of finding an approximate local minimum of an 𝑛-qubit local Hamiltonian 𝑯 under thermal perturbations is universal for quantum computation and is thus classically hard. Local minima under local unitary perturbations Instead of defining local minima in terms of thermal perturbations, one could consider local unitary perturbations instead. Local unitary perturbations are mathematically simple and commonly considered in adaptive variational quantum eigensolvers [35, 77, 143]. However, as we now explain, the task of finding a local minimum with respect to unitary perturbations is classically easy and, therefore, does not fulfill our goal of identifying a problem that separates quantum from classical computation. Local unitary perturbations are short-time unitary evolutions under a sum of few-body Hermitian operators (e.g., a quantum circuit consisting of near-identity two-qubit gates). A local unitary perturbation of an 𝑛-qubit pure state|𝜓⟩ is given by ! 𝑚 ∑︁ |𝜓⟩ → exp −i (local unitary perturbation): 𝛼𝑎 𝒉𝑎 |𝜓⟩, 𝑎=1 where 𝒉𝑎 is a Hermitian operator acting on a few qubits, 𝑚 = poly(𝑛) is the number Í of such Hermitian operators, and 𝜶 = 𝑎 𝛼𝑎 𝒆ˆ 𝑎 ∈ R𝑚 is a vector close to zero. This definition is inspired by adaptive VQE [35, 77, 143], and is the state version of the Riemannian geometry of quantum computation defined in [140]. When one variationally minimizes the energy by applying unitary gates, one finds a local minimum under local unitary perturbations. We prove that for any Hamiltonian 𝑯, a random 𝑛-qubit pure state is almost always a local minimum of 𝑯 under local unitary perturbations; see Lemma 4.4.1 in Methods. Hence, there are exp(exp(Ω(𝑛))) many suboptimal local minima in the energy landscape, i.e., a large barren plateau. Because the number of local minima is enormous, the properties of these local minima concentrate on fixed values, making this problem classically easy. Proposition 4.2.1 (Classically easy to find local minima under local unitary perturbations; informal). The problem of finding approximate local minima of 𝑛-qubit local Hamiltonian 𝑯 under local unitary perturbations is classically easy. 203 See Proposition 4.9.1 for the detailed statement and Appendix 4.9 for its proof. Aside from being classically easy, the landscape defined by local unitary perturbation is not physically well motivated because a physical system in contact with a thermal bath cools by open system dynamics rather than by unitary dynamics acting on the system. 4.3 Discussion We have good reasons for believing that scalable fault-tolerant quantum computers will be more powerful than classical computers, but for what problems of practical interest should we expect a superpolynomial quantum advantage? Quantum computers might substantially speed up the task of characterizing properties of ground states for some local Hamiltonians that arise in physics, chemistry, and materials science, but it is not clear how to identify particular problems for which such speedups occur [109]. In some cases, classical methods provide good solutions, while in other cases, the problem is hard even for quantum computers. Here, we have focused on an easier problem, namely finding local minima rather than global minima of a Hamiltonian. This problem is well motivated physically because the task of finding a local minimum under thermal perturbations is routinely carried out by actual physical systems when in contact with a cold thermal bath. We showed that this problem is solved efficiently by a proposed quantum optimization algorithm, the quantum thermal gradient descent algorithm. Furthermore, we showed that finding a local minimum is classically hard in general (assuming that BPP ≠ BQP). Hence, the local minimum problem is a quantumly tractable alternative to the ground state problem for which superpolynomial quantum advantage can be achieved for some problem instances. There have been other proposals for solving classically hard problems by finding suitable quantum states, such as designing a gapped adiabatic path for Hamiltonians to find ground states [3], engineering Lindbladians to have rapid dissipative evolution towards steady states [174] and performing quantum phase estimation on an initial state with high ground-state overlap [70]. These approaches draw inspiration from physics to motivate algorithms for solving problems on analog and digital quantum devices but do not emulate naturally occurring physical processes. In contrast, the problem of finding a local minimum is motivated by ubiquitous physical processes in Nature that produce the low-energy states studied in physics, chemistry, and materials science. Furthermore, the local minima problem enjoys the robustness of 204 thermodynamics: one merely needs to specify macroscopic bath quantities 𝛽 and 𝜏 without worrying about microscopic details, and the choice of jump operators can be flexible since adding more jumps (even unwanted ones) only removes suboptimal local minima. There are a plethora of classical algorithms for minimizing energies of quantum systems based on classical variational ansatzes for quantum states, such as tensor networks [1, 13, 52, 82, 93, 95, 106, 145, 156, 162, 163, 172, 173, 177, 178, 183, 187] and neural quantum states [33, 46, 47, 59, 60, 67, 74, 88, 120, 131, 141, 175]. These classical algorithms find a local minimum within a family of states defined by the classical variational ansatz. However, such a local minimum for the classical ansatz might not be a local minimum under thermal perturbations. If not, we can load the state found by the classical algorithm into a quantum computer and find a lower energy state by running the quantum thermal gradient descent algorithm. A corollary of our main results states the following. Corollary 4.3.1 (Quantum advantage in finding lower-energy state; informal). Assume that not all polynomial-size quantum circuits can be efficiently simulated classically. Then there are 2D geometrically local Hamiltonians such that given any classical ansatz that allows efficient estimation of single-qubit observables and an output state 𝝆 # of any efficient classical algorithm that optimizes the classical ansatz, running quantum thermal gradient descent starting at 𝝆 # strictly lowers the energy. A formal statement is in Corollary 4.9.1, and its proof is in Appendix 4.9. In many cases, we can check if the classically optimized state 𝝆 # is a local minimum under thermal perturbation by executing an efficient classical computation to compute the energy gradient. A negative energy gradient confirms that a quantum algorithm starting from 𝝆 # could outperform the classical algorithm. Furthermore, we may be able to do better than quantum thermodynamics by sharpening our quantum algorithms. Indeed, we may modify the thermal Lindbladians to have nicer analytic properties or lower algorithmic cost [39]. While these synthetic Lindbladians may not simulate Nature, they constitute a broader class of Quantum Markov Chain Monte Carlo algorithms [39, 41, 61, 149, 167] that may improve upon Nature. Apart from Lindbladians, other families of perturbations, such as unitary perturbations accompanied by mid-circuit measurements and/or qubit resets, may also yield nice bowl-shaped energy landscapes without suboptimal local minima. Progress 205 in this direction could lead to more efficient quantum optimization algorithms for finding low-energy states or for other applications. Many other interesting and challenging questions remain open. Sometimes, when a system performs a random walk over a large plateau of suboptimal local minima for a sufficiently long time, the system escapes the plateau and reaches the true ground state (see, e.g., Case 1 in section 4.13). Could we characterize when ground states can be found efficiently despite having many suboptimal local minima? Our conclusion that finding local minima under thermal perturbations is classically hard relied on the complexity-theoretic conjecture that BPP ≠ BQP. Can we prove unconditionally that finding local minima is hard for classical algorithms, perhaps within a black-box oracle model? We have shown that there is a quantum advantage in finding local minima of quantum systems. Might there also be a quantum advantage in finding better local minima in classical optimization problems under some variant of quantum thermal gradient descent? While ground state problems are hard to solve in general, many experimentally observed quantum systems efficiently relax to their ground states when cooled. This physical phenomenon suggests that perhaps many Hamiltonians of interest in physics, chemistry, and materials science have no suboptimal local minima. We have shown in Theorem 4.2.2 that a particular family of BQP-hard Hamiltonians has no suboptimal local minima under thermal perturbation. An important future goal is to characterize broader classes of Hamiltonians that have a similarly good energy landscape. Our proposed negative gradient condition suffices to rule out suboptimal local minima (Lemma 4.8.3), but checking this condition for a general Hamilton involves highly complex calculations. It would be helpful to develop more general-purpose and efficient methods to verify this property for specified physical Hamiltonians over spins, fermions, or bosons. We hope the ideas and techniques presented here will yield a deeper understanding of the energy landscapes of quantum systems and point toward promising opportunities for achieving quantum advantage for physically relevant problems. Data availability We do not analyse or generate any datasets, because our work proceeds within a theoretical and mathematical approach. 206 Code availability We do not have any computer code, because our work proceeds within a theoretical and mathematical approach. All algorithms are analyzed mathematically in the Supplementary Information. Acknowledgments: The authors thank Anurag Anshu, Ryan Babbush, Fernando Brandao, Garnet Chan, Sitan Chen, Soonwon Choi, Jordan Cotler, Jarrod R. McClean, and Mehdi Soleimanifar for their valuable input. HH thanks Patrick Coles, Gavin Crooks, and Faris Sbahi for the inspiring discussions and for sharing their recent works on classical thermodynamics for AI applications [6, 50]. CFC is supported by the AWS Center for Quantum Computing internship. HH is supported by a Google PhD fellowship and a MediaTek Research Young Scholarship. HH acknowledges the visiting associate position at the Massachusetts Institute of Technology. LZ acknowledges funding from the Walter Burke Institute for Theoretical Physics at Caltech. JP acknowledges support from the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research (DE-NA0003525, DE-SC0020290), the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator, and the National Science Foundation (PHY-1733907). The Institute for Quantum Information and Matter is an NSF Physics Frontiers Center. 4.4 Methods Local minima under thermal perturbations A central concept that enables us to understand the energy landscape and establish the computational complexity of finding local minima under thermal perturbations in Theorem 4.2.1 and 4.2.3 is the energy gradient operator, (energy gradient operator): 𝑚 ∑︁ †𝛽,𝜏,𝑯 L𝑎 (𝑯) 𝒆ˆ 𝑎 , 𝑎=1 where the adjoint L † is the Heisenberg-picture Lindbladian, i.e., tr(L † [𝑶] 𝝆) = tr(𝑶L [𝝆]). The energy gradient operator is a vector of individual gradient operators2 associated with each jump operator 𝑨𝑎 . Indeed, the energy gradient operator naturally 2This is similar to the spin operator 𝝈 = 𝜎 𝑥 𝑥ˆ + 𝜎 𝑦 𝑦ˆ + 𝜎 𝑧 𝑧ˆ, which is a vector of Hermitian observables. 207 emerges by taking an infinitesimal perturbation, i.e., the gradient of the energy tr(𝑯𝝆), ! ! 𝑚 𝑚   ∑︁ ∑︁ 𝛽,𝜏,𝑯 †𝛽,𝜏,𝑯 tr 𝑯 exp 𝛼𝑎 L 𝑎 ( 𝝆) = tr(𝑯𝝆) + 𝜶 · tr L𝑎 (𝑯) 𝝆 𝒆ˆ 𝑎 + O (∥𝜶∥ 2 ). 𝑎=1 𝑎=1 In Appendix 4.8, we describe the formal definition and some properties of the energy gradient. We provide a concrete example by showing the sets of local minima for ferromagnetic Ising chains under different longitudinal field strengths in Appendix 4.13. Quantum thermal gradient descent To establish Theorem 4.2.1 showing that finding local minima under thermal perturbations is quantumly easy, we propose an algorithm called quantum thermal gradient descent. Gradient descent is necessary when the inverse temperature 𝛽 and time scale 𝜏 are not infinite. When 𝛽 = 𝜏 = ∞, the energy gradient L𝑎†∞,∞,𝑯 (𝑯) ⪯ 0 is nonpositive. In this case, the algorithm can just perform a random walk along random directions because no perturbations increase energy, and the use of gradient descent is not necessary. But when 𝛽 and 𝜏 are finite, the energy gradient can be positive. To find a local minimum that is a minimum under all thermal perturbations, the algorithm needs to carefully walk in directions with negative energy gradients. To prove the convergence of quantum thermal gradient descent to a local minimum, we show that every small gradient step starting from a state that is not an approximate local minimum will decrease the energy. Because the energy of a quantum system cannot go unboundedly to negative infinity, the algorithm must terminate at some point, which means it will find an approximate local minimum. To establish this claim, we derive analytic properties of thermal Lindbladians based on a smoothness bound on the second derivatives in [39]. To implement a gradient step based on thermal perturbations on a quantum computer, we build on a recently developed efficient quantum algorithm that simulates thermal Lindbladian evolution using a quantum circuit augmented by mid-circuit measurements [39]. All local minima are global We now highlight the proof idea for Theorem 4.2.2 showing that all approximate local minima are close to ground states in a family of Hamiltonians with classically-hard ground states. The family of geometrically local 𝑛-qubit Hamiltonians {𝑯𝐶 } in a 208 2D lattice studied in this work is a modification of Kitaev’s circuit-to-Hamiltonian construction [100, 142] where the ground state encodes the computation of a quantum circuit 𝑼𝐶 = 𝑼𝑇 . . . 𝑼1 . In particular, we design the ground state of 𝑯𝐶 to be 𝑇 √︁ ∑︁ 𝑛
𝑡 𝑇−𝑡 𝜉𝑡 𝑼𝑡 · · · 𝑼1 |0 ⟩ ⊗ 1 0 , where 𝑡=0   1 𝑇 𝜉𝑡 := 𝑇 . 2 𝑡 The binomial coefficient 𝜉𝑡 is our modification of Kitaev’s construction and is chosen to ensure that desired properties hold for the spectrum and the energy gradients.3 Estimating local properties of the ground state of 𝑯𝐶 is equivalent to simulating the quantum circuit 𝐶, which is BQP-hard. Hence, under the standard assumption that not all quantum circuits can be simulated by classical computations, estimating local properties of the ground state of 𝑯𝐶 is classically hard. Given the Hamiltonian 𝑯𝐶 , showing that all of its approximate local minima under thermal perturbations are also approximate global minima seems daunting due to the 𝛽,𝜏,𝑯 complex expression for the thermal Lindbladian L𝑎 and the doubly exponentially large space of possible quantum states. Previous studies on circuit-to-Hamiltonian mappings mainly focused on the lowest energy states. Here, we need to worry about potential local minima in all excited states in any superposition. To make progress, we propose a sufficient condition in Appendix 4.8 that captures the nice landscape of 𝑯𝐶 and rules out the presence of any suboptimal local minimum. Let 𝑷𝐺 (𝑯) be the projector onto the ground state space of 𝑯. Assume there exists a unit vector 𝜶ˆ ∈ R𝑚 ≥0 and 𝑟 > 0 with (negative gradient condition): − 𝑚 ∑︁ †𝛽,𝜏,𝑯 𝛼ˆ 𝑎 L𝑎 (𝑯) ⪰ 𝑟 ( 𝑰 − 𝑷𝐺 (𝑯)). 𝑎=1 This negative gradient condition implies that any state with a small ground state overlap must experience a substantially negative energy gradient, i.e., it must not be a local minimum. To prove that 𝑯𝐶 satisfies the negative gradient condition, we propose a series of lemmas and mathematical techniques for characterizing energy gradients in few-qubit systems, in commuting Hamiltonians, and in subspaces of the Hamiltonian, which are stated in Appendix 4.12 and proven in Appendix 4.16. These new techniques build on the operator Fourier transform, and the secular approximation given in [39] 3The binomial distribution ensures the Bohr-frequency gap is sufficiently large, which is central to the robustness of energy gradients under errors due to finite temperature and small perturbations. We believe that the standard circuit-to-Hamiltonian construction also has a large Bohr-frequency gap, but the proof seems more difficult. 209 for systematically handling energy uncertainty in thermal Lindbladians, which we review and adapt for our purpose in Appendix 4.15. Using these techniques, we analyze the energy gradient of the entire system perturbatively by considering a sequence of Hamiltonians 𝑯1 → 𝑯2 → 𝑯3 = 𝑯𝐶 with refining ground spaces where 𝑷1 ⊃ 𝑷2 ⊃ 𝑷3 , ∥𝑯1 ∥ ≫ ∥𝑯2 − 𝑯1 ∥ ≫ ∥𝑯3 − 𝑯2 ∥. Through these perturbations, we sequentially rule out local minima in excited states of the Hamiltonian 𝑯1 , 𝑯2 and, finally, 𝑯3 = 𝑯𝐶 . For example, we show the first Hamiltonian 𝑯1 satisfies the negative gradient condition and that the gradient is stable under perturbation going from 𝑯1 → 𝑯2 → 𝑯3 . Controlling perturbations of the energy gradient is surprisingly challenging, and it is not a priori clear why this stability property should hold due to multiple (possibly competing) energy scales, including 𝛽−1 , 𝜏 −1 , the spectral gap, and the Bohr-frequency gap.4 The perturbative errors are not suppressed by the spectral gap of the Hamiltonian as seen in standard settings, but instead by the Bohr-frequency gap, which can be much smaller (see Theorem 4.16.1). These techniques allow us to establish the robustness of energy gradients when perturbing a degenerate Hamiltonian with a sufficiently large Bohr-frequency gap. We emphasize that while we proved that 𝑯𝐶 has no suboptimal local minima when 𝐶 is a polynomial-size quantum circuit, the same is not true for general local Hamiltonians. Finding the ground state of a local Hamiltonian is a QMAhard problem; hence, we do not expect it to be solved efficiently by the quantum thermal gradient descent algorithm or by any other quantum algorithm. In the case of a quantum circuit that verifies the witness for a problem in QMA, Kitaev’s corresponding local Hamiltonian contains a term, often denoted 𝑯i𝑛 , which specifies some of the input qubits and leaves the input qubits corresponding to the witness unspecified, and a term, often denoted 𝑯o𝑢𝑡 , which checks whether the witness is accepted. Due to the unspecified witness qubits in 𝑯i𝑛 , the energy landscape contains a significant number of local minima corresponding to all possible witnesses. Furthermore, most of these local minima correspond to rejected witnesses and are suboptimal because of the energy penalty from 𝑯o𝑢𝑡 . For these QMA-complete Hamiltonians, quantum thermal gradient descent is likely to remain stuck for a long time at a suboptimal local minimum. In 𝑯𝐶 , the term 𝑯i𝑛 specifies all input qubits, 4 Recall that spectral gap is the minimum nonzero difference between energy eigenvalues. Bohr-frequency gap is the minimum nonzero difference between the difference of energy eigenvalues. 210 and the term 𝑯o𝑢𝑡 is absent, which greatly simplifies the energy landscape, enabling quantum thermal gradient descent to find the global minimum efficiently. Local minima under local unitary perturbations We present the central idea for proving Proposition 4.2.1 that establishes the classical easiness for finding local minima under local unitary perturbations. To understand how easy the problem of finding local minima under local unitary perturbations is, we need to characterize the energy landscape. The following lemma provides a universal characterization of the structure of the energy landscape under local unitary perturbations. The formal statement is given in Lemma 4.7.1, and the proof is given in Appendix 4.7. Lemma 4.4.1 (Barren plateau; informal). Given any 𝑛-qubit local Hamiltonian 𝑯. A random pure 𝑛-qubit state |𝜓⟩ is an approximate local minimum of 𝑯 under local unitary perturbations. The proof of the above lemma illustrates the following physical picture: the energy landscape in the pure state space defined in terms of local unitary perturbations consists of a large barren plateau [129] with doubly-exponentially many approximate local minima having exponentially small energy gradient. Additionally, almost all of the local minima have local properties that are exponentially close to that of the maximally mixed state. As a result, while finding ground states is classically hard, finding local minima under local unitary perturbations is classically trivial. 4.5 Appendix:Notations and Preliminaries Before we begin stating and proving our results formally in the rest of the appendices, we present some notations used throughout the paper. We also give a brief review of key concepts in quantum information theory that we utilize in this work. 211 Notations This section recapitulates notations, and the reader may skim through this and return as needed. ∑︁ 𝑯 := 𝐸𝑖 |𝜓𝑖 ⟩⟨𝜓𝑖 | Hamiltonian and the eigendecomposition 𝑖 Spec(𝑯) := {𝐸𝑖 } the spectrum of the Hamiltonian 𝜈 ∈ 𝐵(𝑯) := {𝐸𝑖 − 𝐸 𝑗 | 𝐸𝑖 , 𝐸 𝑗 ∈ Spec(𝑯)} Δ𝜈 (𝑯) := min{|𝜈1 − 𝜈2 | : 𝜈1 ≠ 𝜈2 ∈ 𝐵(𝑯)} 𝑨(𝑡) := ei𝑯𝑡 𝑨e−i𝑯𝑡 the set of Bohr frequencies the Bohr-frequency gap Heisenberg evolution for operator 𝑨 𝑚: the number of jump operators { 𝑨𝑎 } 𝑚 𝑎=1 : the set of jump operators 𝝆: the density matrix L: a Lindbladian in the Schrodinger Picture 𝛽: the inverse temperature ∫ ∞ 1 𝑨ˆ 𝑓 (𝜔) := √ 𝑓 (𝑡)e−i𝜔𝑡 𝑨(𝑡)d𝑡 2𝜋 −∞ 1 𝑓𝜏 (𝑡) := √ · 1 (|𝑡| ≤ 𝜏/2) 𝜏 ∫ ∞ ˆ𝑓 (𝜔) = √1 e−i𝜔𝑡 𝑓 (𝑡)d𝑡 2𝜋 ∑︁ −∞ 𝑨𝜈 := 𝑷 𝐸2 𝑨𝑷 𝐸1 Operator Fourier transform of 𝑨 under 𝑓 the normalized window function with width 𝜏 Fourier transform of a scalar function 𝑓 (𝑡) operator 𝑨 at exact Bohr frequency 𝜈 𝐸 2 −𝐸 1 =𝜈 𝑰: the identity operator ∫ ∞ | 𝑓 (𝑡)| 𝑝 d𝑡) 1/𝑝 the 𝑝-norm of a function ⟨𝜙|𝑶|𝜓⟩ |𝜓⟩,|𝜙⟩ ∥|𝜓⟩∥ · ∥|𝜙⟩∥ the operator norm of a matrix 𝑶 ∥ 𝑓 ∥ 𝑝 := ( −∞ ∥𝑶 ∥ := sup ∥𝑶 ∥ 𝑝 := (tr|𝑶| 𝑝 ) 1/𝑝 ∥L [𝑶] ∥ 𝑝 ∥L ∥ 𝑝−𝑝 := sup ∥𝑶 ∥ 𝑝 𝑶 the Schatten p-norm of a matrix 𝑶 the induced 𝑝 − 𝑝 norm of a superoperator L We write scalars, functions and vectors in normal font, and natural constants e, i, 𝜋 are particularly in Roman font. We write matrices in bold font 𝑶 and super-operators in curly font L. Furthermore, we define the indicator function 1 (𝑆) which is 1 if the statement 𝑆 is true and 0 otherwise. For any orthogonal projector 𝑷, we denote 𝑷⊥ = 𝑰 − 𝑷. 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O(⌧ 1 <latexit 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Spec(H) ) µ <latexit 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!0 <latexit 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kV k B(H 0 ) <latexit 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0 ⌫ 2 B(H) <latexit 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B(H) <latexit 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⌫ Figure 4.2: The Bohr frequencies 𝜈 ∈ 𝐵(𝑯) = {𝐸𝑖 − 𝐸 𝑗 | 𝐸𝑖 , 𝐸 𝑗 ∈ Spec(𝑯)} are the differences of energy (eigenvalues of the Hamiltonian 𝑯). 𝐸 say 𝑨 ≈ 𝑩 when ∥ 𝑨 − 𝑩∥ ≤ 𝐸. ˆ To simplify the notation, we often drop 𝑓 as a subscript 𝑨ˆ 𝑓 (𝜔) ≡ 𝑨(𝜔), by which we have chosen the window function 𝑓 (𝑡) = 𝑓𝜏 (𝑡). Lindbladians Completely Positive Trace-Preserving (CPTP) maps, also called quantum channels and quantum processes in the literature, correspond to all possible physical operations that could transform quantum states into other quantum states. Lindbladians are infinitesimal generators of CPTP maps. That is, they map density operators to density operators (even if the map is tensored with the identity) I ⊗ eL𝑡 [·] : S → S for each 𝑡 ≥ 0. In the Schrodinger Picture, a Lindbladian always has the following structure:  ∑︁  1 † † L [𝝆] = −i[𝑯, 𝝆] + 𝑳 𝑗 𝝆 𝑳 − {𝑳 𝑗 𝑳 𝑗 , 𝝆} , 2 | {z } 𝑗 ∈𝐽 | {z }𝑗 | {z } coherent term transition rate decay rate where the commutator is shorthanded by [ 𝑨, 𝑩] = 𝑨𝑩− 𝑩 𝑨 and the anti-commutator by { 𝑨, 𝑩} = 𝑨𝑩 + 𝑩 𝑨. The operator 𝑯 can be any Hermitian matrix, and the set of Lindblad operators {𝑳 𝑗 } 𝑗 ∈𝐽 can be arbitrary as the second term always ensures trace-preserving. Thermal Lindbladians In this section, we describe the basic parameters that define a thermal Lindbladian, i.e., Lindbladian originating from generic system-bath interactions under a Markovian, 213 weak-coupling assumption [135]. Consider an 𝑛-qubit quantum system governed by a Hamiltonian 𝑯 and a heat bath with inverse temperature 𝛽 and time scale 𝜏. The bath interacts with the system via a set of local interaction terms acting on the 𝑎 system { 𝑨1 , . . . , 𝑨𝑚 } = { 𝑨𝑎 } 𝑚 𝑎=1 , where each operator 𝑨 acts on a constant number of qubits. Each operator 𝑨𝑎 can be arbitrary (𝑨𝑎 does not need to be Hermitian nor unitary), but the set should be closed under Hermitian conjugate, 𝑎† 𝑚 { 𝑨𝑎 } 𝑚 𝑎=1 = { 𝑨 } 𝑎=1 . Each 𝑨𝑎 is referred to as a jump operator and induces changes in energy (in the 𝑛-qubit system). For simplicity, we will enforce the following normalization for the interaction strengths, ∥ 𝑨𝑎† 𝑨𝑎 ∥ ∞ ≤ 1 for each 𝑎 = 1, . . . , 𝑚. (4.2) For example, we may consider 𝑚 = 3𝑛 and 𝑨1 , . . . , 𝑨𝑚 to be all single-qubit Pauli observables 𝑿𝑖 , 𝒀𝑖 , 𝒁𝑖 for 𝑖 = 1, . . . , 𝑛, which have an interaction strength ∥ 𝑨𝑎† 𝑨𝑎 ∥ ∞ = 1. The above parameters determine the thermal Lindbladian governing the equation of motion for the density operator, also referred to as the coarse-grained master equation [135] 𝑚 ∑︁ d𝝆 𝛽,𝜏,𝑯 = −i[𝑯, 𝝆] + 𝛼𝑎 L 𝑎 ( 𝝆), d𝑡 𝑎=1 (4.3) The term −i[𝑯, 𝝆] corresponds to the Hamiltonian dynamics governed by the system Hamiltonian 𝑯, the (closed system) Schrodinger’s equation. The effects of system bath interaction are captured by a weighted average of the thermal Lindbladian 𝛽,𝜏,𝑯 L𝑎 , defined by each local jump operator 𝑨𝑎 , the Hamiltonian 𝑯, and parameters of the bath 𝛽, 𝜏. The weighting is captured by the nonnegative vector 𝜶 ∈ R𝑚 ≥0 . By varying the 𝑚-dimensional nonnegative vector 𝜶 ∈ R𝑚 ≥0 , the open system dynamics in Eq. (4.3) have the freedom to tune the interaction strengths for the jump operators. Each 𝛼𝑎 corresponds to the interaction strength of a jump operator 𝑨𝑎 and can be effectively absorbed into the set of jump operators by considering √ 𝛼𝑎 𝑨 𝑎 𝑎 . The interaction strength 𝛼𝑎 ≥ 0 determines how much contribution each thermal 𝛽,𝜏,𝑯 Lindbladian L𝑎 provides, and can be regarded as a probabilistic mixture. For 214 example, if 𝛼2 is set to 0, one removes the jump operator 𝑨2 from the system-bath interaction. This flexibility lets us study a (convex) set of thermal perturbations due to system-bath interaction by considering all 𝜶 ∈ R𝑚 ≥0 . As the system is weakly Í coupled to the bath, 𝜶 is considered to be a vector with a small ∥𝜶∥ 1 = 𝑎 𝛼𝑎 . 𝛽,𝜏,𝑯 For each local interaction term 𝑨𝑎 , the corresponding thermal Lindbladian L𝑎 is an open system evolution with Lindblad jump operators { 𝑨ˆ 𝑎 (𝜔)}𝜔 for all possible energy differences 𝜔 ∈ (−∞, ∞). Each Lindblad jump operator 𝑨ˆ 𝑎 (𝜔) is a restricted version of the system-bath interaction term 𝑨𝑎 that only contains transitions between eigenstates of 𝑯 whose associated eigenvalues, i.e., energies, differ by approximately 𝜔. The inverse temperature 𝛽 sets the transition weight 𝛾 𝛽 (𝜔), which determines the probability of occurrence for each Lindblad operator 𝑨ˆ 𝑎 (𝜔). For 𝛽 > 0, the transition weight 𝛾 𝛽 (𝜔) favors cooling (𝜔 < 0) over heating (𝜔 > 0) transitions. The timescale ˆ 𝜏 sets the resolution (1/𝜏) at which 𝑨(𝜔) identifies the energy differences between the eigenstates. The exact form of thermal Lindbladians is relatively complex, so we defer further discussion to Appendix 4.10 when needed for the full technical proof. 4.6 Appendix:Local minima in quantum systems In this appendix, we will introduce local minima in classical optimization, extend the definition to quantum systems, and formalize the problem of finding a local minimum in quantum systems. Local minima in classical optimization In this subsection, we describe the definition of local minima in finite-dimensional Euclidean spaces, introduce a direct generalization to geometries with tangent spaces and exponential maps (such as circles and spheres), and discuss the concept of approximate local minima. Local minima in Euclidean space In classical optimization, one considers a real-valued function ℎ(𝒙) : X → R over a domain X ⊆ R𝑛 consisting of 𝑛-dimensional vectors, and the goal is to find the global minimum of ℎ(𝒙), 𝒙 ∗ = arg min ℎ(𝒙). 𝒙∈X Finding the global minimum is already NP-hard even when ℎ(𝒙) is a quadratic function [144]. Instead of finding a global minimum, one typically resorts to finding a local minimum 𝒙 # , which is the minimum in a neighborhood around 𝒙 # . The 215 definition of a local minimum 𝒙 # is that there exists a distance 𝛿 > 0, such that ℎ(𝒙 # + 𝜶) ≥ ℎ(𝒙 # ), for all ∥𝜶∥ ≤ 𝛿 and 𝒙 # + 𝜶 ∈ X. (4.4) Here the vector 𝜶 is of the same dimension as 𝒙 # . We will refer to the above as an exact local minimum because all points in the neighborhood have to be at least ℎ(𝒙 # ). When there is an 𝜶 such that ℎ(𝒙 # + 𝜶) is only lower than ℎ(𝒙 # ) by an extremely small value, 𝒙 # is still not an exact local minimum. We will also define the approximate local minimum that relaxes this in Appendix 4.6. Local minima in general geometrical spaces The concept of a local minimum can be directly generalized to any geometry with tangent spaces and exponential maps, such as spheres, density matrices, unitaries, and more general Riemannian manifolds. Consider the tangent space 𝑇𝒙 and the exponential map exp𝒙 of a point 𝒙. In a physical picture, the tangent space 𝑇𝒙 is the space consisting of all vectors 𝜶 that describe the direction 𝜶ˆ and magnitude ∥𝜶∥ for a particle moving at point 𝒙 on a manifold, and the exponential map exp𝒙 is a function that takes in the vector 𝜶 ∈ 𝑇𝒙 encompassing the direction and magnitude and outputs the point after moving 𝒙 in the direction 𝜶ˆ with a magnitude ∥𝜶∥.5 To visualize these concepts, we give two warm-up examples in the following. Euclidean space: In an 𝑚-dimensional Euclidean space, ∀𝒙 ∈ X = R𝑚 , the tangent space is 𝑇𝒙 = {𝜶 ∈ R𝑚 }. Given 𝜶 ∈ 𝑇𝒙 , when we move 𝒙 in the direction 𝜶ˆ with a magnitude ∥𝜶∥, we obtain exp𝒙 (𝜶) = 𝒙 + 𝜶. We can see that this matches our physical picture. Particle moving counter-clockwise on a circle: As another warm-up, let us consider a unit circle X = {𝒙 ∈ R2 | ∥𝒙∥ = 1} where a particle can only move counter-clockwise. In this example, the tangent space 𝑇𝒙 of a unit vector 𝒙 ∈ R2 with ∥𝒙∥ = 1 is the set of one-dimensional rays, 𝑇𝒙 = {𝛼 ∈ R | 𝛼 ≥ 0}. 5 Strictly speaking, to define the exponential map, we need to know how to “transport” the vector 𝛼 along itself. Fortunately, this is natural for all cases we consider. 216 The condition 𝛼 ≥ 0 comes from the constraint that the particle can only move counter-clockwise (unidirectional rather than bidirectional). When we move 𝒙 according to 𝛼 ∈ 𝑇𝒙 , we obtain !! ! 0 −𝛼 cos 𝛼 − sin 𝛼 exp𝒙 (𝛼) = exp 𝒙= 𝒙. 𝛼 0 sin 𝛼 cos 𝛼 The larger 𝛼 is, the bigger the rotation is. Using the language of tangent spaces and exponential maps, an exact local minimum 𝒙 # ∈ X of a function ℎ is equivalent to the statement that there exists 𝛿 > 0, such that ℎ(exp𝒙 # (𝜶)) ≥ ℎ(𝒙 # ), for all 𝜶 ∈ 𝑇𝒙 # , ∥𝜶∥ ≤ 𝛿. (4.5) For the case of optimizing over 𝑚-dimensional Euclidean space, the condition of Eq. (4.5) becomes the same as Eq. (4.4) noting exp𝒙 # (𝜶) = 𝒙 # + 𝜶. However, the condition can be quite different when the tangent space changes. For example, consider a 2-dimensional Euclidean space and the function ℎ(𝒙) = ∥𝒙∥ 2 . In general, there is only one exact local minimum 𝒙 # = 0. However, if the particle can only move to the right, the tangent space becomes 𝑇𝒙 = {𝜶 ∈ R2 | 𝜶1 ≥ 0} and every point 𝒙 with 𝒙 1 ≥ 0 and 𝒙2 = 0 is an exact local minimum. Modifying the tangent space changes the definition of neighborhood. Hence, the set of local minima would be changed accordingly. We will consider the most suitable norm ∥𝜶∥ for each context. Approximate local minima While global minima are computationally hard to find, exact local minima are not much easier. If there is an 𝜶 such that ℎ(exp𝒙 # (𝜶)) is lower than ℎ(𝒙 # ) by an extremely small value, 𝒙 # is not consider to be an exact local minimum. The requirement to resolve an extremely small value in exact local minima leads to the fact that finding an exact local minimum is still computationally hard [5]. Furthermore, exact local minima are very sensitive to small perturbations to the function ℎ. Therefore, it is desirable to define approximate local minima to promote computational efficiency and robustness to small perturbations. We consider the following principle for defining 𝜖-approximate local minima: if a function ℎ∗ is very close to ℎ, then an exact ˜ The formal definition is local minimum of ℎ∗ is an approximate local minimum of ℎ. given below. Definition 4.6.1. (𝜖-approximate local minima) Given a space X with tangent spaces 𝑇𝑥 and exponential maps exp𝑥 for all 𝑥 ∈ X, and a function ℎ: X → R. 𝒙 # is an 217 𝜖-approximate local minimum of ℎ if 𝒙 # is the exact local minimum of some function ℎ∗ , where Δ(𝒙) := ℎ∗ (𝒙) − ℎ(𝒙) satisfies |Δ(𝒙)| ≤ 𝜖 for each 𝒙 ∈ X, (𝜖-bounded), Δ(exp𝒙 # (𝜶)) ≤ 𝜖 ∥𝜶∥ for each 𝜶 ∈ 𝑇𝒙 # , (𝜖-Lipschitz around 𝑥 # ). A (𝜖 = 0)-approximate local minimum of ℎ is an exact local minimum of ℎ. Under this definition, 𝒙 # is an approximate local minimum of ℎ if there is an 𝒙 in the neighborhood of 𝒙 # such that ℎ(𝒙) is lower than ℎ(𝒙 # ) by an extremely small value. We also give the following equivalent characterization based on looking at the local neighborhood. Proposition 4.6.1. (An equivalent characterization of 𝜖-approximate local minima) 𝒙 # ∈ X is an 𝜖-approximate local minimum of the function ℎ if and only if there exists a distance 𝛿 > 0, ℎ(exp𝒙 # (𝜶)) ≥ ℎ(𝒙 # ) − 𝜖 ∥𝜶∥ for each 𝜶 ∈ 𝑇𝒙 # , ∥𝜶∥ ≤ 𝛿, (4.6) i.e., all the neighboring points can at most be 𝜖 ∥𝜶∥ lower than the point 𝒙 # . Proof. For the “only if” statement, we recall the definition of an exact local minimum that there exists 𝛿 > 0, such that ℎ∗ (exp𝒙 # (𝜶)) − ℎ∗ (𝒙 # ) ≥ 0 for all 𝜶 ∈ 𝑇𝒙 # and ∥𝜶∥ ≤ 𝛿. From the 𝜖-Lipschitz condition around 𝑥 # for the function Δ(𝒙), we have 0 ≤ ℎ∗ (exp𝒙 # (𝜶)) − ℎ∗ (𝒙 # ) = ℎ(exp𝒙 # (𝜶)) − ℎ(𝒙 # ) + Δ(exp𝒙 # (𝜶)) ≤ ℎ(exp𝒙 # (𝜶)) − ℎ(𝒙 # ) − 𝜖 ∥𝜶∥. This concludes the “only if” statement. For the “if” statement, consider 𝛿 to be of at most 1 and let Δ(𝒙) :=    ℎ(𝒙 # ) − ℎ(𝒙),  if 𝒙 = exp𝒙 # (𝜶) for some 𝜶 ∈ 𝑇𝒙 # , ∥𝜶∥ ≤ 𝛿,   0,  otherwise. We have 𝒙 # is an exact local minimum for ℎ∗ (𝒙) := ℎ(𝒙) + Δ(𝒙). Furthermore, because ℎ(𝒙 # ) − ℎ(exp𝒙 # (𝜶)) ≤ 𝜖 ∥𝜶∥ ≤ 𝜖, both 𝜖-bounded and 𝜖-Lipschitz around 𝑥 # are satisfied by Δ(𝒙). ■ 218 Defining local minima in quantum systems To define local minima, we need to consider the domain X of elements 𝒙 ∈ X, the optimization function ℎ(𝒙), the tangent space 𝑇𝒙 consisting of all possible directions and magnitudes to move an element 𝒙, where 𝜶 ∈ 𝑇𝒙 encompass the direction 𝜶ˆ and the magnitude ∥𝜶∥, and the exponential map exp𝒙 (𝜶) that describes the resulting element after moving 𝒙 under 𝜶. In the following, we present two settings. The first setting in Appendix 4.6 considers general quantum states that can evolve under thermodynamic processes induced by interacting with a low-temperature heat bath. This setting defines local minima under thermal perturbations. The second setting in Appendix 4.6 considers pure quantum states that can move under any unitary generated by a set of local Hermitian operators (e.g., all two-qubit Pauli observables 𝑷𝑖 ⊗ 𝑸 𝑗 , where 𝑷, 𝑸 ∈ {𝑿, 𝒀, 𝒁}). This setting defines local minima under local unitary perturbations. Definition based on thermal perturbations In quantum mechanics, the central optimization problem considers a function ℎ defined by the Hamiltonian 𝑯 of an 𝑛-qubit quantum system, ℎ( 𝝆) = tr(𝑯𝝆), which is the average energy of an 𝑛-qubit quantum state 𝝆. The ground states 𝝆 (𝑔) of 𝑯 are the global minima of the optimization over ℎ( 𝝆) = tr(𝑯𝝆) in the quantum state space, i.e., the set of density operators (trace-one positive semidefinite matrices), S2𝑛 := {𝝆 ∈ C2 ×2 | 𝝆 † = 𝝆, 𝝆 ⪰ 0, tr( 𝝆) = 1}. 𝑛 𝑛 When the quantum system is placed in a heat bath with inverse temperature 𝛽 ∈ [0, ∞], time scale 𝜏 ∈ [0, ∞], and system-bath interactions based on 𝑚 local jump operators6 𝑨1 , . . . , 𝑨𝑚 , the system dynamics are effectively described by the thermal 𝛽,𝜏,𝑯 Lindbladians L𝑎 , 𝑚 ∑︁ d𝝆(𝑡) 𝛽,𝜏,𝑯 = −i[𝑯, 𝝆] + 𝛼𝑎 L 𝑎 [𝝆], d𝑡 𝑎=1 (4.7) where 𝛼𝑎 ≥ 0 for each 𝑎. After time 𝑡, the initial quantum state 𝝆 will evolve to ! 𝑚 ∑︁ 𝛽,𝜏,𝑯 𝝆(𝑡) = exp −i𝑡 [𝑯, ·] + 𝑡𝛼𝑎 L𝑎 ( 𝝆). 𝑎=1 6A local operator 𝑨 𝑎 acts on O (1) qubits, but the set of qubits that 𝑨 𝑎 acts on may not be geometrically close. 219 𝛽,𝜏,𝑯 Each term L𝑎 is the thermal Lindbladian associated with a local jump operator 𝑨𝑎 (recall that local operator 𝑨𝑎 acts on a constant number of qubits). See Appendix 4.5 for a brief review of thermal Lindbladians, and Appendix 4.10 for the exact form of thermal Lindbladians. The coefficient 𝛼𝑎 ≥ 0 corresponds to the interaction strength of each jump operator 𝑨𝑎 . As 𝛼𝑎 < 0 is equivalent to reversing time, we cannot have 𝛼𝑎 < 0 since thermodynamic processes are irreversible in general. Different interaction strength vector 𝜶 corresponds to a different system-bath interaction, and the thermodynamics could be different. Because 𝜶 describes the probability of each jump occurring, the natural norm ∥𝜶∥ for the interaction strength vector 𝜶 is ∥𝜶∥ 1 . We denote 𝜶ˆ = 𝜶/∥𝜶∥ 1 as the unit vector. The thermodynamics equation in Eq. (4.7) consists of a fast-rotating term −i[𝑯, ·] due to the system Hamiltonian 𝑯 that keeps the energy tr(𝑯𝝆) invariant and the Í 𝛽,𝜏,𝑯 thermal perturbation term 𝑎 𝛼𝑎 L𝑎 due to the heat bath that cools the system. Because −i[𝑯, ·] keeps the energy constant, only the thermal perturbation term Í 𝛽,𝜏,𝑯 is relevant for minimizing the energy ℎ( 𝝆) = tr(𝑯𝝆). For notational 𝑎 𝛼𝑎 L 𝑎 simplicity, we will only consider contributions from the thermal perturbations and absorb the 𝑡 dependence in 𝑡𝛼𝑎 into 𝛼𝑎 since 𝜶 is an arbitrary nonnegative vector. Together, the thermal perturbation on 𝝆 due to a heat bath with inverse temperature 𝛽, time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 can be written as ! 𝑚 ∑︁ 𝛽,𝜏,𝑯 𝝆 → exp 𝛼𝑎 L 𝑎 ( 𝝆) 𝑎=1 for a nonnegative vector 𝜶 ∈ R𝑚 ≥0 that combines the interaction strength vector and time 𝑡. A dictionary between all the relevant functions and variables for optimizing tr(𝑯𝝆) in 𝑛-qubit quantum systems under a heat bath with inverse temperature 𝛽 and time scale 𝜏 and optimizing ℎ(𝒙) in an 𝑛-dimensional Euclidean space is given as follows. X = R𝑛 ↔ X = S2𝑛 , (domain) 𝒙 ∈ R𝑛 ↔ 𝝆 ∈ S2𝑛 , (an element) ℎ(𝒙) ↔ ℎ( 𝝆) = tr(𝑯𝝆), 𝑇𝒙 = {𝜶 ∈ R𝑛 } ↔ {𝜶 ∈ R𝑚 ≥0 }, exp𝒙 (𝜶) = 𝒙 + 𝜶 ↔ exp 𝑚 ∑︁ 𝑎=1 (optimization function) (tangent space) ! 𝛽,𝜏,𝑯 𝛼𝑎 L 𝑎 ( 𝝆) (exponential map). 220 The formal definition of tangent spaces and exponential maps via Lindbladians is given below. Definition 4.6.2 (Tangent spaces of quantum states in a heat bath). Consider an 𝑛-qubit quantum state 𝝆, an 𝑛-qubit Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑚 𝑎=1 , 𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑚 𝑎=1 and parameters 𝛽, 𝜏 ≥ 0. The tangent space 𝑇𝝆 under a heat bath with an inverse temperature 𝛽, a time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 is defined as  𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑚 𝑎=1 𝑇𝝆 := 𝜶 ∈ R𝑚 ≥0 , 𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 which is independent of 𝛽, 𝜏, 𝑯, { 𝑨𝑎 } 𝑚 𝑎=1 . The exponential map exp 𝝆 defined as ! 𝑚 ∑︁ 𝑎 𝛽,𝜏,𝑯,{ 𝑨 } 𝑎 𝛽,𝜏,𝑯 𝛼𝑎 L 𝑎 ( 𝝆). exp 𝝆 (𝜶) := exp is (4.8) 𝑎=1 With the definition of tangent spaces and exponential maps, we can define 𝜖approximate local minimum similar to the classical case in Eq. (4.6). We consider the natural choice of ∥·∥ 1 for the nonnegative vector 𝜶 encompassing the probability of each jump. Our results remain qualitatively the same for other reasonable vector norms, such as Euclidean norm ∥·∥ 2 or ℓ 𝑝 norm ∥·∥ 𝑝 . Definition 4.6.3 (Local minima under thermal perturbations). Given an 𝑛-qubit Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑚 𝑎=1 , and parameters 𝛽, 𝜏 ≥ 0, an 𝑛-qubit state 𝝆 ∈ S2𝑛 is an 𝜖-approximate local minimum of 𝑯 under thermal perturbations with an inverse temperature 𝛽, a time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 if there is a 𝛿 > 0 such that   𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 tr 𝑯 exp 𝝆 (𝜶) ≥ tr(𝑯𝝆) − 𝜖 ∥𝜶∥ 1 for each 𝜶 ∈ R𝑚 ≥0 , ∥𝜶∥ 1 ≤ 𝛿, i.e., all the neighboring points can at most be 𝜖 ∥𝜶∥ 1 lower than the point 𝝆. A central concept we will be using for characterizing local minima under thermal perturbations is the energy gradient. The energy gradient at an 𝑛-qubit state 𝝆 under thermal perturbation is determined by the following state-independent operator: (energy gradient operator): 𝑚 ∑︁ †𝛽,𝜏,𝑯 L𝑎 (𝑯) 𝒆ˆ 𝑎 , 𝑎=1 †𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 where we denote L𝑎 to be the Hermitian conjugate of L𝑎 . The energy gradient operator is a vector of Hermitian observables. The terminology stems from 221 the fact that evaluating the energy gradient operator on a state 𝝆 gives the energy gradient at the state 𝝆, ! ! 𝑚 𝑚   ∑︁ ∑︁ 𝛽,𝜏,𝑯 †𝛽,𝜏,𝑯 tr 𝑯 exp −i 𝛼𝑎 L 𝑎 ( 𝝆) = tr(𝑯𝝆)+𝜶· tr L𝑎 (𝑯) 𝝆 𝒆ˆ 𝑎 +O (∥𝛼∥ 2 ). 𝑎=1 𝑎=1 In Appendix 4.8, we provide more discussions about the energy gradient. Thermal perturbations depend on how the quantum system is interacting with the heat bath. Local minima defined above are local minima of the Hamiltonian 𝑯 under thermal perturbations induced by all system-bath interactions generated by the jump operators { 𝑨𝑎 } 𝑎 . Remark 4.6.0.1 (Thermodynamics at local minima). Given a specific systembath interaction, inverse temperature 𝛽, and time scale 𝜏, there could still be thermodynamics at a local minimum. For example, when 𝛽 is not infinitely large, a local minimum could still move to other higher-energy states due to thermal fluctuations. Another example is when the local minimum is on a large and flat plateau, then the local minimum can still perform a random walk on the plateau. Definition based on local unitary perturbations Inspired by variation quantum eigensolvers [35, 77], another natural definition for tangent spaces, exponential maps, and local minima considers pure states and local unitary perturbation. Given 𝑚 local Hermitian operators 𝒉1 , . . . , 𝒉𝑚 with ∥ 𝒉𝑎 ∥ ∞ = 1. Here, local means that each operator 𝒉𝑎 only acts on a constant number of qubits. We can consider all possible local unitary perturbations formed by performing time evolution under a Hamiltonian generated by the set {𝒉𝑎 } 𝑎 of local Hermitian operators, 𝑚 ∑︁ 𝛼𝑎 𝒉 𝑎 , 𝑎=1 for any 𝜶 ∈ R𝑚 . Since the time evolution under a Hamiltonian is always reversible, there is no additional requirement that 𝜶 must be in the nonnegative orthant. Similar to thermal perturbations, we will absorb the contribution of evolution time 𝑡 into the arbitrary vector 𝜶. Consider the following dictionary between all the relevant functions and variables for optimizing ⟨𝜓|𝑯|𝜓⟩ in 𝑛-qubit pure state |𝜓⟩ under local 222 unitary perturbation and optimizing ℎ(𝒙) in an 𝑛-dimensional Euclidean space. X = R𝑛 ↔ 𝑛 X = {|𝜓⟩ ∈ C2 |⟨𝜓|𝜓⟩ = 1}, 𝒙 ∈ R𝑛 ↔ |𝜓⟩ ∈ C , ⟨𝜓|𝜓⟩ = 1, ℎ(𝒙) ↔ ℎ(|𝜓⟩) = ⟨𝜓|𝑯|𝜓⟩, 𝑇𝒙 = {𝜶 ∈ R𝑛 } ↔ {𝜶 ∈ R𝑚 }, exp𝒙 (𝜶) = 𝒙 + 𝜶 ↔ (domain) 2𝑛 exp 𝑚 ∑︁ (an element) (optimization function) (tangent space) ! 𝛼𝑎 𝒉𝑎 |𝜓⟩ (exponential map). 𝑎=1 The tangent space and the exponential map can be formally defined as follows. Definition 4.6.4 (Tangent spaces of pure quantum states under local unitaries). Given an 𝑛-qubit pure quantum state |𝜓⟩ and 𝑚 local Hermitian operators {𝒉𝑎 } 𝑎 . The tangent space 𝑇𝜓 is defined as 𝑇𝜓{𝒉 𝑎 } 𝑎 := R𝑚 , and the exponential map exp𝜓 is defined as ! exp𝜓{𝒉 𝑎 } 𝑎 (𝜶) := exp −𝑖 ∑︁ 𝛼𝑎 𝒉𝑎 |𝜓⟩. 𝑎 When the set {𝒉𝑎 } 𝑎 is the set of all two-qubit Pauli observables, the tangent space 𝑇𝜓 and exponential map exp𝜓 define a Riemannian manifold that connects all 𝑛-qubit pure states through unitary evolutions. This Riemannian manifold is the state version of the manifold over quantum unitaries defined in a seminal work on the geometry of quantum computation [140]. The optimization function is ℎ(|𝜓⟩) = ⟨𝜓|𝑯|𝜓⟩, the average energy of the Hamiltonian 𝑯 for the pure state |𝜓⟩. Performing gradient descent on this pure state Riemannian manifold to minimize ⟨𝜓|𝑯|𝜓⟩ is equivalent to performing adaptive variational quantum optimization [77] to minimize the Hamiltonian 𝑯. The local minima can be defined similarly as before. To be consistent with local minima under thermal perturbations, we consider the ℓ1 -norm ∥𝜶∥ 1 . All of our results remain qualitatively the same for other reasonable vector norms, such as the Euclidean norm or ℓ 𝑝 norm. Definition 4.6.5 (local minima under local unitary perturbations). Given an 𝑛-qubit Hamiltonian 𝑯, and 𝑚 local Hermitian operators {𝒉𝑎 } 𝑎 . A pure state |𝜓⟩ is an 223 𝜖-approximate local minimum of 𝑯 under local unitary perturbations generated by {𝒉𝑎 } 𝑎 if exp𝜓{𝒉 𝑎 } 𝑎 (𝜶) † 𝑯 exp𝜓{𝒉 𝑎 } 𝑎 (𝜶) ≥ ⟨𝜓|𝑯|𝜓⟩−𝜖 ∥𝜶∥ 1 , for each 𝜶 ∈ 𝑇𝜓{𝒉 𝑎 } 𝑎 , ∥𝜶∥ 1 ≤ 𝛿. for some 𝛿 > 0. This is also a valid definition of local minima in quantum systems. However, we will later show that the optimization landscape defined in this way always has a very large barren plateau. Hence, the problem of finding a local minimum defined in this way will be a trivial problem. The problem of finding a local minimum in quantum systems With these definitions of local minima, we can define the task of finding a local minimum in a straightforward manner. To formulate the problem to have purely classical output, we focus on outputting a simple property, such as the expectation value of a local observable 𝑶, of an approximate local minimum 𝝆. Furthermore, we only consider Hamiltonians 𝑯 that can be written as a sum of local observables, commonly referred to as local Hamiltonians in the literature. While there can be many approximate local minima, we consider the algorithm to be successful if the algorithm outputs the property of any one of the local minima. Definition 4.6.6 (Finding a local minimum under low-temperature thermal perturbations). Given error 𝜖 > 0, inverse temperature 𝛽 ≥ 0, time scale 𝜏 ≥ 0, an 𝑛-qubit local Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑎 , and a local observable 𝑶 with ∥𝑶 ∥ ∞ ≤ 1. Output a real value 𝑣 ∈ [−1, 1], such that 𝑣 is 𝜖-close to tr(𝑶 𝝆) for an 𝜖-approximate local minimum 𝝆 of 𝑯 under thermal perturbations with an inverse temperature 𝛽, a time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 . Definition 4.6.7 (Finding a local minimum under local unitary perturbations). Given error 𝜖 > 0, an 𝑛-qubit local Hamiltonian 𝑯, 𝑚 local Hermitian operators {𝒉𝑎 } 𝑎 , and a local observable 𝑶 with ∥𝑶 ∥ ∞ ≤ 1. Output a real value 𝑣 ∈ [−1, 1], such that 𝑣 is 𝜖-close to ⟨𝜓|𝑶|𝜓⟩ for an 𝜖-approximate local minimum |𝜓⟩ of the Hamiltonian 𝑯 under local unitary perturbations generated by {𝒉𝑎 } 𝑎 . Ideally, we would like the two problems to be quantumly easy and classically hard. However, we will show that only the first problem based on thermal perturbations is both quantumly easy and classically hard. The second problem based on local unitary 224 perturbation is classically trivial due to the presence of too many local minima in an exponentially large barren plateau. The importance of irreversible perturbations Suppose that the perturbations P𝜶 parameterized by a polynomial-size vector 𝜶 are reversible P−𝜶 = P𝜶−1 and are smooth. The following argument shows that the energy landscape must have doubly-exponentially many approximate local minima. Given any 𝑛-qubit state 𝝆 and any 𝑛-qubit Hamiltonian 𝑯 with ∥𝑯∥ ∞ = poly(𝑛). Consider a gradient descent algorithm that starts at 𝝆. Because ∥𝑯∥ ∞ = poly(𝑛), after a polynomial number of steps 𝑇, the gradient descent algorithm can find an approximate local minimum 𝝆 # of 𝑯, 𝝆 # = P𝜶𝑇 . . . P𝜶1 ( 𝝆). From the reversibility of the perturbations, we have 𝝆 = P−𝜶1 . . . P−𝜶𝑇 ( 𝝆 # ). Consider a covering net N for the set of approximate local minima of 𝑯. Because the packing net for all 𝑛-qubit states is of size exp(exp(Ω(𝑛))), and the covering net for the perturbations is of size exp(poly(𝑛)). We have the following relationship, exp(𝑇 · poly(𝑛))|N | = exp(exp(Ω(𝑛))). Hence, we can see that |N | = exp(exp(Ω(𝑛)) − poly(𝑛)) = exp(exp(Ω(𝑛))) since exp(Ω(𝑛)) grows much faster than poly(𝑛). 4.7 Appendix:Characterizing local minima under local unitary perturbations Now that we have defined local minima in quantum systems, we present a set of results characterizing properties of local minima in quantum systems in this and the next appendix. These results provide a further understanding of local minima 225 in quantum systems and are essential to establishing the main theorems given in Appendix 4.9. We begin by looking at the energy landscape defined by local unitary perturbations. We will prove a central lemma portraying the energy landscapes defined by local unitary perturbations for pure quantum systems. The lemma states that most pure quantum states |𝜓⟩ are local minima under local unitary perturbations with an expectation value close to tr(𝑶)/2𝑛 = tr(𝑶 ( 𝑰/2𝑛 )) for any local observable 𝑶. Furthermore, the proof shows that the gradient at a randomly sampled local minimum |𝜓⟩ is exponentially close to zero. Lemma 4.7.1 and its proof provide the following physical picture. In the energy landscape defined by local unitary perturbations, there is an overwhelmingly large barren plateau consisting of local minima with almost equal energy as their neighbors. Furthermore, these local minima behave like the maximally mixed state 𝑰/2𝑛 , which makes the task of predicting properties for a local minimum under local unitary perturbations classically trivial to solve. Given 𝑚 local Hermitian operators {𝒉𝑎 } 𝑎 and 𝜶 ∈ R𝑚 . By applying Taylor’s theorem in Prop. 4.8.1 to the one-dimensional function ˆ † 𝑯 exp𝜓{𝒉 𝑎 } 𝑎 (𝑡 𝜶) ˆ 𝑔(𝑡) = exp𝜓{𝒉 𝑎 } 𝑎 (𝑡 𝜶) for 0 ≤ 𝑡 ≤ ∥𝜶∥ 1 and 𝜶ˆ = 𝜶/∥𝜶∥ 1 , we can obtain the following proposition. Proposition 4.7.1 (Taylor’s theorem for local unitary perturbations). Given an 𝑛-qubit Hamiltonian 𝑯, 𝜶 ∈ R𝑛 , 𝑚 local Hermitian operators {𝒉𝑎 } 𝑎 , and an 𝑛-qubit pure state |𝜓⟩. We have " # 𝑚 ∑︁ exp𝜓 (𝜶) † 𝑯 exp𝜓 (𝜶) = ⟨𝜓|𝑯|𝜓⟩ − i⟨𝜓| 𝑯, 𝛼𝑎 𝒉𝑎 |𝜓⟩ 𝑎=1 − 𝑚 𝑚 1 ∑︁ ∑︁ 2 𝑎=1 𝑎′ =1 ˆ † [[𝑯, 𝒉𝑎 ], 𝒉𝑎′ ] exp𝜓 (𝜂 𝜶), ˆ 𝛼𝑎 𝛼𝑎′ exp𝜓 (𝜂 𝜶) for some 0 ≤ 𝜂 ≤ ∥𝜶∥ 1 . Lemma 4.7.1 (A random state is a local minimum under local unitary perturbations; Restatement of Lemma 4.4.1). Consider a large problem size 𝑛. Given error 𝜖 ≥ 1/2𝑛/4 , an 𝑛-qubit local Hamiltonian 𝑯 with ∥𝑯∥ ∞ = poly(𝑛), 𝑚 local Hermitian operators {𝒉𝑎 } 𝑎 with 𝑚 = poly(𝑛) and ∥ 𝒉𝑎 ∥ ∞ = 1, and a local observable 226 𝑛/4 𝑶 with ∥𝑶 ∥ ∞ ≤ 1. With probability at least 1 − 1/22 , an 𝑛-qubit state |𝜓⟩ sampled uniformly at random is an 𝜖-approximate local minimum of 𝑯 under local unitary perturbations generated by {𝒉𝑎 } 𝑎 and ⟨𝜓|𝑶|𝜓⟩ is 𝜖-close to tr(𝑶)/2𝑛 . Proof. From Lemma III.5 in [87], for any Pauli operator 𝑸 ∈ {𝑰, 𝑿, 𝒀, 𝒁} ⊗𝑛 \ {𝑰 ⊗𝑛 } and a random 𝑛-qubit pure state |𝜓⟩ sampled uniformly, we have  𝑛 2 2 𝛿 Pr [|⟨𝜓|𝑸|𝜓⟩| > 𝛿] ≤ 2 exp − , |𝜓⟩ 10 for any 0 ≤ 𝛿 ≤ 1. Let 𝛿 = 1/2𝑛/3 . Then, we have   𝑛/3   2 1 . Pr |⟨𝜓|𝑸|𝜓⟩| > 𝑛/3 ≤ 2 exp − |𝜓⟩ 10 2 Recall that any Hermitian operator has a unique Pauli decomposition: ∑︁ 𝑯= 𝛼 𝑷 (𝑯)𝑷, 𝑷∈{𝑰,𝑿,𝒀,𝒁} ⊗𝑛 ∑︁ 𝑶= 𝛼 𝑷 (𝑶)𝑷, 𝑷∈{𝑰,𝑿,𝒀,𝒁} ⊗𝑛 ∑︁ 𝒉𝑎 = 𝛼 𝑷 (𝒉𝑎 )𝑷, 𝑷∈{𝑰,𝑿,𝒀,𝒁} ⊗𝑛 where the Pauli coefficients 𝛼 𝑷 (·) satisfy ∑︁ 𝛼2𝑷 (𝑯) ≤ ∥𝑯∥ 2∞ = poly(𝑛), (4.9) 𝑷∈{𝑰,𝑿,𝒀,𝒁} ⊗𝑛 ∑︁ 𝛼2𝑷 (𝑶) ≤ ∥𝑶 ∥ 2∞ = 1. (4.10) 𝛼2𝑷 (𝒉𝑎 ) ≤ ∥ 𝒉𝑎 ∥ 2∞ = 1. (4.11) 𝑷∈{𝑰,𝑿,𝒀,𝒁} ⊗𝑛 ∑︁ 𝑷∈{𝑰,𝑿,𝒀,𝒁} ⊗𝑛 Let 𝑆0 be the set of Pauli operator 𝑷 with nonzero Pauli coefficients 𝛼 𝑷 in the Pauli decompositions of either 𝑯 or 𝑶,  𝑆0 = 𝑷 ∈ {𝑰, 𝑿, 𝒀, 𝒁} ⊗𝑛 \ {𝑰 ⊗𝑛 } | 𝛼 𝑷 (𝑯) ≠ 0 or 𝛼 𝑷 (𝑶) ≠ 0 , and 𝑆 𝐸 be the set of Pauli operator 𝑷 with nonzero Pauli coefficients 𝛼 𝑷 in the Pauli decompositions of 𝒉𝑎 for some 𝑎,  𝑆 𝐸 = 𝑷 ∈ {𝑰, 𝑿, 𝒀, 𝒁} ⊗𝑛 \ {𝑰 ⊗𝑛 } | ∃1 ≤ 𝑎 ≤ 𝑚, 𝛼 𝑷 (𝒉𝑎 ) ≠ 0 . 227 Because 𝑯 is a local Hamiltonian and 𝑶 is a local observable, we have |𝑆0 | = poly(𝑛). Because 𝒉𝑎 is a local observable, we have |𝑆 𝐸 | = O (𝑚) = poly(𝑛). We then define n o 𝑆 = 𝑷′ ∈ {𝑰, 𝑿, 𝒀, 𝒁} ⊗𝑛 \ {𝑰 ⊗𝑛 } | ∃𝑸 ∈ 𝑆0 , 𝑷 ∈ 𝑆 𝐸 , tr(𝑷′ [𝑸, 𝑷]) ≠ 0 ∪ 𝑆0 . Because |𝑆0 | = poly(𝑛) and [𝑸, 𝑷] is another Pauli observable up to a phase, we have |𝑆| ≤ |𝑆 𝐸 ||𝑆1 | = poly(𝑛). The union bound yields the following probabilistic statement:    𝑛/3  1 2 poly(𝑛) 1 1 − Pr |⟨𝜓|𝑸|𝜓⟩| < 𝑛/3 , ∀𝑸 ∈ 𝑆 ≤ 2|𝑆| exp − ≤ 𝑛/3 < 𝑛/4 , 2 /10 2 |𝜓⟩ 10 2 2 2 where the last inequality holds for any large 𝑛 since 22 /10−2 grows much faster than any polynomial of 𝑛. We condition on the event for the random state |𝜓⟩ that 𝑛/3 |⟨𝜓|𝑸|𝜓⟩| < 1 2𝑛/3 for all 𝑛/4 𝑸 ∈ 𝑆, referred to as event 𝐸 ∗ . We can obtain the following from Cauchy-Schwarz inequality: ∑︁ |⟨𝜓|[𝑯, 𝒉𝑎 ] |𝜓⟩| ≤ 𝛼𝑸 (𝑯) |𝛼 𝑷 (𝒉𝑎 )||⟨𝜓| [𝑸, 𝑷] |𝜓⟩| 𝑸,𝑷∈{𝑰,𝑿,𝒀,𝒁} ⊗𝑛 √︄ ≤ ∑︁ √︂ 2 (𝑯)𝛼 2 (𝒉 ) 𝛼𝑸 𝑎 𝑷 𝑸,𝑷∈{𝑰,𝑿,𝒀,𝒁} ⊗𝑛 |𝑆| poly(𝑛) ≤ ,(4.12) 2𝑛/3 2 2𝑛/3 where the second inequality uses the conditioning on event 𝐸 ∗ and [𝑸, 𝑷] ≠ 0 =⇒ 𝑸, 𝑷 ≠ 𝑰 ⊗𝑛 , and the third inequality uses |𝑆| = poly(𝑛) and Eq. (4.9), (4.11). Similarly, we also have |⟨𝜓|[[𝑯, 𝒉𝑎 ], 𝒉𝑎′ ]|𝜓⟩| ≤ poly(𝑛) . 2𝑛/3 Using Eq. (4.10) instead of Eq. (4.9), we can similarly obtain |⟨𝜓|𝑶|𝜓⟩ − 𝛼 𝐼 ⊗𝑛 (𝑶)| = ⟨𝜓|𝑶|𝜓⟩ − tr(𝑂) poly(𝑛) 1 ≤ < 𝑛/4 ≤ 𝜖 𝑛 𝑛/3 2 2 2 (4.13) for any large problem size 𝑛 since 2𝑛/3 grows much faster than any polynomial in 𝑛. We now show that |𝜓⟩ is an 𝜖-approximate local minimum of 𝑯 under local unitary perturbations. To establish this claim, from Def. 4.6.5, we need to prove that exp𝜓 (𝜶) † 𝑯 exp𝜓 (𝜶) ≥ ⟨𝜓|𝑯|𝜓⟩ − 𝜖 ∥𝜶∥ 1 , for each 𝜶 ∈ 𝑇𝜓 , ∥𝜶∥ 1 ≤ 𝛿 228 for some 𝛿 > 0. Recall from Lemma 4.7.1 based on Taylor’s theorem (Prop. 4.8.1), we have " # 𝑚 ∑︁ † exp𝜓 (𝜶) 𝑯 exp𝜓 (𝜶) = ⟨𝜓|𝑯|𝜓⟩ − i⟨𝜓| 𝑯, 𝛼𝑎 𝒉𝑎 |𝜓⟩ 𝑎=1 − 𝑚 𝑚 1 ∑︁ ∑︁ 2 𝑎=1 𝑎′ =1 ˆ † [[𝑯, 𝒉𝑎 ], 𝒉𝑎′ ] exp𝜓 (𝜂 𝜶), ˆ 𝛼𝑎 𝛼𝑎′ exp𝜓 (𝜂 𝜶) for some 0 ≤ 𝜂 ≤ ∥𝜶∥ 1 . For the linear term, from Eq. (4.12) bounding |⟨𝜓|[𝑯, 𝒉𝑎 ] |𝜓⟩|, we have " # 𝑚 𝑚 ∑︁ ∑︁ poly(𝑛) poly(𝑛) −i⟨𝜓| 𝑯, 𝛼𝑎 𝒉𝑎 |𝜓⟩ ≤ |𝛼𝑎 | 𝑛/3 ≤ ∥𝜶∥ 1 , 2 2𝑛/3 𝑎=1 𝑎=1 where the last inequality uses 𝑚 = poly(𝑛). For the quadratic residual term, we have 𝑚 𝑚 1 ∑︁ ∑︁ ˆ † [[𝑯, 𝒉𝑎 ], 𝒉𝑎′ ] exp𝜓 (𝜂 𝜶) ˆ ≤ 2∥𝜶∥ 21 ∥𝑯∥ ∞ = poly(𝑛)∥𝜶∥ 21 . 𝛼𝑎 𝛼𝑎′ exp𝜓 (𝜂 𝜶) 2 𝑎=1 𝑎′ =1 Together, we can combine the inequalities to get poly(𝑛) ∥𝜶∥ 1 − poly(𝑛)∥𝜶∥ 21 2𝑛/3 ≥ ⟨𝜓|𝑯|𝜓⟩ − 0.5𝜖 ∥𝜶∥ 1 − poly(𝑛)∥𝜶∥ 21 , exp𝜓 (𝜶) † 𝑯 exp𝜓 (𝜶) ≥ ⟨𝜓|𝑯|𝜓⟩ − where the second inequality holds for any large problem size 𝑛 since 𝜖 ≥ 1/2𝑛/4 decays much slower than poly(𝑛)/2𝑛/3 . For any ∥𝜶∥ < 𝛿 := 0.5𝜖/poly(𝑛), we have exp𝜓 (𝜶) † 𝑯 exp𝜓 (𝜶) ≥ ⟨𝜓|𝑯|𝜓⟩ − 0.5𝜖 ∥𝜶∥ 1 − 0.5𝜖 ∥𝜶∥ 1 = ⟨𝜓|𝑯|𝜓⟩ − 𝜖 ∥𝜶∥ 1 , which shows that |𝜓⟩ is an 𝜖-approximate local minimum of 𝑯 under local unitary perturbations. Finally, because the event 𝐸 ∗ occurs with probability at least 0.99, by combining Eq. (4.13) and the above, we establish the claim that, with high probability, a random 𝑛-qubit state |𝜓⟩ sampled uniformly is an 𝜖-approximate local minimum of 𝑯 under local unitary perturbations and ⟨𝜓|𝑶|𝜓⟩ is 𝜖-close to tr(𝑂)/2𝑛 . ■ 4.8 Appendix:Characterizing local minima under thermal perturbations In this appendix, we characterize local minima under thermal perturbations. In particular, we will focus on the gradients of the energy landscape, conditions of local minima, and conditions on the Hamiltonian 𝑯 that ensure approximate local minima are approximate global minima, i.e., there are no suboptimal local minima in the energy landscape. 229 Energy gradients The energy landscape is much more nontrivial when defined under thermal perturbations. We can study the energy landscape by looking at the energy gradients. Recall 𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 the exponential map exp 𝝆 in Eq. (4.8) and consider the one-dimensional function ! ! 𝑚   ∑︁ 𝑎 𝛽,𝜏,𝑯,{ 𝑨 } 𝑎 𝛽,𝜏,𝑯 ˆ = tr 𝑯 exp 𝑔(𝑡) = tr 𝑯 exp 𝝆 (𝑡 𝜶) 𝑡 𝛼ˆ 𝑎 L𝑎 ( 𝝆) 𝑎=1 for 0 ≤ 𝑡 ≤ ∥𝜶∥ 1 , 𝜶ˆ = 𝜶/∥𝜶∥ 1 . We have the following derivatives: ! #! " 𝑚 ∑︁ ∑︁ 𝑑𝑔 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 (𝑡) = tr 𝑯 𝑡 𝛼ˆ 𝑎 L𝑎 ( 𝝆) , 𝛼ˆ 𝑎 L𝑎 exp 𝑑𝑡 𝑎 𝑎=1 " " ! ##! 𝑚 2 ∑︁ ∑︁ ∑︁ 𝑑 𝑔 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 𝛼ˆ 𝑎 𝛼ˆ 𝑎′ L𝑎′ (𝑡) = tr 𝑯 L𝑎 exp 𝑡 𝛼ˆ 𝑎 L𝑎 ( 𝝆) . 𝑑𝑡 2 𝑎 𝑎′ 𝑎=1 Recall Taylor’s theorem with the Lagrange form of the remainder from standard single-variate calculus. By applying Taylor’s theorem in Prop. 4.8.1 to 𝑔(𝑡), we can obtain Prop. 4.8.2. Proposition 4.8.1 (Taylor’s theorem). Let 𝑔(𝑡) : R → R be twice differentiable on the open interval between 0 and 𝑡 and 𝑔′ (𝑡) continuous on the closed interval between 0 and 𝑡. Then 1 𝑔(𝑡) = 𝑔(0) + 𝑔′ (0)𝑡 + 𝑔′′ (𝜂)𝑡 2 , 2 for some real number 𝜂 between 0 and 𝑡. Proposition 4.8.2 (Taylor’s theorem for thermal perturbations). Given an 𝑛-qubit Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑎 , parameters 𝛽, 𝜏 ≥ 0, 𝜶 ∈ R𝑚 ≥0 , and an 𝑛-qubit state 𝝆 ∈ S2𝑛 .   ∑︁ 𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 𝛽,𝜏,𝑯 tr 𝑯 exp 𝝆 (𝜶) = tr(𝑯𝝆) + 𝛼𝑎 tr(𝑯L𝑎 [𝝆]) 𝑎 + 1 ∑︁ ∑︁ 2 𝑎  𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 ˆ 𝛼𝑎 𝛼𝑎′ tr 𝑯L𝑎′ [L𝑎 [exp 𝝆 (𝜂 𝜶)]] 𝑎′ for some 0 ≤ 𝜂 ≤ ∥𝜶∥ 1 . We define the energy gradients as follows. We separately consider a positive and a negative energy gradient. The motivation of the definition is that the positive  230 (negative) energy gradient should determine the direction of the thermodynamics that causes the energy of the state to increase (decrease). Because our goal is to understand local minima, we will focus on the negative energy gradient. When one studies local maxima, one will focus on the positive energy gradient. Definition 4.8.1 (Energy gradients of a state under thermal perturbations). Given an 𝑛-qubit Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑎 , and parameters 𝛽, 𝜏 ≥ 0, the gradients of an 𝑛-qubit state 𝝆 ∈ S2𝑛 under thermal perturbations with inverse temperature 𝛽, time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 is defined as 𝑚     ∑︁ 𝛽,𝜏,𝑯 + ∇ 𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆) := max + tr 𝑯L𝑎 [𝝆] , 0 𝒆ˆ 𝑎 , (positive energy gradient) 𝑎=1 ∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆) := 𝑚 ∑︁     𝛽,𝜏,𝑯 max − tr 𝑯L𝑎 [𝝆] , 0 𝒆ˆ 𝑎 , (negative energy gradient) 𝑎=1 ∇ 𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆) := 𝑚   ∑︁ 𝛽,𝜏,𝑯 tr 𝑯L𝑎 [𝝆] 𝒆ˆ 𝑎 , (energy gradient) 𝑎=1 where 𝒆ˆ 𝑎 is the unit vector along the 𝑎-th coordinate. Since the set of jump operators { 𝑨𝑎 } 𝑎 will be fixed, we will sometimes drop the dependence on { 𝑨𝑎 } 𝑎 for notational simplicity. The positive/negative energy gradient belongs to the tangent space R𝑚 ≥0 , but the energy gradient ∇ 𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆) = ∇+𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆) − ∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆) may not be in the tangent space due to negative values. So, in general, one could not move in the direction of the energy gradient. However, one could move in the direction of the positive or negative energy gradient. It is instructive to think about the Heisenberg picture and define the energy gradient operator. Definition 4.8.2 (Energy gradient operator). Given an 𝑛-qubit Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑎 , inverse temperature 𝛽 ≥ 0, and time scale 𝜏 ≥ 0, the energy gradient operators under thermal perturbations is 𝑚 ∑︁ †𝛽,𝜏,𝑯 L𝑎 [𝑯] 𝒆ˆ 𝑎 , 𝑎=1 which is a vector of 𝑛-qubit Hermitian operators. 231 We can provide an upper and lower bound to the energy gradients by combining Prop. 4.10.2 and Prop. 4.10.3 to obtain the following proposition. Proposition 4.8.3 (Bound on the energy gradients). Given an 𝑛-qubit Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑎 , inverse temperature 𝛽 ≥ 0, and time scale 𝜏 ≥ 0, †𝛽,𝜏,𝑯 ∥L𝑎 (𝑯)∥ ∞ ≤ 3∥𝑯∥ ∞ for all 𝑎 = 1, . . . , 𝑚. The 𝛽, 𝜏 → ∞ limit (zero temperature heat bath with an infinite time scale) recovers the Davies’ generator L𝑎∞,∞,𝑯 . The Davies’ generator takes an energy eigenvector |𝜓 𝑗 ⟩⟨𝜓 𝑗 | of 𝑯 to energy eigenvectors with equal or lower energy, i.e., for any 𝑡 ≥ 0,  
⟨𝜓 𝑘 | exp 𝑡L𝑎∞,∞,𝑯 |𝜓 𝑗 ⟩⟨𝜓 𝑗 | |𝜓 𝑘 ⟩ = 0 for any 𝑗, 𝑘 such that 𝐸 𝑘 > 𝐸 𝑗 . We can use the above to obtain the following proposition. Proposition 4.8.4 (Vanishing positive energy gradient). For 𝛽 = 𝜏 = ∞, we have L𝑎∞,∞,𝑯† [𝑯] ⪯ 0, for each 𝑎. + Hence, the positive energy gradient vanishes ∇∞,∞ (𝑯, 𝝆) = 0 and ∇∞,∞ (𝑯, 𝝆) = − (𝑯, 𝝆) for all Hamiltonian 𝑯 and state 𝝆. −∇∞,∞ This proposition illustrates that thermal perturbations induced by a zero-temperature heat bath with an infinite time scale should only absorb energy from the quantum system and not cause the energy to increase. Hence, the positive energy gradient must vanish. A sufficient condition and a necessary condition of local minima Using the negative gradient, we can show a necessary condition and a sufficient condition for local minima under thermal perturbations. They differ only slightly (< vs ≤). From the conditions, we can see that local minima are well characterized by the negative energy gradient. Recall that ∥𝒙∥ ∞ = max𝑖 |𝑥𝑖 | is the ℓ∞ norm for a finite-dimensional vector 𝒙. Lemma 4.8.1 (A sufficient condition for local minima under thermal perturbations). Given 𝜖 > 0, an 𝑛-qubit Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑎 , and parameters 𝛽, 𝜏 ≥ 0, an 𝑛-qubit state 𝝆 with a small negative energy gradient, ∥∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆)∥ ∞ < 𝜖, (4.14) 232 is an 𝜖-approximate local minimum 𝝆 of the 𝑛-qubit Hamiltonian 𝑯 under thermal perturbations with inverse temperature 𝛽, time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 . 𝛽,𝜏,𝑯 Proof. Consider 𝐶 𝐿 = max𝑎 ∥L𝑎 have ∑︁ ∑︁ 𝑎 ∥ 1−1 > 0 and 𝐶𝐻 = ∥𝑯∥ ∞ . Given 𝜶 ∈ R𝑚 ≥0 , we 𝛽,𝜏,𝑯 𝛼𝑎 𝛼𝑎′ tr(𝑯L𝑎′ 𝛽,𝜏,𝑯 [L𝑎 [𝝈]]) ≤ 𝐶 𝐿2 𝐶𝐻 ∥𝜶∥ 21 , 𝑎′ for any state 𝝈. Let 𝜖0 := ∥∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆)∥ ∞ < 𝜖. From 𝛼𝑎 ≥ 0 and CauchySchwarz inequality,     ∑︁ ∑︁ 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 |𝛼𝑎 | max − tr 𝑯L𝑖 𝛼𝑎 tr(𝑯L𝑎 [𝝆]) ≥ − [𝝆] , 0 ≥ −∥𝜶∥ 1 𝜖 0 . 𝑎 𝑎 Together, Taylor’s theorem for thermal perturbations (Prop. 4.8.2) implies  𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 tr 𝑯 exp 𝝆 (𝜶)  ∥𝜶∥ 21 2 ≥ tr(𝑯𝝆) − ∥𝜶∥ 1 𝜖0 − 𝐶 𝐿 𝐶𝐻 , 2 2(𝜖−𝜖0 ) 2 for any 𝜶 ∈ R𝑚 ≥0 . From the above, we see that for any ∥𝜶∥ 1 < 𝛿 := 𝐶 2 𝐶 , 𝐿  𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 tr 𝑯 exp 𝝆 (𝜶)  𝐻 𝐶 𝐿2 𝐶𝐻 ≥ tr(𝑯𝝆) − 𝜖 ∥𝜶∥ 1 + ∥𝜶∥ 1 (𝜖 − 𝜖0 ) − ∥𝜶∥ 21 2 ! ≥ tr(𝑯𝝆) − 𝜖 ∥𝜶∥ 1 . So, 𝝆 is an 𝜖-approximate local minimum of 𝑯 under thermal perturbations with inverse temperature 𝛽, time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 . ■ Lemma 4.8.2 (A necessary condition for local minima under thermal perturbations). Given 𝜖 > 0, an 𝑛-qubit Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑎 , and parameters 𝛽, 𝜏 ≥ 0, an 𝜖-approximate local minimum 𝝆 of 𝑯 under thermal perturbations with inverse temperature 𝛽, time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 satisfies ∥∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆)∥ ∞ ≤ 𝜖, which differs only slightly from the condition in Eq. (4.14). 233  ∗ = a𝑟𝑔 𝑚𝑎𝑥 ∇− Proof. Recall that ∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆) ∈ R𝑚 . Let 𝑎 (𝑯, 𝝆) . 𝑎 𝑎 ≥0 𝛽,𝜏,{ 𝑨 } 𝑎  𝑎 𝛽,𝜏,𝑯 If the negative energy gradient vector tr(𝑯L𝑎∗ [𝝆]) is zero, then the claim holds. Hence, we only need to consider the case when the negative energy gradient vector is nonzero. In this case, 0 < ∥∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆)∥ ∞ = ∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆)𝑎∗ = − tr(𝑯L𝑎∗ 𝛽,𝜏,𝑯 [𝝆]). ˆ Consider 𝜶ˆ := 𝒆ˆ 𝑎∗ ∈ R𝑚 ≥0 , which satisfies ∥ 𝜶∥ 1 = 1. We have 𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 lim+ tr(𝑯 exp 𝝆 ˆ − tr(𝑯𝝆) (𝑡 𝜶)) 𝑡 𝑡→0 𝛽,𝜏,𝑯 = tr(𝑯L𝑎∗ [𝝆]) = −∥∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆)∥ ∞ . At the same time, for any 𝑡 > 0, we also have 𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 tr(𝑯 exp 𝝆 ˆ − tr(𝑯𝝆) (𝑡 𝜶)) 𝑡 Together, we obtain the desired claim. ˆ 1 = −𝜖 . ≥ −𝜖 ∥ 𝜶∥ ■ Hamiltonians without suboptimal local minima An important concept in classical optimization is to understand when all local minima are global minima. For example, in convex optimization, checking the convexity of the objective function ℎ(𝒙) ensures that all local minima are global minima. When all local minima are global minima, it is commonly referred to as having no suboptimal local minima in the optimization landscape. For optimizing quantum Hamiltonians, we can define a similar concept. Let us begin with a definition of approximate global minimum. Definition 4.8.3 (Approximate global minimum of Hamiltonians). Given 𝜖, 𝛿 > 0 and an 𝑛-qubit Hamiltonian 𝑯 with minimum energy 𝐸 0 . Let 𝑷𝐺+𝜖 (𝑯) be the projector to the subspace of energy eigenstates of 𝑯 with energy at most 𝐸 0 + 𝜖. An 𝑛-qubit state 𝝆 is an 𝜖-approximate global minimum of 𝑯 with failure probability ≤ 𝛿 if tr(𝑷𝐺+𝜖 (𝑯) 𝝆) ≥ 1 − 𝛿. Definition 4.8.4 (No suboptimal local minima). Given 𝜖 > 0. We say an 𝑛qubit Hamiltonian 𝑯 has no suboptimal 𝜖-approximate local minima with failure probability 𝛿 if any 𝜖-approximate local minimum 𝝆 of 𝑯 is an 𝜖-approximate global minimum of 𝑯 with failure probability ≤ 𝛿, i.e., tr(𝑷𝐺+𝜖 (𝑯) 𝝆) ≥ 1 − 𝛿. 234 While the above definitions apply to any Hamiltonian 𝑯, in this work, we will focus on Hamiltonians with a gap Δ > 0 between the minimum energy and the second minimum energy, also known as the spectral gap. By definition of 𝑷𝐺+𝜖 (𝑯) and spectral gap Δ, we have 𝜖 < Δ =⇒ 𝑷𝐺+𝜖 (𝑯) = 𝑷𝐺 (𝑯). As we will almost always consider 𝜖 < Δ, any 𝜖-approximate global minimum is an 0-approximate global minimum or exact global minimum. Convexity implies all local minima are global in classical optimization. In the following lemma, we present a sufficient condition for ensuring that all local minima are global in quantum systems. As we can see, all we need is to check the negative gradient operator is sufficiently positive in the non-ground-state space 𝑰 − 𝑷𝐺 . We will refer to this as the negative gradient condition. Lemma 4.8.3 (A sufficient condition ensuring all local minima are global). Given 𝜖, 𝛿 > 0, an 𝑛-qubit Hamiltonian 𝑯, 𝑚 local jump operators { 𝑨𝑎 } 𝑎 , and parameters 𝛽, 𝜏 ≥ 0. Let 𝑷𝐺 (𝑯) be the projection onto the ground state space of 𝑯. If there exists 𝜶 ∈ R𝑚 ≥0 with ∥𝜶∥ 1 = 1, such that the negative gradient operator satisfies 2𝜖 ( 𝑰 − 𝑷𝐺 (𝑯)) − 𝜖 𝑰, 𝛿 𝑎 (4.15) then any 𝜖-approximate local minimum 𝝆 of the 𝑛-qubit Hamiltonian 𝑯 under thermal perturbations with inverse temperature 𝛽, time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 is an exact global minimum with failure probability ≤ 𝛿. That is, tr(𝑷𝐺 (𝑯) 𝝆) ≥ 1 − 𝛿. (negative gradient condition): − ∑︁ †𝛽,𝜏,𝑯 𝛼𝑎 L 𝑎 [𝑯] ⪰ Proof. From the necessary condition for local minima in Lemma 4.8.2, any 𝜖approximate local minimum 𝝆 of the 𝑛-qubit Hamiltonian 𝑯 under thermal perturbations with inverse temperature 𝛽, time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 satisfies †𝛽,𝜏,𝑯 − tr(L𝑎 [𝑯] 𝝆) ≤ 𝜖 for each 𝑎 = 1, . . . , 𝑚. Hence, from Eq. (4.15), we have ∑︁ 2𝜖 †𝛽,𝜏,𝑯 𝜖≥− 𝛼𝑎 tr(L𝑎 [𝑯] 𝝆) ≥ (1 − tr(𝑷𝐺 (𝑯) 𝝆)) − 𝜖 . 𝛿 𝑎 This immediately implies that tr(𝑷𝐺 (𝑯) 𝝆) ≥ 1 − 𝛿. ■ 235 4.9 Appendix:Complexity of finding a local minimum in quantum systems In this appendix, we formally present the main results of this paper shown earlier in section 4.2 regarding the computational complexity of finding a local minimum in quantum systems. We separate the results into two parts. First, we look at the problem of finding a local minimum under local unitary perturbations (Def. 4.6.7), showing that the problem is classically trivial to solve. Next, we look at the problem of finding a local minimum under low-temperature thermal perturbations (Def. 4.6.6). We will see that this problem is quantumly easy but classically hard to solve, establishing a promising candidate problem for quantum advantage. Finding a local minimum under local unitary perturbations We begin with the first main result stating the problem of a local minimum under local unitary perturbations is classically trivial. The main issue is that there is a large barren plateau (which consists of many local minima with high energy) in the quantum optimization landscape. Hence, a classical algorithm can efficiently estimate the properties of a single local minimum. Proposition 4.9.1 (Classically easy to find a local minimum under local unitary perturbations; Restatement of Proposition 4.2.1). Consider a large problem size 𝑛. There is a trivial classical algorithm that guarantees the following. Given error 𝜖 = 1/poly(𝑛), an 𝑛-qubit local Hamiltonian 𝑯 with ∥𝑯∥ ∞ = poly(𝑛), 𝑚 local Hermitian operators {𝒉𝑎 } 𝑚 𝑎=1 with 𝑚 = poly(𝑛) and ∥ 𝒉 𝑎 ∥ ∞ = 1, and a local observable 𝑶 with ∥𝑶 ∥ ∞ ≤ 1. The classical algorithm runs in time O (1) and outputs a real value 𝑣 ∈ [−1, 1], such that 𝑣 is 𝜖-close to ⟨𝜓|𝑶|𝜓⟩ for an 𝜖-approximate local minimum |𝜓⟩ of the Hamiltonian 𝑯 under local unitary perturbations generated by {𝒉𝑎 } 𝑎 . Proof. From Lemma 4.7.1 given in Appendix 4.7 characterizing local minima of 𝑯 under local unitary perturbations, with high probability, a state |𝜓⟩ sampled uniformly at random from the space of pure states is an 𝜖-approximate local minimum |𝜓⟩ of the local Hamiltonian 𝑯 under local unitary perturbations, and ⟨𝜓|𝑶|𝜓⟩ is 𝜖-close to tr(𝑶)/2𝑛 . Hence, there exists an 𝜖-approximate local minimum |𝜓⟩ of 𝑯 under local unitary perturbations, and ⟨𝜓|𝑶|𝜓⟩ is 𝜖-close to tr(𝑶)/2𝑛 . This characterization of local minima gives rise to the following trivial classical algorithm. Given a local observable 𝑶, represented by the subset 𝑆 of qubits 𝑶 acts on with |𝑆| = O (1) and a 2|𝑆|×|𝑆| Hermitian matrix 𝑶 ∗ . A classical algorithm can 236 compute tr(𝑶)/2𝑛 efficiently by computing the trace of 𝑶 ∗ and dividing by 2|𝑆| . This trivial classical algorithm runs in time O (1). ■ Finding a local minimum under thermal perturbations We now turn to the second main result of this work, which shows that finding a local minimum under low-temperature thermal perturbations is easy with a quantum computer. This is in contrast to the task of finding the ground state (global minimum), which is hard on quantum computers. The formal statement is given below in Theorem 4.9.1. Theorem 4.9.1 (Quantumly easy to find a local minimum under thermal perturbations; Restatement of Theorem 4.2.1). Let 𝑛 be the problem size. There is a poly(𝑛)-time quantum algorithm that guarantees the following. Given error 𝜖 = 1/poly(𝑛), inverse temperature 0 ≤ 𝛽 ≤ poly(𝑛), time scale 𝜏 = poly(𝑛), an 𝑛-qubit local Hamiltonian 𝑯 with ∥𝑯∥ ∞ = poly(𝑛), 𝑚 local jump operators { 𝑨𝑎 } 𝑚 𝑎=1 with 𝑚 = poly(𝑛), and a local observable 𝑶 with ∥𝑶 ∥ ∞ ≤ 1. The quantum algorithm outputs a real value 𝑣 ∈ [−1, 1], such that 𝑣 is 𝜖-close to tr(𝑶 𝝆) for an 𝜖-approximate local minimum 𝝆 of 𝑯 under thermal perturbations with an inverse temperature 𝛽, a time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 . Proof idea. We consider a version of gradient descent, which we refer to as Quantum thermal gradient descent, that mimics how Nature cools the quantum system when the system is interacting locally and weakly with a low-temperature heat bath. The algorithm starts with an arbitrary initial state 𝝆0 . For each step 𝑡 = 0, 1, 2, . . ., the algorithm considers the current state 𝝆𝑡 and proposes the next state 𝝆𝑡+1 . The tangent 𝛽,𝜏,𝑯 space 𝑇𝝆𝑡 at 𝝆𝑡 is high dimensional with many possible directions/dynamics depending on the system-bath interaction. The algorithm chooses a direction that lowers the energy as fast as possible by computing the gradient of the energy and proposes 𝝆𝑡+1 by performing gradient descent. As long as the current state 𝝆𝑡 is not an 𝜖-approximate local minimum of 𝑯 under thermal perturbations, the energy will decrease by a sufficiently large amount tr(𝑯𝝆𝑡+1 ) < tr(𝑯𝝆𝑡 ) − 1 . poly(𝑛) Because the energy is bounded from below, there are, at most, a polynomial number of steps 𝑡 ≤ poly(𝑛) until the algorithm arrives at an 𝜖-approximate local minimum 237 of 𝑯 under thermal perturbations. The detailed proof of Theorem 4.9.1 is given in Appendix 4.11. ■ Finally, we turn to the third main result establishing the difficulty of finding a local minimum under thermal perturbations using a classical computer. To establish this result, we consider a class of geometrically local Hamiltonians {𝑯𝐶 }𝐶 on 2D lattices. Each Hamiltonian 𝑯𝐶 corresponds to a 2D circuit 𝑼𝐶 = 𝑼𝑇 · · · 𝑼2𝑼1 acting on 𝑛 qubits with 𝑇 = 2𝑡0 + 𝐿 = poly(𝑛) gates as constructed in Fig. 1 of [142] with the additional padding to the construction in [142] such that the first and last 𝑡0 = 𝑐𝐿 2 gates being the identity gates for a constant 𝑐 = O (1). The construction in [142] has the property that each gate of the 2D circuit 𝑼𝐶 is geometrically adjacent to the subsequent gate. Given the 2D circuit 𝑼𝐶 on 𝑛 qubits with 𝑇 gates. The geometrically local Hamiltonian 𝑯𝐶 acts on 𝑛 + 𝑇 qubits on a 2D lattice and has a highly-entangled unique ground state that encodes the quantum computation based on the 2D circuit 𝑼𝐶 , |𝜂0 ⟩ = 𝑇 √︁ ∑︁ 𝑛
𝑡 𝑇−𝑡 𝜉𝑡 𝑼𝑡 · · · 𝑼1 |0 ⟩ ⊗ 1 0 , where 𝑡=0   1 𝑇 𝜉𝑡 := 𝑇 . 2 𝑡 We present the detailed construction of the 2D Hamiltonian 𝑯𝐶 in Definition 4.14.1 in Appendix 4.14. We have the following proposition for estimating single-qubit observables on the ground state of 𝑯𝐶 . Proposition 4.9.2 (BQP-hardness for estimating properties of the ground state of 𝑯𝐶 ). If there is a classical algorithm that can estimate any single-qubit observable on the unique ground state of the geometrically local Hamiltonian 𝑯𝐶 in time polynomial in the number of qubits in 𝑯𝐶 to error 1/4 for any 𝑯𝐶 in the class, then BPP = BQP. Proof. Consider the single-qubit observable 𝒁 𝑗 and let 𝑇 𝑗 be the last time that qubit 𝑗 is acted by a gate in the circuit 𝐶. The ground state expectation of 𝒁 𝑗 is ⟨𝜂0 |𝒁 𝑗 |𝜂0 ⟩ = 𝑇 ∑︁ 𝜉𝑡 ⟨0 |𝑼1† · · · 𝑼𝑡† 𝒁 𝑗 𝑼𝑡 · · · 𝑼1 |0𝑛 ⟩ + 𝑛 𝑡=𝑇 𝑗 +1 𝑇𝑗 ∑︁ 𝜉𝑡 ⟨0𝑛 |𝑼1† · · · 𝑼𝑡† 𝒁 𝑗 𝑼𝑡 · · · 𝑼1 |0𝑛 ⟩ 𝑡=0 = ⟨0𝑛 |𝑼1† · · · 𝑼𝑇† 𝒁 𝑗 𝑼𝑇 · · · 𝑼1 |0𝑛 ⟩𝑃𝑡>𝑇 𝑗 + 𝜖 𝑗 , ∑︁ ∑︁ where 𝑃𝑡>𝑇 𝑗 := 𝜉𝑡 , 𝜖 𝑗 := 𝜉𝑡 ⟨0𝑛 |𝑼1† · · · 𝑼𝑡† 𝒁 𝑗 𝑼𝑡 · · · 𝑼1 |0𝑛 ⟩. 𝑡>𝑇 𝑗 𝑡≤𝑇 𝑗 238 We have used the fact that 𝑈𝑡 for 𝑡 > 𝑇 𝑗 acts like identity on the 𝑗-th qubit. Note that |𝜖 𝑗 | ≤ 1 − 𝑃𝑡>𝑇 𝑗 =: 𝑃𝑡≤𝑇 𝑗 . We can make 𝜖 𝑗 arbitrarily small using a tail bound on the binomial distribution. Given any circuit, one could always pad more identity gates to form an 𝐿-gate circuit, such that the last 3/4 of the 𝐿 gates are identity. Recall that 𝑇 = 2𝑡 0 + 𝐿 = 2𝑐𝐿 2 + 𝐿. Then 𝑇 𝑗 ≤ 𝑐𝐿 2 + 𝐿/4 and we have |𝜖 𝑗 | ≤ 𝑃𝑡≤𝑇 𝑗 ≤ 𝑃𝑡≤𝑐𝐿 2 +𝐿/4 . Using Hoeffding’s inequality, we can bound the probability of sampling a time 𝑡, such that 𝑡 ≤ 𝑐𝐿 2 + 𝐿/4, according to the Binomial distribution {𝜉𝑡 }𝑇𝑡=0 . This yields " 2#  𝐿 1 𝑐𝐿 2 + 𝐿/4 − 8+16𝑐𝐿 = e |𝜖 𝑗 | ≤ exp −2𝑇 − . 2 2𝑐𝐿 2 + 𝐿 By choosing a small constant 𝑐 ≤ 1/(16 ln 18) − 1/(2𝐿), we have |𝜖 𝑗 | ≤ 1/18 and 𝑃𝑡>𝑇 𝑗 ≥ 17/18. Because of the bounds on the error |𝜖 𝑗 | and the probability 𝑃𝑡>𝑇 𝑗 , a classical algorithm satisfying the assumption of the proposition can determine whether ⟨0𝑛 |𝑼1† · · · 𝑼𝑇† 𝑍 𝑗 𝑼𝑇 · · · 𝑼1 |0𝑛 ⟩ > 1/3 or ⟨0𝑛 |𝑼1† · · · 𝑼𝑇† 𝑍 𝑗 𝑼𝑇 · · · 𝑼1 |0𝑛 ⟩ < −1/3, for any 2D circuit 𝑼𝐶 with 𝑇 = 2𝑡0 + 𝐿 = 2𝑐𝐿 2 + 𝐿 gates, where the first 𝑡0 and and the last (3/4)𝐿 + 𝑡0 gates are identity. Because one could think of the circuit 𝑼𝐶 as having 𝐿/4 gates for any 𝐿 = poly(𝑛), this immediately implies that a polynomial-time classical algorithm can decide whether the expectation value of 𝑍 𝑗 on the output state is > 1/3 or < −1/3 for any polynomial-size 2D circuit where all consecutive gates are adjacent in the 2D geometry. A 2D circuit 𝑼𝐶 such that any gate is adjacent to the subsequent gate can be constructed from any quantum circuit without the 2D constraint, such that a singlequbit observable 𝑍𝑖 on the output of the original quantum circuit corresponds to a single-qubit observable 𝑍 𝑗 on the output of the 2D circuit. As a result, any polynomial-time classical algorithm that can determine whether the expectation value of 𝑍 𝑗 on 𝑼𝑡 · · · 𝑼1 |0𝑛 ⟩ is greater than 1/3 or smaller than −1/3 can be used to simulate any polynomial-time quantum algorithm for solving decision problems in classical polynomial time. Hence, BPP = BQP. ■ Using a series of mathematical techniques presented in Appendix 4.12 for characterizing whether all local minima are global minima in a many-body Hamiltonian, 239 we prove that all local minima 𝑯𝐶 are close to the unique ground state |𝜂0 ⟩ in Theorem 4.9.2. This theorem is the most involved technical contribution of this work. Intuitively, one can think of the energy landscape of the 2D Hamiltonian 𝑯𝐶 over the space of 𝑛-qubit density matrices under low-temperature thermal perturbations to have a good bowl shape. This is in stark contrast to the energy landscape under local unitary perturbations, where the landscape always contains an overwhelmingly large barren plateau causing the problem of finding local minima to be classically easy. Furthermore, this theorem shows that a low-temperature cooling can always find a state close to the ground state irrespective of where we initialize the state in the exponentially large quantum state space. Theorem 4.9.2 (All local minima are global in BQP-hard Hamiltonians; Restatement of Theorem 4.2.2). Let 𝑷𝐺 (𝑯𝐶 ) = |𝜂0 ⟩⟨𝜂0 | be the ground state of the 2D Hamiltonian 𝑯𝐶 acting on 𝑛 + 𝑇 = poly(𝑛) qubits. There is a choice of 𝑚 = Poly(𝑛) two-qubit jump operators { 𝑨𝑎 } 𝑎 satisfying the following. Given 0 < 𝛿 < 1. For any small error 𝜖 = 1/Poly(𝑛, 1/𝛿), any 𝜖-approximate local minimum 𝝆 of 𝑯𝐶 under thermal perturbations with a large inverse temperature 𝛽 = Poly(𝑛, 1/𝛿), a large time scale 𝜏 = Poly(𝑛, 1/𝛿), and system-bath interactions generated by { 𝑨𝑎 } 𝑎 is an exact global minimum with high probability, i.e., we have tr(𝑷𝐺 (𝑯𝐶 ) 𝝆) ≥ 1 − 𝛿. The proof of Theorem 4.9.2 is given in Appendix 4.14. To show that the landscape has a good bowl shape, we utilize the negative gradient condition given in Appendix 4.8. However, the negative energy gradient operator is not easy to study. To establish this strong claim, we give a series of techniques in Appendix 4.12 for characterizing negative energy gradient operator in few-qubit systems, in commuting Hamiltonians, and in perturbed Hamiltonians. These technical tools can also be used to understand the energy landscape in other interacting many-body Hamiltonians. While finding a local minimum under local unitary perturbations is classically easy, the characterization of the energy landscape in these BQP-hard Hamiltonians 𝑯𝐶 implies that finding a local minimum under thermal perturbations is universal for quantum computation and is hence classically hard if BPP ≠ BQP. Recall that BPP = BQP implies that all single-qubit measurements of all polynomial-size quantum circuits can be simulated in polynomial time on a classical computer. Since one expects some quantum circuits to be hard to simulate on a classical computer, 240 Theorem 4.9.3 implies that finding a local minimum under thermal perturbations is classically hard. Theorem 4.9.3 (Classically hard to find a local minimum under thermal perturbations; Restatement of Theorem 4.2.3). Let 𝑛 be the problem size. Suppose there is a poly(𝑛)time classical algorithm guaranteeing the following. Given error 𝜖 = 1/poly(𝑛), inverse temperature 0 ≤ 𝛽 ≤ poly(𝑛), time scale 0 ≤ 𝜏 ≤ poly(𝑛), an 𝑛-qubit local Hamiltonian 𝑯 with ∥𝑯∥ ∞ = poly(𝑛), 𝑚 local jump operators { 𝑨𝑎 } 𝑚 𝑎=1 with 𝑚 = poly(𝑛), and a single-qubit observable 𝑶 with ∥𝑶 ∥ ∞ ≤ 1. The classical algorithm outputs a real value 𝑣 ∈ [−1, 1], such that 𝑣 is 𝜖-close to tr(𝑶 𝝆) for an 𝜖-approximate local minimum 𝝆 of the Hamiltonian 𝑯 under thermal perturbations with an inverse temperature 𝛽, a time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 . Then BPP = BQP. Proof. Assuming the existence of a polynomial-time classical algorithm that satisfies the properties stated in the theorem. Apply this classical algorithm to the 2D Hamiltonian 𝑯𝐶 considered in Theorem 4.9.2 with a sufficiently small approximation error 𝜖, such that any 𝜖-approximate local minimum 𝝆 of 𝑯𝐶 under thermal perturbations with polynomially-large 𝛽, 𝜏 and system-bath interactions generated by { 𝑨𝑎 } 𝑎 is an exact global minimum with high probability, i.e., ⟨𝜂0 | 𝝆|𝜂0 ⟩ = tr(𝑷𝐺 (𝑯𝐶 ) 𝝆) ≥ 1 − 1 , 162 where |𝜂0 ⟩ is the unique ground state of 𝑯𝐶 . We further consider 𝜖 to be small enough such that 1 (4.16) 𝜖< . 8 Let 𝝆 be an 𝜖-approximate local minimum of the Hamiltonian 𝑯 under thermal perturbations. Consider the observable 𝑶 𝑗 = 𝒁 𝑗 from the proof of Proposition 4.9.2. Using the Fuchs–van de Graaf inequalities, we have ∥ 𝝆 − |𝜂0 ⟩⟨𝜂0 |∥ 1 ≤ 1 . 8 (4.17) Because the classical algorithm can estimate tr(𝑶 𝑗 𝝆) to error 𝜖, from Eq. (4.16) and (4.17), the classical algorithm can estimate ⟨𝜂0 |𝑶 𝑗 |𝜂0 ⟩ to error 1/4 in time polynomial in the number of qubits in 𝑯𝐶 . From Prop. 4.9.2, this implies that BPP = BQP. ■ In the following, we use the previous theorem to show that quantum machines can improve over any efficient classical algorithm that variationally optimizes a 241 classical ansatz that can efficiently predict local properties by performing lowtemperature cooling. Examples of the classical ansatz include tensor networks with efficient tensor contraction algorithms and neural-network quantum states with fast sampling algorithms. This result provides a physically-relevant problem that yields an advantage in minimizing the energy of a geometrically-local Hamiltonian. Corollary 4.9.1 (Quantum advantage over variationally optimized classical ansatz). Under the conjecture that BPP ≠ BQP, there exists a class of 𝑛-qubit geometricallylocal Hamiltonian 𝑯 on a two-dimensional lattice with ∥𝑯∥ ∞ = O (𝑛) that satisfies the following. Given any classical ansatz of 𝑛-qubit state 𝝆 that can estimate the expectation value of single-qubit observables to 1/poly(𝑛) error in poly(𝑛)-time on classical computers, any poly(𝑛)-time classical algorithm for minimizing the energy tr(𝑯𝝆) using the classical ansatz, and samples of the state 𝝆 represented by the optimized classical ansatz. A quantum machine can find a state 𝝆 # with strictly lower energy than 𝝆 in poly(𝑛) time by running a quantum thermal gradient descent based on low-temperature cooling. Proof. The central claim is that the state 𝝆 found by an efficient classical algorithm cannot be an 𝜖-approximate local minimum under low-temperature thermal perturbations. We establish this claim by contradiction. Suppose that the classical ansatz for 𝝆 found by the efficient classical algorithm is an 𝜖-approximate local minimum. Then the classical algorithm can use the classical ansatz to predict the expectation values of single-qubit observables of an 𝜖-approximate local minimum 𝝆 of 𝑯 to 𝜖 error. From Theorem 4.9.3, this implies that BPP = BQP, which is a contradiction. Because 𝝆 is not an 𝜖-approximate local minimum under low-temperature thermal perturbations, a quantum machine can use samples of 𝝆 to initialize at the state 𝝆 and perform one gradient descent step based on low-temperature cooling. From Lemma 4.8.2 on the necessary condition for local minima, there exists 𝑎 ∈ {1, . . . , 𝑚} 𝛽,𝜏,𝑯 such that tr(𝑯L𝑎 [𝝆]) < −𝜖. From Lemma 4.11.1 on cooling by gradient descent, one can show that a single gradient descent step yields a state 𝝆 (next) with a strictly lower energy than the state 𝝆. Hence, one establishes the desired claim. ■ 4.10 Appendix:Details of thermal Lindbladians In the rest of the appendices, we give the full detailed proofs of Theorems 4.9.1 and 4.9.2 that are central to establishing the computational complexity of finding local minima under thermal perturbations in the previous appendix. To that end, 242 we need to provide the techincal details of thermal Lindbladians that generate such perturbations. We have previously presented a high-level introduction to thermal Lindbladians in Appendix 4.5. This has been sufficient for defining local minima and analyzing some basic properties, but not enough for proving Theorems 4.9.1 and 4.9.2. In this appendix, we present the exact form of thermal Lindbladians, their properties, and the algorithmic primitives for simulating quantum thermodynamics. Exact form The exact form of the thermal Lindbladian depends on a few physical concepts due to the microscopic derivation from a system-bath interaction [135]. For each jump 𝑨𝑎 , we have ∫ ∞ h i 1 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 L𝑎 ( 𝝆) := −i[𝑯 𝐿𝑆,𝑎 , 𝝆]+ 𝛾 𝛽 (𝜔) 𝑨ˆ 𝑎 (𝜔) 𝝆 𝑨ˆ 𝑎 (𝜔) † − { 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔), 𝝆} d𝜔, 2 −∞ | {z } 𝛽, 𝜏,𝑯 :=D𝑎 [𝝆] (4.18) 𝛽,𝜏,𝑯 where D𝑎 is the purely dissipative part of the thermal Lindbladian. Implicitly, the operator 𝑨ˆ 𝑎 (𝜔) also depends on the Hamiltonian 𝑯 and the time scale 𝜏. We now unpack the physical concepts that form the building blocks of this expression. Transition weight. At a fixed inverse temperature 𝛽, the transition weight 𝛾 𝛽 (𝜔) tells us how strong the rate of a transition/jump should be, depending on the energy difference 𝜔. In particular, the transition weight satisfies the following Kubo-Martin-Schwinger (KMS) condition and convenient normalization 𝛾 𝛽 (𝜔)/𝛾 𝛽 (−𝜔) = e−𝛽𝜔 and 0 ≤ 𝛾 𝛽 (𝜔) ≤ 1 for any 𝛽≥0 and any 𝜔 ∈ R, (4.19) which is reminiscent of how detailed balance is enforced in classical Markov chains. We remark that any 𝛾 𝛽 (𝜔) obeying the above KMS condition and normalization also satisfies the following tail bound: max 𝜔𝛾 𝛽 (𝜔) ≤ max 𝜔e−𝛽𝜔 = 𝜔≥Δ 𝜔≥Δ 1 1 e−𝛽Δ/2 max 𝑥e−𝑥 ≤ max e−𝑥/2 = . 𝛽 𝑥≥𝛽Δ 𝛽 𝑥≥𝛽Δ 𝛽 (4.20) For concreteness, we usually adopt the common choice of 𝛾 𝛽 corresponding to Glauber dynamics, with a cut-off frequency Λ0 to regulate the inverse Fourier transform: 2 2 1 e−𝜔 /2Λ0 𝛾 𝛽 (𝜔) = . (4.21) 2 + ln(1 + 𝛽Λ0 ) 1 + e 𝛽𝜔 243 In the zero temperature regime (𝛽 = ∞), the function (1 + e 𝛽𝜔 ) −1 gives a step function (one for negative 𝜔 and zero for positive 𝜔). Based on the choice of bath phenomenology, there are plenty of options for the transition weight, such as ohmic − 𝜔 2 /2Λ2 heating 𝛾 𝛽 (𝜔) = 𝜔10 𝜔e1−e−𝛽 𝜔 0 . However, for simplicity, we will stick to the Glauber dynamics. Unless otherwise stated, we will also choose the cut-off frequency to be Λ0 = 1, since each local jumps 𝑨𝑎 changes the energy by at most O (1) for our usage (and is generally true for local Hamiltonians with bounded degree interaction graph and bounded-norm terms). We do not expect our main conclusion to change under other reasonable choices of 𝛾 𝛽 (𝜔). Operator Fourier transform. Given a jump operator 𝑨𝑎 , we consider the operator Fourier Transform [39] for the Heisenberg-evolved jump operator 𝑨𝑎 (𝑡) characterized by a time scale 𝜏 ∈ R of the heat bath ˆ𝑎 𝑨 (𝜔) := √ 1 ∫ 𝜏/2 ei𝑯𝑡 𝑨𝑎 e−i𝑯𝑡 e−i𝜔𝑡 d𝑡. 2𝜋𝜏 −𝜏/2 | {z } =:𝑨 𝑎 (𝑡) The operator 𝑨ˆ 𝑎 (𝜔) corresponds to matrix elements in 𝑨𝑎 that induce jumps between energy eigenstates with an energy difference approximately 𝜔 ± O ( 1𝜏 ). The bigger 𝜏 is, the more precise 𝜔 corresponds to the true energy difference; see Appendix 4.15 for further details. Physically, 𝜏 is related to microscopic parameters of the bath (the bath correlation time and the weak-coupling strength [135]), but our discussion only requires the single time scale 𝜏 that sets the Fourier transform energy uncertainty. Lamb-shift. The interaction with the heat bath induces an additional correction term in the coherent Hamiltonian dynamics of the 𝑛-qubit system, known as the Lambshift. Given a jump operator 𝑨𝑎 , we have the following Lamb-shift Hamiltonian that depends on the bath correlation function 𝑐 𝛽 (𝑡) and the time scale 𝜏: 𝛽,𝜏,𝑯 𝑯 𝐿𝑆,𝑎 := i √ 2 2𝜋𝜏 ∫ 𝜏/2 ∫ 𝜏/2 sgn(𝑡1 − 𝑡2 )𝑐 𝛽 (𝑡2 − 𝑡 1 ) 𝑨𝑎 (𝑡2 ) 𝑨𝑎 (𝑡1 )d𝑡 2 d𝑡1(4.22) . −𝜏/2 −𝜏/2 While the Lamb-shift term is physically important, for our purposes, it is largely treated as a source of error; the energy gradient contribution comes from the 𝛽,𝜏,𝑯 dissipative part D𝑎 . 244 Bath correlation function. In the Lamb-shift term, the bath correlation function 𝑐 𝛽 (𝑡) is the Fourier transform of the transition weight 𝛾 𝛽 (𝜔), ∫ ∞ 1 𝑐 𝛽 (𝑡) = √ 𝛾 𝛽 (𝜔)e+i𝜔𝑡 d𝑡. 2𝜋 −∞ The prefactor in Eq. (4.21) is chosen such that (see Proposition 4.10.1) ∫ ∞ 1 𝑐 𝛽 (𝑡) d𝑡 ≤ 1. √ 2𝜋 −∞ (4.23) 𝛽,𝜏,𝑯 This normalization sets the strength of ∥𝑯 𝐿𝑆,𝑎 ∥ to be bounded by O (1). Absolute zero 𝛽 = ∞. It is instructive to consider the case of zero temperature 𝛽 = ∞ and infinite time scale 𝜏 = ∞. In this case, the transition weight 𝛾 𝛽 (𝜔) is a step function (1 for 𝜔 < 0 and 0 for 𝜔 > 0) and 𝑨ˆ 𝑎 (𝜔) measures the energy difference perfectly. Thus, all heating transitions (|𝐸⟩ → |𝐸 + 𝜔⟩ for 𝜔 > 0) are forbidden, and all cooling transitions (|𝐸⟩ → |𝐸 + 𝜔⟩ for 𝜔 < 0) will remain. Hence, in the case when 𝛽 = 𝜏 = ∞, the thermal Lindbladian only lowers the energy. This matches our physical intuition that a zero-temperature bath only absorbs energy from the system. 𝛽,𝜏,𝑯 Multiple jumps. The thermal Lindbladian L𝑎 considers merely a single jump 𝑎 operator 𝑨 in the system-bath interaction. When there are multiple jump operators, the total thermal Lindbladian is a weighted sum of the individual thermal Lindbladian 𝛽,𝜏,𝑯 L𝑎 , 𝑚 ∑︁ 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 L = 𝛼𝑎 L 𝑎 , 𝑎=1 where 𝛼𝑎 ≥ 0 is a nonnegative weight. Again, the interaction strength vector 𝜶 ∈ R𝑚 ≥0 weights the contribution of each thermal Lindbladian. Thus, the total equation of motion under multiple jumps reads 𝑑𝝆 = −i[𝑯, 𝝆] + L 𝛽,𝜏,𝑯 ( 𝝆) 𝑑𝑡 " # 𝑚 𝑚 ∑︁ ∑︁ 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 = −i 𝑯 + 𝛼𝑎 𝑯 𝐿𝑆,𝑎 , 𝝆 + 𝛼𝑎 D𝑎 ( 𝝆), 𝑎=1 𝑎=1 which consists of a coherent part and a purely dissipative part. (4.24) 245 Calculation for normalization of 𝑐 𝛽 (𝑡). We now give a supplemental calculation that shows our choice of 𝛾 𝛽 (𝜔) in Equation 4.21 satisfies the condition in Equation 4.23. Proposition 4.10.1. For −𝜔2 /2Λ20 e 𝑓ˆ(𝜔) := , 1 + e 𝛽𝜔 we have that 1 √ ∥ 𝑓 ∥ 1 ≤ 2 + ln(1 + 𝛽Λ0 ). 2𝜋 Proof. We want to bound the 1-norm of 𝑓 (𝑡) in the time domain. To do so, we bound the moments in the time domains ∫ ∞ √ √ 2 2 2𝜋∥ 𝑓 ∥ ∞ ≤ ∥ 𝑓ˆ∥ 1 ≤ e−𝜔 /2Λ0 d𝜔 = Λ0 2𝜋. −∞ √ d ˆ 𝑓 ∥ 1 = 2 · ∥ 𝑓ˆ∥ ∞ ≤ 2. 2𝜋∥𝑡 𝑓 (𝑡)∥ ∞ ≤ ∥ d𝜔 √ d2 d ˆ 𝑓 ∥∞ ≤ 𝑇 . 2𝜋∥𝑡 2 𝑓 (𝑡)∥ ∞ ≤ ∥ 2 𝑓ˆ∥ 1 ≤ 4 · ∥ d𝜔 d𝜔 The second line uses the fact that 𝑓ˆ is increasing and then decreasing (from −∞ to ∞). The third line evaluates the derivative ! 2 2 2 2 2 2 −e−𝜔 /2Λ0 𝜔/Λ20 e−𝜔 /2Λ0 𝛽e 𝛽𝜔 d e−𝜔 /2Λ0 4 =4 − d𝜔 1 + e 𝛽𝜔 1 + e 𝛽𝜔 (1 + e 𝛽𝜔 ) 2   1 ≤4 √ + 𝛽 =: 𝑇 . eΛ0 Thus, we may partition into three integrals to optimize the bound ! ∫ ∫ ∫ | 𝑓 (𝑡)|d𝑡 ∥ 𝑓 ∥1 = + + |𝑡|≤Λ0−1 ∫ 𝑇 ≥|𝑡|≥Λ0−1 |𝑡|≥𝑇 ∫ ∫ 2 1 𝑇 1 ≤ Λ0 d𝑡 + √ d𝑡 + √ d𝑡 −1 −1 |𝑡|≤Λ0 2𝜋 𝑇 ≥|𝑡|≥Λ0 |𝑡| 2𝜋 |𝑡|≥𝑇 𝑡 2 4 2 ≤ 2 + √ ln(Λ0𝑇) + √ 2𝜋 2𝜋 √ 4 2 + 2 2𝜋 + 4 ln( √e + 4𝛽Λ0 ) ≤ √ 2𝜋 √ 2 + 2 2𝜋 + 8 ln(2) + 4 ln(1 + 𝛽Λ0 ) √ ≤ ≤ 2𝜋(2 + ln(1 + 𝛽Λ0 )), √ 2𝜋 √ where in the last line, we used 1/ e ≤ 1 among other numerical bounds. ■ 246 Properties of thermal Lindbladians From the exact forms of the thermal Lindbladians, we have the following propositions. Proposition 4.10.2 (Norm for the dissipative part [39]). Any purely dissipative Í 𝛽,𝜏,𝑯 Lindbladian 𝑎 D𝑎 defined in Eq. (4.18) for any set of jump operators { 𝑨𝑎 } 𝑚 𝑎=1 and any transition weight satisfying Eq. (4.19) have bounded superoperator norms 𝑚 ∑︁ †𝛽,𝜏,𝑯 𝛼𝑎 D𝑎 = 𝑚 ∑︁ ∞−∞ 𝑎=1 𝛽,𝜏,𝑯 𝛼𝑎 D𝑎 ≤ 2∥ 1−1 𝑎=1 𝑚 ∑︁ 𝛼𝑎 𝑨𝑎† 𝑨𝑎 ∥. 𝑎=1 The first equality is the duality between the 1 − 1 and ∞ − ∞ superoperator norms. Proposition 4.10.3 (Properties of the Lamb-shift term [39]). The sum of Lambshift term (4.22) for any set of jump operators { 𝑨𝑎 } 𝑚 𝑎=1 under a normalized bath correlation function 𝑐 𝛽 (𝑡) given by Eq. (4.23) satisfies that7 𝑚 ∑︁ 𝑚 𝛽,𝜏,𝑯 𝛼𝑎 𝑯 𝐿𝑆,𝑎 𝑎=1 ∥ 𝑚 ∑︁ 1 ∑︁ ≤ ∥ 𝛼𝑎 𝑨𝑎† 𝑨𝑎 ∥ 2 𝑎=1 𝛽,𝜏,𝑯 𝛼𝑎 [𝑯 𝐿𝑆,𝑎 , 𝑯] ∥ ≤ O 𝑎=1 ! 𝑚 ∥𝑯∥ 3/4 ∑︁ 𝑎† 𝑎 ∥ 𝛼𝑎 𝑨 𝑨 ∥ . 𝜏 1/4 𝑎=1 For large enough 𝜏, the Lamb-shift term almost commutes with the Hamiltonian. From Prop. 4.10.2 and Prop. 4.10.3, we have the following norm bound for thermal Lindbladians. Proposition 4.10.4 (Norm of thermal Lindbladians). Given a Hamiltonian 𝑯, an inverse temperature 𝛽 ≥ 0, a time scale 𝜏 ≥ 0, 𝑚 local jump operators { 𝑨𝑎 } 𝑚 𝑎=1 , a transition weight 𝛾 𝛽 (𝜔) satisfying Eq. (4.19), a normalized bath correlation function Í 𝛽,𝜏,𝑯 𝑐 𝛽 (𝑡) satisfying Eq. (4.23). The associated thermal Lindbladian 𝑚 has 𝑎=1 𝛼𝑎 L 𝑎 bounded superoperator norms 𝑚 ∑︁ 𝑎=1 †𝛽,𝜏,𝑯 𝛼𝑎 L 𝑎 = ∞−∞ 𝑚 ∑︁ 𝑎=1 𝛽,𝜏,𝑯 𝛼𝑎 L 𝑎 ≤ 3∥ 1−1 𝑚 ∑︁ 𝛼𝑎 𝑨𝑎† 𝑨𝑎 ∥, 𝑎=1 which is controlled by the interaction strength vector 𝜶 under the normalization of 𝑨𝑎 in Eq. (4.2). 7 Implicitly, the Lamb-shift term has units of energy yet do not scale with ∥𝑯∥. 247 Algorithmic primitives for simulating thermal Lindbladians In this subsection, we review existing algorithmic primitives for simulating thermal Lindbladians [39], estimating energy and expectation value of observables using block-encoding and quantum singular value transform (QSVT). See [126] for a tutorial on block encoding and QSVT. We begin with a definition of a block-encoding for Hermitian matrices, i.e., observables. Definition 4.10.1 (Block-encoding for Hermitian matrices). We say that a unitary 𝑼 is a block-encoding for a Hermitian matrix 𝑶 if ( 0𝑑 ⊗ 𝑰) · 𝑼 · ( 0𝑑 ⊗ 𝑰) = 𝑶 for 𝑑 ∈ Z+ . Recall the following result stating that expectation values can be estimated using block-encoding. Proposition 4.10.5 (Measuring observable using block-encoding). Given a blockencoding 𝑼𝑶 for a Hermitian matrix 𝑶 and samples of a state 𝝆. One could estimate tr(𝑶 𝝆) to small error 0 < 𝜖 < 0.5 using only Õ (1/𝜖 2 ) queries to the unitary 𝑼𝑶 . Proof. Consider |0⟩⟨0| ⊗ 𝝆 and apply the Hadamard test to sample tr[𝑼|0⟩⟨0| ⊗ 𝝆] = tr[𝑶 𝝆] 8. ■ Linear combinations of unitaries allow us to make efficient block-encodings of Hamiltonians presented as a sum of local terms. This fact results in the following proposition. Proposition 4.10.6 (Block-encoding for Hamiltonian; see [19, 45, 126]). Any 𝑛-qubit Hamiltonian 𝑯 has an efficient block-encoding 𝑼 𝑯/𝜆1 for some scalar 𝜆 1 being the 1-norm of Pauli expansion coefficients. From [39], we have the following for the Lamb-shift term 𝑯 𝐿𝑆 from Eq. (4.22). Conveniently, the Lamb-shift term is already normalized (Proposition 4.10.3). Proposition 4.10.7 (Block-encoding for Lamb-shift term; see [39]). The Lamb-shift term 𝑯 𝐿𝑆 has an efficient block-encoding 𝑼 𝐿𝑆 . We define the block-encoding for a Lindbladian without the coherent commutator term −i[𝑯, 𝝆]. 8We thank Yu Tong for discussions on this argument. 248 Definition 4.10.2 (Block-encoding for Lindblad operators [39]). Given a purely irreversible Lindbladian  ∑︁  1 † 1 † † L [𝝆] := 𝑳 𝑗 𝝆𝑳 𝑗 − 𝑳 𝑗 𝑳 𝑗 𝝆 − 𝝆𝑳 𝑗 𝑳 𝑗 , 2 2 𝑗 ∈𝐽 we say that a unitary 𝑼 is a block-encoding for Lindblad operators {𝑳 𝑗 } 𝑗 ∈𝐽 if 9 ∑︁ | 𝑗⟩ ⊗ 𝑳 𝑗 for 𝑏 ≤ 𝑐 ∈ Z+ . ( 0𝑏 ⊗ 𝑰) · 𝑼 · (|0𝑐 ⟩ ⊗ 𝑰) = 𝑗 ∈𝐽 Theorem 4.10.1 (Linear-time Lindbladian simulation [39]). Suppose the jumps 𝑨𝑎 can be block-encoded by a unitary 𝑽 𝑗𝑢𝑚 𝑝 using 𝑐 ∈ Z ancillas qubits. Then, we can simulate the map e𝑡L for (4.24) to 𝜖 ≤ 1/2 precision in the diamond norm using Õ ((𝑐 + 1)) resettable ancilla, Õ ((𝑡 + 1)𝜏) controlled Hamiltonian simulation time, Õ ((𝑡 + 1)(𝑐 + 1)) and other two-qubit gates, queries to 𝑾, 𝒑𝒓𝒆 𝒑 𝑐 𝛽 (𝑡) ,𝑷𝒓𝒆 𝒑′𝑐 𝛽 (𝑡¯) , and 𝑽 𝑗𝑢𝑚 𝑝 Õ (𝑡 + 1) where Õ (·) absorbed poly-logarithmic dependences on 𝑡, ∥𝑯∥, 𝜖, 𝜏, 𝛽. Furthermore, a block-encoding of the purely irreversible Lindbladian D 𝛽,𝜏,𝑯 with discretized frequency labels can be implemented efficiently. The above uses the following circuit components required for implementation: the controlled Hamiltonian simulation ∑︁ | 𝑡¯⟩⟨𝑡¯| ⊗ e±i𝑡¯𝑯 , 𝑡¯∈𝑆𝑡0 the unitary gates for preparing the bath correlation function in superposition ∑︁ √︃ ∑︁ 𝑐 𝛽 ( 𝑡¯) | 𝑡¯⟩, 𝑷𝒓𝒆 𝒑 𝑐 𝛽 (𝑡¯) : 0̄ → 𝑐 𝛽 ( 𝑡¯) | 𝑡¯⟩ and 𝑷𝒓𝒆 𝒑′𝑐 𝛽 (𝑡¯) : 0̄ → √︃ 𝑡¯∈𝑆𝑡0 𝑡¯∈𝑆𝑡0 𝑐 𝛽 ( 𝑡¯) and the controlled rotation for transition weights ! √︁ √︁ ∑︁ 𝛾( 𝜔) ¯ − 1 − 𝛾( 𝜔) ¯ √︁ √︁ 𝑾 := ⊗ | 𝜔⟩⟨ ¯ 𝜔|. ¯ 1 − 𝛾( 𝜔) ¯ 𝛾( 𝜔) ¯ 𝜔∈𝑆 ¯ 𝜔 0 9 In the first register, we could use any orthonormal basis, sticking to computational basis elements | 𝑗⟩ is just for ease of presentation. Intuitively one can think about 𝑏 as the number of ancilla qubits used for implementing the operators 𝑳 𝑗 , while typically 𝑎 − 𝑏 ≈ log |𝐽 |. 249 Indeed, the above implementation uses discrete labels for the time 𝑡¯ ∈ 𝑆𝑡0 and frequencies 𝜔¯ ∈ 𝑆𝜔0 corresponding to d𝑡 and d𝜔; these dominate the ancilla use. For conceptual simplicity, we focus on the continuous integral everywhere else and emphasize that the discretization is merely for implementation and introduces a negligible error; see [39]. The controlled Hamiltonian simulation can be implemented efficiently for any 𝑛-qubit local Hamiltonian 𝑯 [19, 45, 126]. The other operations 𝑾, 𝒑𝒓𝒆 𝒑 𝑐 𝛽 (𝑡) ,𝑷𝒓𝒆 𝒑′𝑐 𝛽 (𝑡¯) can all be implemented efficiently [39] with the physically-motivated choice considered in Appendix 4.10. Proposition 4.10.8 (Gradient of an observable under Lindbladian evolution; adapted from [39]). Given block-encodings 𝑼 for a purely irreversible Lindbladian (Def. 4.10.2) and 𝑼𝑶 for a Hermitian observable 𝑶, we get a block-encoding of   ∑︁ 𝑳 †𝑗 𝑶 𝑳 𝑗 via 𝑽 := (𝒀 1 ⊗ 𝑼 † ⊗ 𝑰 𝑑 ) · 2|0𝑏+1 ⟩⟨0𝑏+1 | ⊗ 𝑰 − 𝑰 ⊗ 𝑼𝑶 · (𝒀 1 ⊗ 𝑼 ⊗ 𝑰 𝑑 ), 2 𝑗 ∈𝐽 2 ! √ 1 −1 where |±⟩ := (|0⟩ ± |1⟩)/ 2 and 𝒀 1 := √1 . 2 1 2 1 Proof. We calculate ( 0𝑐+1 ⊗ 𝑰 ⊗ 0𝑑 ) · 𝑽 · ( 0𝑐+1 ⊗ 𝑰 ⊗ 0𝑑 )       𝑑 † 𝑑 𝑏+1 𝑏+1 𝑐 𝑐 = ⟨−| ⊗ (⟨0 | ⊗ 𝑰)𝑼 ⊗ 0 · 2|0 ⟩⟨0 | ⊗ 𝑰 − 𝑰 ⊗ 𝑼𝑶 · |+⟩ ⊗ 𝑼(|0 ⟩ ⊗ 𝑰) ⊗ 0       𝑐 † 𝑑 𝑐 𝑑 𝑏+1 𝑏+1 = ⟨−| ⊗ (⟨0 | ⊗ 𝑰)𝑼 ⊗ 0 · 2|0 ⟩⟨0 | ⊗ 𝑰 ⊗ 𝑼𝑶 · |+⟩ ⊗ 𝑼(|0 ⟩ ⊗ 𝑰) ⊗ 0 = (⟨0𝑐 | ⊗ 𝑰) · 𝑼 † · (|0𝑏 ⟩⟨0𝑏 | ⊗ 𝑰) ⊗ 𝑶 · (|0𝑐 ⟩ ⊗ 𝑰) ! ! ∑︁ ∑︁ ∑︁ † ⟨𝑗| ⊗ 𝑳𝑗 𝑰 ⊗ 𝑶 | 𝑗 ′⟩ ⊗ 𝑳 𝑗 ′ = = 𝑳 †𝑗 𝑶 𝑳 𝑗 . 𝑗 ∈𝐽 𝑗 ′ ∈𝐽 ■ 𝑗 ∈𝐽 Corollary 4.10.1 (Block-encoding the gradient of the Hamiltonian). Given a blockencoding for a purely irreversible Lindbladian L and a Hamiltonian 𝑯, there is an efficient block-encoding for 1 † L [𝑯], 2 which is a Hermitian operator corresponding to the gradient of 𝑯 under L. 250 Proof. Apply Proposition 4.10.8 for Lindbladian L, Hermitian observables 𝑯 and Í Í 𝑰 to obtain block-encodings for 𝑗 ∈𝐽 𝑳 †𝑗 𝑯𝑳 𝑗 and 𝑗 ∈𝐽 𝑳 †𝑗 𝑳 𝑗 . Then, use quantum singular value transform (QSVT) for products and sums of block-encoding to obtain the block-encoding for 1 ∑︁ † 1 † 1 ∑︁ † 1 ∑︁ † L [𝑯] = 𝑳 𝑗 𝑯𝑳 𝑗 − 𝑳𝑗𝑳𝑗𝑯 − 𝑯 𝑳 𝑳𝑗 2 2 𝑗 ∈𝐽 4 𝑗 ∈𝐽 4 𝑗 ∈𝐽 𝑗 at high precision. ■ From all of the above propositions, corollaries, and theorems, we can obtain the following. Lemma 4.10.1 (Measuring energy gradient). Given an 𝑛-qubit Hamiltonian 𝑯, inverse temperature 𝛽 ≥ 0, time scale 𝜏 ≥ 0, samples of an 𝑛-qubit state 𝝆, a thermal Lindbladian L 𝛽,𝜏,{ 𝑨𝑎 } 𝑎 from Eq. (4.24). The energy gradient tr(𝑯L 𝛽,𝜏,𝑯 ( 𝝆)) = tr(L †𝛽,𝜏,𝑯 (𝑯) 𝝆) can be estimated to error 𝜖 using time and samples of 𝝆 polynomial in 𝑛, 1/𝜖, ∥𝑯∥, 𝛽, 𝜏. Proof. From the form of thermal Lindbladians (4.24) and dropping the scripts 𝛽,𝜏,𝑯 L 𝛽,𝜏,𝑯 = L, 𝑯 𝐿𝑆 = 𝑯 𝐿𝑆 , D 𝛽,𝜏,𝑯 = D we have L † (𝑯) = i[𝑯 𝐿𝑆 , 𝑯] + D † (𝑯). Our goal is to create the block-encoding for L † (𝑯). First, we use quantum singular value transform (QSVT) for products and sums of block-encoding to obtain the blockencoding for i[𝑯 𝐿𝑆 , 𝑯] from block-encoding for 𝑯 and 𝑯 𝐿𝑆 in Propositions 4.10.6 and 4.10.7. Next, using the block-encoding for the purely irreversible Lindbladian D from Theorem 4.10.1 and the block-encoding for 𝑯, we can apply Corollary 4.10.1 to obtain efficient block-encoding for D † (𝑯). To obtain the block-encoding for L † (𝑯), we use QSVT for sums of block-encoding to add i[𝑯 𝐿𝑆 , 𝑯] and D † (𝑯). Finally, using Prop. 4.10.5, we can estimate tr(L † (𝑯) 𝝆) efficiently. All the above QSVT manipulations operate at high precision, and the discrete Fourier transform well-approximates the continuum at poly-logarithmic costs [39]. ■ 4.11 Appendix:A polynomial-time quantum algorithm for finding a local minimum under thermal perturbations (Proof of Theorem 4.9.1) In this appendix, we present the proof of Theorem 4.9.1 by giving a polynomial-time quantum algorithm for finding local minima under thermal perturbations. We 251 refer to the efficient quantum algorithm as Quantum thermal gradient descent as the algorithm performs gradient descent using thermal Lindbladians induced by a heat bath. The algorithm uses the properties of thermal Lindbladians presented in Appendix 4.10. Cooling by gradient descent The central idea of quantum thermal gradient descent is the following. When we are not at a local minimum under thermal perturbations, the negative energy gradient will be sufficiently large, and we can decrease the energy by following a direction with a negative energy gradient. This is characterized by the following lemma. We will use this lemma to design the gradient descent algorithm for finding a local minimum. Lemma 4.11.1 (Cooling by gradient descent). Given parameters 0 < 𝜖˜ < 0.5, 𝐵 ≥ 1, 𝛽, 𝜏 ≥ 0, an 𝑛-qubit Hamiltonian 𝑯 with ∥𝑯∥ ∞ ≤ 𝐵, and 𝑚 local jump operators { 𝑨𝑎 } 𝑎 . Consider 𝑎 = 1, . . . , 𝑚 with an approximate energy gradient 𝑔𝑎 satisfying   𝛽,𝜏,𝑯 𝑔𝑎 − tr 𝑯L𝑎 [𝝆 (𝑡−1) ] < 0.01𝜖˜. Suppose there exist 𝑎 ∗ ∈ {1, . . . , 𝑚} with sufficiently negative approximate energy gradient, 𝑔𝑎∗ < −0.99𝜖˜. The state after evolving 𝝆 along the direction 𝒆ˆ 𝑎∗ for a small step 𝑠 = |𝑔𝑎∗ |/(9𝐵2 ) > 0, 𝛽,𝜏,𝑯,{ 𝑨 𝑎 } 𝑎 𝝆 (next) := exp 𝝆 (𝑠 𝒆ˆ 𝑎∗ ) guarantees the following energy decrease,   𝜖˜2 tr 𝑯𝝆 (next) < tr(𝑯𝝆) − . 20𝐵2 Proof. From Prop. 4.8.2 on Taylor’s theorem, we have the following identity   𝑠2 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 tr 𝑯𝝆 (next) = tr(𝑯𝝆) + 𝑠 tr(𝑯L𝑎∗ [𝝆]) + tr(𝑯L𝑎∗ [L𝑎∗ [𝝈]]) 2 for some 𝑛-qubit state 𝝈. We will separately control the linear term and the quadratic term. Linear term. From the definition of the energy gradient vector ∇ 𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆), we have 98 98 𝛽,𝜏,𝑯 tr(𝑯L𝑎∗ [𝝆]) < 𝑔𝑎∗ + 0.01𝜖˜ < 𝑔𝑎∗ = − |𝑔𝑎∗ |. 99 99 ∗ The second inequality follows from 𝑔𝑎 < −0.99𝜖, ˜ hence 0.01𝜖˜ < −(1/99)𝑔𝑎∗ . 252 Quadratic term. We can bound the quadratic term as follows: 1 1 †𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 †𝛽,𝜏,𝑯 tr(𝑯L𝑎∗ [L𝑎∗ [𝝈]]) ≤ L𝑎∗ ∥L𝑎∗ (𝑯)∥ ∞ . ∞−∞ 2 2 From Prop. 4.8.3 and Prop. 4.10.4 that bounds the norm of these objects, we have 1 †𝛽,𝜏,𝑯 †𝛽,𝜏,𝑯 L𝑎 ∥L𝑎′ (𝑯)∥ ∞ ≤ 4.5∥𝑯∥ 2∞ ≤ 4.5𝐵2 . ∞−∞ 2 Combining the linear and quadratic terms with 𝑠 = |𝑔𝑎∗ |/(9𝐵2 ) > 0, we have     98 1 |𝑔𝑎∗ | 2 |𝑔𝑎∗ | 2 (next) tr 𝑯𝝆 ≤ tr(𝑯𝝆) − − < tr(𝑯𝝆) − 0.054 . 99 × 9 18 𝐵2 𝐵2 We can use |𝑔𝑎∗ | 2 > 0.992 𝜖˜2 to obtain the desired claim. ■ Quantum thermal gradient descent Given error 𝜖 = 1/poly(𝑛), norm bound 𝐵 = poly(𝑛), inverse temperature 0 ≤ 𝛽 ≤ poly(𝑛), time scale 𝜏 = poly(𝑛), an 𝑛-qubit local Hamiltonian 𝑯 with ∥𝑯∥ ∞ ≤ 𝐵, 𝑚 local jump operators { 𝑨𝑎 } 𝑎 with 𝑚 = poly(𝑛), and a local observable 𝑶 with ∥𝑶 ∥ ∞ ≤ 1. We consider a coordinate-wise gradient descent algorithm that implements the following. The initial state 𝝆 (0) is arbitrary as long as copies of the state can be prepared on the quantum computer. For example, we can set 𝝆 (0) to be the maximally mixed state 2𝑰𝑛 . The total number of steps is 42𝐵3 𝑇 := 2 . 𝜖 (4.27) For each time step 𝑡 from 1 to 𝑇, the algorithm does the following. 1. For each direction 𝑎 = 1, . . . , 𝑚, estimate an approximate energy gradient 𝑔𝑎(𝑡) satisfying   †𝛽,𝜏,𝑯 (𝑡) (𝑡−1) 𝑔𝑎 − tr L𝑎 (𝑯) 𝝆 < 0.0099𝜖 . (4.28) The energy gradient can be estimated efficiently using Lemma 4.10.1 given copies of 𝝆 (𝑡−1) prepared through Eq. (4.29) and Theorem 4.10.1. From the bound on energy gradients in Prop. 4.8.3, we have |𝑔𝑎(𝑡) | ≤ 3𝐵 + 0.0099𝜖. If 𝑔𝑎(𝑡) < −0.99𝜖, set 𝑎 (𝑡) := 𝑎 and terminate the for-loop over 𝑎. 2. If 𝑎 (𝑡) is not found, set 𝝆 (𝑇) := 𝝆 (𝑡−1) and terminate the for-loop over 𝑡. Otherwise evolve 𝝆 (𝑡−1) under the direction 𝒆ˆ 𝑎 (𝑡 ) for a small step 𝑠 (𝑡) := 253 |𝑔𝑎(𝑡) |/(9𝐵2 ), 𝝆 (𝑡)  := exp 𝑠 (𝑡) 𝛽,𝜏,𝑯 L𝑎 (𝑡 )  𝝆 (𝑡−1)  = 𝑡 Ö    𝛽,𝜏,𝑯 exp 𝑠 (𝑡) L𝑎 (𝑡 ) 𝝆 (0) . (4.29) 𝑡 ′ =1 Because 0 ≤ 𝑠 (𝑡) ≤ 1/(2𝐵), a single copy of 𝝆 (𝑡) can be prepared in polynomialtime using the thermal Lindbladian simulation algorithm in [39]; see Theorem 4.10.1. We will show that the state 𝝆 (𝑇) created by the gradient descent algorithm is an 𝜖-approximate local minimum of 𝑯 under thermal perturbations. Furthermore, using the thermal Lindbladian simulation algorithm, a quantum machine can efficiently create many copies of 𝝆 (𝑇) . Proof of Theorem 4.9.1 The central idea in the proof of Theorem 4.9.1 is the following lemma. The lemma combines the key results characterizing local minima in Appendix 4.8. Lemma 4.11.2 (Gradient descent finds a local minimum). 𝝆 (𝑇) from Eq. (4.29) is an 𝜖-approximate local minimum of 𝑯 under thermal perturbations with inverse temperature 𝛽, time scale 𝜏, and system-bath interactions generated by { 𝑨𝑎 } 𝑎 . (𝑡) Proof. Suppose the algorithm  terminates at some  time step 𝑡 < 𝑇, then 𝑔𝑎 ≥ −0.99𝜖. †𝛽,𝜏,𝑯 From Eq. (4.28), we have tr L𝑎 (𝑯) 𝝆 (𝑡−1) ≥ −0.9999𝜖. Hence, ∥∇−𝛽,𝜏,{ 𝑨𝑎 } 𝑎 (𝑯, 𝝆 (𝑡−1) )∥ ∞ ≤ 0.9999𝜖 < 𝜖 . From the sufficient condition for local minima given in Lemma 4.8.1, we have 𝝆 (𝑇) = 𝝆 (𝑡−1) is an 𝜖-approximate local minimum 𝝆 of the 𝑛-qubit Hamiltonian 𝑯 under thermal perturbations. We now show by contradiction that the algorithm must terminate early. Assume that the algorithm did not terminate early. Then, we can use Lemma 4.11.1 with 𝜖˜ = 0.99𝜖 for cooling by gradient descent to obtain tr(𝑯𝝆 (𝑇) ) ≤ tr(𝑯𝝆 (𝑇−1) )− 0.992 𝜖 2 0.992 𝜖 2 0.992 𝜖 2 (0) ≤ . . . ≤ tr(𝑯𝝆 )− 𝑇 ≤ ∥𝑯∥ − 𝑇. ∞ 20𝐵2 20𝐵2 20𝐵2 From the definition of 𝑇 in Eq. (4.27) and ∥𝑯∥ ∞ ≤ 𝐵, we have tr(𝑯𝝆 (𝑇) ) ≤ ∥𝑯∥ ∞ − 0.992 𝜖 2 42𝐵3 ≤ ∥𝑯∥ ∞ − 2.05𝐵 ≤ −1.05𝐵. 20𝐵2 𝜖 2 254 At the same time, because ∥ 𝝆 (𝑇) ∥ 1 = 1, tr(𝑯𝝆 (𝑇) ) ≥ −∥𝑯∥ ∞ ≥ −𝐵. This is a contradiction. Hence the algorithm must terminate early. ■ The polynomial-time quantum algorithm for establishing Theorem 4.9.1 is as follows. The algorithm runs quantum thermal gradient descent to find a local minimum 𝝆 (𝑇) of 𝑯 under thermal perturbations. Recall that 𝐵 is the upper bound on ∥𝑯∥ ∞ , and is equal to poly(𝑛), and 1/𝜖 = poly(𝑛). Because every step can be done in polynomial time, and there are at most 𝑇 = 42𝐵3 /𝜖 2 = poly(𝑛) time steps, quantum thermal gradient descent runs in time polynomial in 𝑛. Now, given any observable 𝑶. The quantum algorithm prepares O (1/𝜖 2 ) = poly(𝑛) copies of 𝝆 (𝑇) in poly(𝑛) time, then measures 𝑶 on the O (1/𝜖 2 ) copies of 𝝆 (𝑇) to estimate tr(𝑶 𝝆 (𝑇) ) to 𝜖 error. This concludes the proof of Theorem 4.9.1. 4.12 Appendix:Characterizing energy gradients in low-temperature heat bath Recall from Appendix 4.8 on certifying Hamiltonians without suboptimal local minima, if there exists 𝜶 ∈ R𝑚 ≥0 with ∥𝜶∥ 1 = 1, such that the negative gradient condition holds, ∑︁ 2𝜖 †𝛽,𝜏,𝑯 ( 𝑰 − 𝑷𝐺 (𝑯)) − 𝜖 𝑰, − 𝛼𝑎 L 𝑎 [𝑯] ⪰ 𝛿 𝑎 then any 𝜖-approximate local minimum 𝝆 of the 𝑛-qubit Hamiltonian 𝑯 under thermal perturbations is an exact global minimum of 𝑯 with failure probability ≤ 𝛿, i.e., tr(𝑷𝐺 (𝑯) 𝝆) ≥ 1 − 𝛿, where 𝑷𝐺 (𝑯) is the projection onto the ground state space. To understand when the above condition holds, it is imperative to characterize the †𝛽,𝜏,𝑯 energy gradients, L𝑎 [𝑯]. In this appendix, we present various lemmas and theorems characterizing the energy gradients, which will be used in our proof of Theorem 4.9.2 in section 4.14 for showing that a certain family of Hamiltonians has no suboptimal local minima. We remark that the proofs of many formal statements in this appendix require concepts and results that won’t be shown till later in Appendices 4.15 and 4.16, and we recommend the first-time reader to freely skip the proofs and return later. For simplicity, we will focus on the nonnegative vector 𝜶 being uniform over a subset 𝑆 for the remaining appendices. We will show that this is sufficient for our purposes 255 even though having the ability to choose 𝜶 is more powerful. We define the following Lindbladian with uniform weights over a subset 𝑆 ⊆ {1, . . . , 𝑚}: ∑︁ 𝛽,𝜏,𝑯 L := L𝑎 . (4.30) 𝑎∈𝑆 †𝛽,𝜏,𝑯 Recall from Appendix 4.10 that each L𝑎 corresponds to a jump operator 𝑨𝑎 satisfying the normalization condition ∥ 𝑨𝑎 ∥ ∞ ≤ 1. If we let 𝑟 := 2𝑚𝜖/𝛿, 𝜖 ′ = 𝑚𝜖 and 𝑆 = {1, . . . , 𝑚}, then the negative gradient condition becomes (negative gradient condition) : −L † [𝑯] ⪰ 𝑟 ( 𝑰 − 𝑷𝐺 ) − 𝜖 ′ 𝑰. (4.31) This will be the central inequality we would like to establish for the remaining appendices. Throughout the proofs, we will consider different subsets 𝑆 and show a relation similar to Eq. (4.31) for subset 𝑆. Basic properties of energy gradients in low-temperature bath We show a few basic properties of energy gradients under a low-temperature, long-time-scale bath. First, we show that the energy gradient, at large 𝛽 (i.e., low temperatures), is negative semi-definite up to controllable error. Intuitively, this can be seen from the KMS condition in Eq. (4.19), 𝛾 𝛽 (𝜔) = 𝛾 𝛽 (−𝜔)e−𝛽𝜔 : the heating transition is suppressed by the Boltzmann weight, allowing energy to increase by 𝜔 ∼ 𝛽−1 . Another source of error is the uncertainty in energy 𝜏 −1 . Lemma 4.12.1 (Almost negative gradients). Consider the thermal Lindbladian Í 𝛽,𝜏,𝑯 L = 𝑎∈𝑆 L𝑎 with jump operators { 𝑨𝑎 } 𝑎∈𝑆 where ∥ 𝑨𝑎 ∥ ≤ 1, and 𝛾 𝛽 (𝜔) satisfying Eq. (4.19). Then,    ∥𝑯∥ 3/4 1 1 † L [𝑯] ⪯ O |𝑆| + + · 𝑰. 𝜏 𝛽 𝜏 1/4 Proof. Rewrite the energy gradient with an error controlled by Proposition 4.10.3 and Lemma 4.16.1 gives ∑︁ ∫ ∞ † L [𝑯] ≈ 𝛾 𝛽 (𝜔)𝜔 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔)d𝜔 𝑎∈𝑆 = −∞ ∑︁ ∫ ∞ 𝑎∈𝑆 0 ˆ𝑎 † ˆ𝑎 𝛾 𝛽 (𝜔)𝜔 𝑨 (𝜔) 𝑨 (𝜔)d𝜔 + ∑︁ ∫ 0 𝑎∈𝑆 −∞ 𝛾 𝛽 (𝜔)𝜔 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔)d𝜔 256 and bound the positive operator ∑︁ ∫ ∞ |𝑆| ∥ 𝛾 𝛽 (𝜔)𝜔 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔)d𝜔∥ ∞ ≤ |𝑆| max 𝛾 𝛽 (𝜔)𝜔 ≤ . 𝜔≥0 𝛽 0 𝑎∈𝑆 The second inequality used the tail bound in Eq. (4.20) with Δ = 0. ■ Second, we show that the energy gradient operator is nearly diagonal in the energy basis. The intuition is that for any operator 𝑨, the product ˆ 𝑨ˆ † (𝜔) 𝑨(𝜔) is nearly diagonal in the energy basis for large 𝜏. Lemma 4.12.2 (Energy gradient is almost diagonal). In the setting of Lemma 4.12.1, assume that for any two well-isolated energy eigensubspaces 𝑷1 and 𝑷2 such that the two sets of eigenvalues have at least distance 𝛿. Then,    ∥𝑯∥ 3/4 1 ∥𝜃 𝛽 ∥ ∞ † ∥𝑷1 L [𝑯] 𝑷2 ∥ ≤ O |𝑆| , + + √ 𝜏 𝜏 1/4 𝛿𝜏 where 𝜃 𝛽 (𝜔) := 𝛾 𝛽 (𝜔)𝜔. Proof. Formally, approximate the energy gradient by dropping the Lamb-shift term (Proposition 4.10.3) and applying Lemma 4.16.1, ∑︁ ∫ ∞ † L [𝑯] ≈ 𝛾 𝛽 (𝜔)𝜔 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔)d𝜔. 𝑎∈𝑆 −∞ We then apply the secular approximation for 𝜇 = 𝛿/2 (Corollary 4.15.3) such that the transition amplitudes vanishes between the subspaces 𝑷1 𝑺ˆ 𝑎𝜇 (𝜔) † 𝑺ˆ 𝑎𝜇 (𝜔)𝑷2 = 0 for each 𝜔 ∈ R and 𝑎 ∈ 𝐴. Combining the errors in each of the approximations leads to the claimed result. ■ Next, we show that the finite-𝜏 Lindbladian can be approximated by the infinite-𝜏 version under certain conditions. The latter, known as the Davies’ generator [57], has a simpler form that is more amenable for analysis in some situations. Lemma 4.12.3 (Recovering Davies’ generator). Consider the dissipative part of the 𝛽,𝜏,𝑯 thermal Lindbladian D𝑎 with the jump operators 𝑨𝑎 where ∥ 𝑨𝑎 ∥ ≤ 1, and any 𝛾 𝛽 such that ∥𝛾 𝛽 ∥ ∞ ≤ 1. Suppose the Bohr-frequency gap is Δ𝜈 (𝑯), then ! ∫ ∞ 1 2 †𝛽,𝜏,𝑯 †𝛽,∞,𝑯 ∥D𝑎 − D𝑎 ∥ ∞−∞ ≤ O max 𝛾 𝛽 (𝜈) − 𝛾 𝛽 (𝜔) 𝑓ˆ(𝜔 − 𝜈) d𝜔 + √︁ . 𝜈 −∞ Δ𝜈 (𝑯)𝜏 257 †𝛽,𝜏,𝑯 Therefore, the Bohr-frequency gap sets a timescale ∼ Δ−1 𝜈 such that the map D𝑎 stabilized. †𝛽,𝜏,𝑯 Proof. It suffices to consider D𝑎 [𝑶] acting on arbitrary operator 𝑶 such that ∥𝑶 ∥ = 1: ∫ ∞ i h 1 ˆ𝑎 †𝛽,𝜏,𝑯 † ˆ𝑎 𝑎 † 𝑎 ˆ ˆ D𝑎 [𝑶] = 𝛾 𝛽 (𝜔) 𝑨 (𝜔) 𝑶 𝑨 (𝜔) − { 𝑨 (𝜔) 𝑨 (𝜔), 𝑶} d𝜔 2 ∫−∞∞ h i 𝐸1 1 ≈ 𝛾 𝛽 (𝜔) 𝑺ˆ 𝑎 (𝜔) † 𝑶 𝑺ˆ 𝑎 (𝜔) − { 𝑺ˆ 𝑎 (𝜔) † 𝑺ˆ 𝑎 (𝜔), 𝑶} d𝜔 2 −∞ ∫ ∞ ∑︁  1 𝑎† 𝑎† 𝑎 𝑎 𝑨𝜈′ 𝑶 𝑨𝜈 − { 𝑨𝜈′ 𝑨𝜈 , 𝑶} = 𝛾 𝛽 (𝜔) 𝑓ˆ𝜇∗ (𝜔 − 𝜈′) 𝑓ˆ𝜇 (𝜔 − 𝜈)d𝜔 2 −∞ 𝜈,𝜈 ′ ∈𝐵(𝑯) ∫  ∞ ∑︁  1 2 = 𝛾 𝛽 (𝜔) 𝑓ˆ𝜇 (𝜔 − 𝜈) d𝜔 𝑨𝜈𝑎† 𝑶 𝑨𝜈𝑎 − { 𝑨𝜈𝑎† 𝑨𝜈𝑎 , 𝑶} 2 −∞ 𝜈∈𝐵(𝑯)   𝐸 2 ∑︁ 1 ≈ 𝑨𝜈𝑎† 𝑶 𝑨𝜈𝑎 − { 𝑨𝜈𝑎† 𝑨𝜈𝑎 , 𝑶} 𝛾 𝛽 (𝜈). 2 𝜈∈𝐵(𝑯) The approximation errors are bounded by   1 𝐸 1 ≤ 2∥𝑶 ∥∥𝛾 𝛽 ∥ ∞ ∥ 𝑨 ∥∥ 𝑓𝜏 · (1 − 𝑠ˆ 𝜇 )∥ 2 ∥ 𝑓𝜏 ∥ 2 = O √ , 𝜇𝜏 ∫ ∞ 2 𝑎 𝐸 2 ≤ 2∥𝑶 ∥∥ 𝑨 ∥ max 𝛾 𝛽 (𝜈) − 𝛾 𝛽 (𝜔) 𝑓ˆ𝜇 (𝜔 − 𝜈) d𝜔 𝜈 −∞   ∫ ∞ ∥𝛾 𝛽 ∥ ∞ 2 . ≤ O max 𝛾 𝛽 (𝜈) − 𝛾 𝛽 (𝜔) 𝑓ˆ(𝜔 − 𝜈) d𝜔 + 𝜈 𝜇𝜏 −∞ Combine the error bounds to conclude the proof. Note that since the bound becomes √ √ vacuous at 𝜇𝜏 = Ω(1), we have that O (1/ 𝜇𝜏 + 1/(𝜇𝜏)) = O (1/ 𝜇𝜏). ■ 𝑎 Lemma 4.12.4. For 𝛾 𝛽 (𝜔) defined in Eq. (4.21) with Λ0 = Θ(1), we have that   ∫ ∞ 1 + 𝛽 2 max 𝛾 𝛽 (𝜈) − 𝛾 𝛽 (𝜔) 𝑓ˆ(𝜔 − 𝜈) d𝜔 ≤ O ln 𝜏 . 𝜈∈R 𝜏 −∞ Proof. Recall the integration-by-part trick for expectation integral ∫ ∞ ∫ ∞ 𝑥=∞ 𝑓 (𝑥) 𝑝(𝑥)d𝑥 = −[ 𝑓 (𝑥)𝑃(𝑥)] 𝑥=0 + 𝑓 ′ (𝑥)𝑃(𝑥)d𝑥 where 0 ∫ ∞ 𝑃(𝑥) := 0 𝑝(𝑦)d𝑦. 𝑥 Then, ∫ ∞ ∫ ∞ ∫ ∞ 𝛾 𝛽 (𝜈 + 𝑥) + 𝛾 𝛽 (𝜈 − 𝑥) 2 2 2 𝛾 𝛽 (𝜔) 𝑓ˆ(𝜔 − 𝜈) d𝜔 = 𝛾 𝛽 (𝜈 + 𝑥) 𝑓ˆ(𝑥) d𝑥 = 2 𝑓ˆ(𝑥) d𝑥 2 −∞ −∞ 0 | {z } =:𝑝(𝑥) ∫ ∞ 𝛾 ′ (𝜈 + 𝑥) + 𝛾 ′ (𝑥 − 𝜈) 𝛽 = 𝛾 𝛽 (𝜈) + 0 𝛽 2 𝑃(𝑥)d𝑥, 258 where in the last line we applied the integration-by-part and used 𝛾(±∞) = 0 and 𝑃(0) = 1. The error term can be bounded as follows: ∫ ∞ 𝛾 ′ (𝜈 + 𝑥) + 𝛾 ′ (𝑥 − 𝜈) 𝛽 𝛽 𝑃(𝑥)d𝑥 2 0 ∫ 1/𝜏 𝛾 ′ (𝜈 + 𝑥) + 𝛾 ′ (𝜈 − 𝑥) ∫ ∞ 𝛾 ′ (𝜈 + 𝑥) + 𝛾 ′ (𝜈 − 𝑥) 𝛽 𝛽 𝛽 𝛽 = 𝑃(𝑥)d𝑥 + 𝑃(𝑥)d𝑥 2 2 0 1/𝜏 1+𝛽 1+𝛽 ) + O( ln 𝜏). 𝜏 𝜏 In the last line, we control the first term by 𝑃(𝑥) ≤ 1 and noting by the product rule we have ! !! 2 2 2 2 2 2 e−𝜔 /2Λ0 𝜔/Λ20 e−𝜔 /2Λ0 𝛽e 𝛽𝜔 d e−𝜔 /2Λ0 ′ 𝛾 𝛽 (𝜔) = O =O − − d𝜔 1 + e 𝛽𝜔 1 + e 𝛽𝜔 (1 + e 𝛽𝜔 ) 2 = O( ≤ O( 1 + 𝛽). Λ0 4 The second term uses the tail bound 𝑃(𝑥) ≤ 𝜋𝑥𝜏 from Eq. (4.93) and that 𝛾 ′𝛽 (𝜈 ± 𝑥) are each rapidly decaying outside an 𝑥 ∈ [∓𝜈 − Λ0 , ∓𝜈 + Λ0 ] window so that the ∫Λ integral over 1𝑥 d𝑥 only contributes at most O ( 1/𝜏0 1𝑥 d𝑥) = O (log(𝜏Λ0 )). ■ Relating subspace and local gradients to global gradients As a method of proof, we will often analyze a Lindbladian by its constituents, and here we present a few useful relations. First, when studying gradients, the gradient acting on a subspace is often conceptually simpler. The following lemma relates the energy gradient in a subspace and the full energy gradient. This is a direct consequence of Lemma 4.12.1 and Lemma 4.12.2 above. Lemma 4.12.5 (Subspace gradient and global gradient). In the setting of Lemma 4.12.2, suppose 𝑷 projects onto a set of eigenstates of 𝑯 separated by the rest by a gap of at least 𝛿. Then,    ∥𝑯∥ 3/4 1 1 ∥𝜃 𝛽 ∥ ∞ † † −L [𝑯] ⪰ −𝑷L [𝑯] 𝑷 − O |𝑆| + + + √ · 𝑰. 𝛽 𝜏 𝜏 1/4 𝛿𝜏 Proof. Let 𝑳 = L † [𝑯]. We have, 𝑳 = 𝑷𝑳𝑷 + 𝑷⊥ 𝑳𝑷⊥ + 𝑷𝑳𝑷⊥ + 𝑷⊥ 𝑳𝑷. Using Lemma 4.12.1 establishing the almost negativity of the energy gradient,    ∥𝑯∥ 3/4 1 1 −𝑳 ⪰ −O |𝑆| + + 𝑰, 𝜏 𝛽 𝜏 1/4 259 we have  ∥𝑯∥ 3/4 1 1 −𝑷 𝑳𝑷 ⪰ −O |𝑆| + + 𝑷⊥ . 1/4 𝜏 𝛽 𝜏    ∥𝑯∥ 3/4 1 ∥𝜃 𝛽 ∥ ∞ ⊥ ⊥ ∥𝑷𝑳𝑷 + 𝑷 𝑳𝑷∥ ≤ O |𝑆| . + + √ 𝜏 𝜏 1/4 𝛿𝜏 ⊥  ⊥  (Lemma 4.12.1) (Lemma 4.12.2) Putting the bounds together yields the advertised result. ■ Next, we provide a lemma that gives a simplified expression of the energy gradient operator when restricted to a subspace of low-energy eigenstates. Lemma 4.12.6 (Gradient in a subspace). In the setting of Lemma 4.12.1, suppose 𝑯 has a subspace of low-energy eigenstates with corresponding projector 𝑸 that is separated from the higher energy eigenstates by an excitation gap Δ𝑸 . Let Δ𝜈 = min𝜈1 ≠𝜈2 ∈𝐵(𝑯|𝑸 ) |𝜈1 − 𝜈2 | be the Bohr-frequency gap of 𝑯 restricted to the subspace. Assuming Δ𝜈 /2 < Δ𝑸 , then the energy gradient operator in the subspace can be approximated using † 𝑸L [𝑯]𝑸 − ∑︁ ∑︁ 𝑎∈𝑆 𝜈∈𝐵(𝑯|𝑸 ) ∫ 0 𝑸 𝑨𝜈𝑎† 𝑸 𝑨𝜈𝑎 𝑸 𝛾 𝛽 (𝜔)𝜔| 𝑓ˆ𝜇 (𝜔 − 𝜈)| 2 d𝜔 ≤ 𝜖 −∞ where 𝜇 = Δ𝜈 /2 and  ∥𝑯∥ 3/4 1 1 ∥𝜔𝛾 𝛽 (𝜔)∥ ∞ + + + . 𝜖 ≤ |𝑆|O √ 𝜏 𝛽 𝜏 1/4 Δ𝜈 𝜏  Proof. We invoke a series of approximations to rewrite in terms of the exact Bohr frequencies on the subspace 𝑷I . 𝐸1 L † [𝑯] ≈ D † [𝑯] ∫ ∞ 𝐸 2 ∑︁ ≈ 𝛾 𝛽 (𝜔)𝜔 𝑨𝑎 (𝜔) † 𝑨𝑎 (𝜔)d𝜔 𝑎∈𝑆 𝐸3 ≈ 𝑎∈𝑆 𝐸4 ≈ = −∞ ∑︁ ∫ 0 𝛾 𝛽 (𝜔)𝜔 𝑨𝑎 (𝜔) † 𝑨𝑎 (𝜔)d𝜔 −∞ ∑︁ ∫ 0 𝛾 𝛽 (𝜔)𝜔𝑺 𝑎 (𝜔) † 𝑺 𝑎 (𝜔)d𝜔 𝑎∈𝑆 −∞ ∑︁ ∑︁ 𝑎∈𝑆 𝜈 ′ ,𝜈∈𝐵(𝑯) ∫ 0 −∞ ∗ 𝛾 𝛽 (𝜔)𝜔 𝑨𝜈𝑎†′ 𝑨𝜈𝑎 𝑓ˆ𝜇 (𝜔 − 𝜈′) 𝑓ˆ𝜇 (𝜔 − 𝜈)d𝜔 =: 𝑿. (4.44) 260 The errors are 𝐸 1 = O (|𝑆|∥𝑯∥ 3/4 /𝜏 1/4 ), 𝐸 2 = O (|𝑆|/𝜏), 𝐸 3 = O (|𝑆|/𝛽), and √ 𝐸 4 = O (|𝑆| × ∥𝜔𝛾 𝛽 (𝜔)∥ ∞ / 𝜇𝜏). In particular, 𝐸 3 arises from dropping the positive integral range with error bounded by max𝜔≥0 𝜔𝛾 𝛽 (𝜔) ≤ 1/𝛽. In other words, our choice of 𝛾 𝛽 (𝜔) allows the energy to increase by at most O (1/𝛽). Sandwiching Eq. (4.12) with 𝑸 further simplifies the expression as it restricts to transitions in the subspace. Specifically, we have 𝑸 𝑿𝑸 = ∑︁ ∑︁ 𝑎∈𝑆 𝜈 ′ ,𝜈∈𝐵(𝑯) = ∑︁ ∑︁ 𝑎∈𝑆 𝜈∈𝐵(𝑯|𝑸 ) ∫ 0 ∗ 𝛾 𝛽 (𝜔)𝜔𝑸 𝑨𝜈𝑎†′ 𝑸 𝑨𝜈𝑎 𝑸 𝑓ˆ𝜇 (𝜔 − 𝜈′) 𝑓ˆ𝜇 (𝜔 − 𝜈)d𝜔 −∞ ∫ 0 2 𝛾 𝛽 (𝜔)𝜔𝑸 𝑨𝜈𝑎† 𝑸 𝑨𝜈𝑎 𝑸 𝑓ˆ𝜇 (𝜔 − 𝜈) d𝜔. −∞ The first line inserts an additional projector 𝑸 between 𝑨𝜈𝑎†′ and 𝑨𝜈𝑎 because any transition to excited states require 𝜈, 𝜈′ > Δ𝑸 , but this is forbidden by the restrictions that 𝜔 ≤ 0 (from the integral) and that |𝜈 − 𝜔|, |𝜈′ − 𝜔| < 𝜇 < Δ𝑸 (from the secular approximation). In the second line, since the Bohr frequencies in 𝐵(𝑯| 𝑸 ) are at least Δ𝜈 = 2𝜇 apart, we must have that 𝑓ˆ𝜇∗ (𝜔 − 𝜈′) 𝑓ˆ𝜇 (𝜔 − 𝜈) = 0 for all 𝜔 ∈ R, unless Combining the above with Eq. (4.12) to conclude the proof. 𝜈′ = 𝜈. ■ When the Hamiltonian is local, thinking about the gradient “locally” is sometimes useful. The following lemma gives a sufficient condition that guarantees a global gradient. Since the consequence is strong, the premise is also more stringent; it is only helpful when the Hamiltonian is frustration-free. Í Lemma 4.12.7 (Local-to-global gradient condition). Suppose 𝑯 = 𝑖 𝒉𝑖 , where each term 𝒉𝑖 ⪰ 0. Then for any (not necessarily thermal) Lindbladian L, −L † [𝒉𝑖 ] ⪰ 𝑟𝑖 𝒉𝑖 =⇒ −L † [𝑯] ⪰ 𝑟 𝑯, where 𝑟 = min𝑖 𝑟𝑖 . Í Í Proof. By linearity, we have −L † [𝑯] = 𝑖 −L † [𝒉𝑖 ] ⪰ 𝑖 𝑟𝑖 𝒉𝑖 . Since 𝑟𝑖 𝒉𝑖 ⪰ 𝑟 𝒉𝑖 , Í we have −L † [𝑯] ⪰ 𝑟 𝑖 𝒉𝑖 = 𝑟 𝑯, concluding the proof. ■ 261 Gradients for commuting Hamiltonians When we are given a commuting Hamiltonian, the energy gradient induced by any local jump operator can be understood by restricting the system to its neighborhood. In this situation, the negative gradient condition for the overall Hamiltonian can be decomposed into conditions that can be checked locally. This gives an efficient method to show a commuting Hamiltonian has a negative gradient for all its excited states, which we elucidate in this section of the appendix. Recall the thermal Lindbladian L := L 𝛽,𝜏,𝑯 defined in Eq. (4.18) for a local jump operator 𝑨𝑎 , whose Heisenberg picture is †𝛽,𝜏,𝑯 L𝑎 𝛽,𝜏,𝑯 †𝛽,𝜏,𝑯 [𝑶] = i[𝑯 𝐿𝑆,𝑎 , 𝑶] + D𝑎 [𝑶], where †𝛽,𝜏,𝑯 D𝑎 [𝑶] = ∫ ∞ h i 1 𝛾 𝛽 (𝜔) 𝑨ˆ 𝑎 (𝜔) † 𝑶 𝑨ˆ 𝑎 (𝜔) − { 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔), 𝑶} d𝜔. 2 −∞ Note 𝑨ˆ 𝑎 (𝜔) is the operator Fourier transform of 𝑨𝑎 (𝑡) = ei𝑯𝑡 𝑨𝑎 e−i𝑯𝑡 , and 𝑯 𝐿𝑆,𝑎 is a Lamb-shift term defined in Eq. (4.22). 𝛽,𝜏,𝑯 When 𝑯 is a commuting Hamiltonian (e.g., [31, 96]), an important observation is that 𝑨𝑎 (𝑡) only depends on the part of 𝑯 that does not commute with 𝑨𝑎 . In particular, the energy gradient for each jump operator only depends on the neighborhood of 𝑨𝑎 . Lemma 4.12.8 (Commuting Hamiltonian and localized Lindblad operators). Suppose Í 𝑯 = 𝑒 𝒉 𝑒 is a commuting Hamiltonian. For any jump operator 𝑨𝑎 , the associated energy gradient simplifies to †𝛽,𝜏,𝑯 L𝑎 where 𝑯∋𝑎 = Í †𝛽,𝜏,𝑯 ∋𝑎 [𝑯] = L𝑎 [𝑯∋𝑎 ], 𝑎 𝑒: [𝒉𝑒 , 𝑨 𝑎 ]≠0 𝒉 𝑒 is the part of 𝑯 does not commute with 𝑨 . Proof. When 𝑯 is commuting, we have 𝑨𝑎 (𝑡) = ei𝑯𝑡 𝑨𝑎 e−i𝑯𝑡 = ei𝑯 ∋𝑎 𝑡 𝑨𝑎 e−i𝑯 ∋𝑎 𝑡 , so †𝛽,𝜏,𝑯 †𝛽,𝜏,𝑯 ∋𝑎 the Lindbladian superoperator only depends on 𝑯∋𝑎 , i.e., L𝑎 = L𝑎 . Let 𝑯∌𝑎 = 𝑯 − 𝑯∋𝑎 be the part of 𝑯 that commutes with 𝑨𝑎 . Since [𝑯∌𝑎 , 𝑯∋𝑎 ] = 0, 𝛽,𝜏,𝑯 we have [ 𝑨𝑎 (𝑡), 𝑯∌𝑎 ] = 0 for each 𝑡, which implies [𝑯 𝐿𝑆,𝑎 , 𝑯∌𝑎 ] = [ 𝑨ˆ 𝑎 (𝜔), 𝑯∌𝑎 ] = †𝛽,𝜏,𝑯 0. Thus we have L𝑎 [𝑯∌𝑎 ] = 0. ■ 262 Negative gradient condition under perturbations to Hamiltonians We next look at how the negative energy gradient condition changes under perturbations to the 𝑛-qubit Hamiltonian 𝑯. See Appendix 4.16 for the proof of the following theorem. Theorem 4.12.1 (Monotonicity of gradient under level splitting). Consider a Í highly degenerate Hamiltonian 𝑯 = 𝐸¯ 𝐸¯ 𝑷 𝐸¯ with Bohr-frequency gap Δ𝜈 := min𝜈1 ≠𝜈2 ∈𝐵(𝑯) |𝜈1 − 𝜈2 | of 𝑯, and add a perturbation 𝑯′ := 𝑯 + 𝑽. Let 𝑷 = 𝑷 𝐸¯ be a projector to an energy subspace and 𝑷′ the corresponding perturbed subspace. Suppose the perturbation is weaker than the Bohr-frequency gap, ∥𝑽 ∥ ≤ 18 Δ𝜈 . For Í Í 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 ′ any 𝛽, 𝜏 > 0, let L = 𝑎∈𝑆 L𝑎 , L ′ = 𝑎∈𝑆 L𝑎 be thermal Lindbladians with jumps { 𝑨𝑎 } 𝑎∈𝑆 , where ∥ 𝑨𝑎 ∥ ≤ 1 and the transition weight 𝛾 𝛽 (𝜔) is given by Eq. (4.21). Then we have the monotone property that −L † [𝑯] ⪰ 𝑟 ( 𝑰 − 𝑷) − 𝜖 𝑰 implies ′ − L † [𝑯′] ⪰ 𝑟 ( 𝑰 − 𝑷′) − 𝜖 ′ 𝑰, where ! 2/3 3/4 −𝛽Δ𝜈 /4 Λ
∥𝑯∥ Λ + 𝑟 1 Λ e 0 0 0 + 1+ ∥𝑽 ∥(4.48) 𝜖 ′ ≤ 𝜖 + |𝑆| · O + + 1/3 +√ + 𝜏 𝛽 Δ𝜈 𝜏 1/4 𝜏 Δ𝜈 𝜏 Finally, we look at how the negative energy gradient condition changes when restricted to a subspace. See Appendix 4.16 for proofs of the following two corollaries. Corollary 4.12.1 (Monotonicity of gradient on a subspace). Consider a Hamiltonian Í 𝑯 = 𝐸¯ 𝐸¯ 𝑷 𝐸¯ and its perturbation 𝑯′ := 𝑯 + 𝑽. Let 𝑷 be the ground space projector for 𝑯 and 𝑷′ be the corresponding perturbed eigensubspace of 𝑯′. Let 𝑸 be a lowÍ energy eigensubspace projector of 𝑯 (i.e., 𝑸 = 𝐸 ≤𝐸𝑸 𝑷 𝐸 for 𝐸 𝑸 ∈ Spec(𝑯)) with 1 excitation gap Δ𝑸 . Assume ∥𝑽Δ∥ ∥𝑯∥ ≤ 144 Δ𝜈 where Δ𝜈 := min𝜈1 ≠𝜈2 ∈𝐵(𝑯|𝑸 ) |𝜈1 − 𝜈2 | 𝑸 is the Bohr-frequency gap of 𝑯 within the subspace 𝑸. For any 𝛽, 𝜏 > 0, let Í Í 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 ′ L = 𝑎∈𝑆 L𝑎 , L ′ = 𝑎∈𝑆 L𝑎 be thermal Lindbladians with jumps { 𝑨𝑎 } 𝑎∈𝑆 , where ∥ 𝑨𝑎 ∥ ≤ 1 and the transition weight 𝛾 𝛽 (𝜔) is given by Eq. (4.21). Then we have the monotone property that −𝑸L † [𝑯]𝑸 ⪰ 𝑟𝑸( 𝑰 − 𝑷) − 𝜖 𝑰 implies − 𝑸 ′ L ′† [𝑯′]𝑸 ′ ⪰ 𝑟𝑸 ′ ( 𝑰 − 𝑷′) − 𝜖 ′ 𝑰, 263 where 𝑸 ′ projects onto the perturbed eigensubspace of 𝑯′ identified with 𝑸, and 2/3 1 ∥𝑯∥ 3/4 Λ0 Λ0 Λ0 e−𝛽Δ𝜈 /4 e−𝛽Δ𝑸 /4 𝜖 ≤ 𝜖 + |𝑆| · O + + + + + + √ √︁ 𝜏 𝛽 𝛽 𝜏 1/4 𝜏 1/3 Δ𝜈 𝜏 Δ𝑸 𝜏     ∥𝑽 ∥ ∥𝑽 ∥  Λ0 ∥𝑽 ∥∥𝑯∥ + 1+ +𝑟 + . (4.49) Δ𝜈 Δ𝑸 Δ𝑸 Δ𝜈  ′ Corollary 4.12.2 (Monotonicity of gradient on a subspace under off-block-diagonal ∥ ∥𝑽 ∥ perturbation). In the setting of Corollary 4.12.1, instead assume ∥𝑽 Δ𝜈 , Δ𝑸 ≤ (𝑐𝑜𝑛𝑠𝑡.), and that the perturbation is off-block-diagonal, i.e., 𝑸𝑽𝑸 = ( 𝑰 − 𝑸)𝑽 ( 𝑰 − 𝑸) = 0. Then, −𝑸L † [𝑯]𝑸 ⪰ 𝑟𝑸( 𝑰 − 𝑷) − 𝜖 𝑰 implies − 𝑸 ′ L ′† [𝑯′]𝑸 ′ ⪰ 𝑟𝑸 ′ ( 𝑰 − 𝑷′) − 𝜖 ′ 𝑰, where 2/3 1 ∥𝑯∥ 3/4 Λ0 Λ0 e−𝛽Δ𝜈 /4 e−𝛽Δ𝑸 /4 Λ0 + + + 𝜖 ≤ 𝜖 + |𝑆| · O + + + √ √︁ 𝜏 𝛽 𝛽 𝜏 1/4 𝜏 1/3 Δ𝜈 𝜏 Δ𝑸 𝜏  ∥𝑯 ∥∥𝑽 ∥ ∥𝑽 ∥ 2   ∥𝑽 ∥ ∥𝑽 ∥ 2   ∥𝑽 ∥ 2 𝑸 + + ∥𝑯𝑸 ∥ · + +𝑟 + . Δ𝑸 Δ𝑸 Δ 𝜈 Δ𝑸 Δ 𝜈 Δ𝑸 Δ𝑸 Δ 𝜈 ′  264 4.13 Appendix:Energy landscape of an Ising chain In this appendix, we take a brief aside to characterize the energy landscape of the one-dimensional ferromagnetic Ising chain under thermal perturbations. This provides a basic example on how the definition of local minima under thermal perturbations is related to the physical picture. We will see that this system has many suboptimal local minima in the absence of an external field with a lifetime polynomial in the system size. Once an external field is added, however, the system essentially has no suboptimal local minima and can quickly cool to the ground state where all spins are aligned. This observation corresponds to the following physical phenomena: when there is no external magnetic field, a ferromagnetic system will often be stuck in a configuration with many domain walls, and an externally applied magnetic field can quickly magnetize the system. The Hamiltonian for the ferromagnetic Ising chain on a periodic boundary condition is 𝑛 𝑛 ∑︁ ∑︁ 𝑯=− 𝒁 𝑗 𝒁 𝑗+1 − ℎ 𝒁𝑗, 𝑗=1 𝑗=1 where we identify 𝒁𝑛+1 ≡ 𝒁1 . Intuitively, this system energetically favors configurations where adjacent spins are aligned. When ℎ = 0, we have two degenerate ground states, |00 · · · 0⟩ and |11 · · · 1⟩, which are the global minima. This degeneracy is broken when ℎ ≠ 0, and these two states split by energy 2𝑛ℎ. The system also has many excited states with domain walls, i.e., locations where adjacent spins are anti-aligned such as |01⟩ and |10⟩. In what follows, we study the energy landscape of the above system under thermal perturbations with jump operator { 𝑨 𝑗 = 𝑿 𝑗 }𝑛𝑗=1 , setting 𝜏 = ∞ for simplicity. We analyze three cases. Case 1: no external field (ℎ = 0). In this case, we will see that any bit string state with domain walls sufficiently far from each other, e.g. |· · · 0001111000 · · · ⟩ is a suboptimal local minimum. Indeed, there is no local operation to strictly decrease the energy of such states; the jump operators {𝑿 𝑗 } can only displace the domain walls by one site, which does not change the energy. We can see this more formally by computing the energy gradient operator. Since 𝑯 is a commuting Hamiltonian, we may apply Lemma 4.12.8 and study the gradient induced by a single jump operator 𝑿 𝑗 by restricting the Hamiltonian to its neighborhood, i.e., 𝑯∋ 𝑗 = −𝒁 𝑗−1 𝒁 𝑗 − 𝒁 𝑗 𝒁 𝑗+1 . 265 Observe 𝑯∋ 𝑗 has three degenerate eigenspaces 𝑷 𝐸𝑗 with energy 𝐸 as follows: ∑︁ ∑︁ −2 2 𝑷𝑗 = |𝒔⟩⟨𝒔| 𝑗−1, 𝑗, 𝑗+1 , 𝑷𝑗 = |𝒔⟩⟨𝒔| 𝑗−1, 𝑗, 𝑗+1 , 𝒔∈{010,101} 𝒔∈{000,111} ∑︁ 𝑷0𝑗 = and |𝒔⟩⟨𝒔| 𝑗−1, 𝑗, 𝑗+1 . 𝒔∈{001,100,011,110} Then the negative Bohr-frequencies and the associated jumps are 2 𝑨𝜈1 = 𝑷−2 𝑗 𝑿 𝑗 𝑷 𝑗 = (|000⟩⟨010| + |111⟩⟨101|) 𝑗−1, 𝑗, 𝑗+1 , 𝜈1 = −4, 𝑗 0 𝑨𝜈2 = 𝑷0𝑗 𝑿 𝑗 𝑷2𝑗 + 𝑷−2 𝑗 𝑿 𝑗 𝑷 𝑗 = 0, 𝜈2 = −2. 𝑗 Hence, the energy gradient operator associated with jump 𝑿 𝑗 is †𝛽,∞,𝑯 D𝑗 †𝛽,∞,𝑯 ∋ 𝑗 [𝑯] = D 𝑗 [𝑯∋ 𝑗 ] = Í 𝑗† 𝑗 𝜈∈𝐵(𝑯 ∋ 𝑗 ) 𝜈𝛾 𝛽 (𝜈) 𝑨𝜈 𝑨𝜈 = 𝜃 0 · (|010⟩⟨010| + |101⟩⟨101|) 𝑗−1, 𝑗, 𝑗+1 + O (𝑒 −4𝛽 ), where 𝜃 0 = −4𝛾 𝛽 (−4) = −Ω(1). As we can see, the energy gradient is essentially 0 when the domain walls are more than distance 1 apart, and only becomes significant when two domain walls are next to each other, as in |· · · 010 · · · ⟩ or |· · · 101 · · · ⟩. This implies the presence of exponentially many suboptimal local minima; for example, choose whether or not to have a domain wall every 2 sites. Despite the presence of many suboptimal local minima, we now argue that they have a lifetime polynomial in the system size 𝑛 when the system evolves under thermal perturbations. We may understand the dynamics of the system as a random walk of domain walls, and two domain walls annihilate each other when they meet. Since two domain walls at distance ℓ apart moving under diffusive dynamics take O (ℓ 2 ) time to meet, a suboptimal local minimum with 𝑘 domain walls decays to a lower energy state after approximately O (𝑛2 /𝑘 2 ) time. Case 2: weak external field (0 < ℎ < 2). In this case, the ground state of 𝑯 is uniquely |0𝑛 ⟩, as all spins are slightly favored to be in the |0⟩ state instead of the |1⟩ state. When the domain walls are far apart, e.g. |· · · 0001111000 · · · ⟩, the applied external field causes an attraction across the domain of 1’s, which energetically favors the domain walls to move closer together. The presence of the field ℎ removes all the suboptimal local minima that were in the previous case. The state |1𝑛 ⟩, which was a ground state in the previous case, becomes now the only suboptimal local minimum. We now more formally characterize the energy landscape of 𝑯 using the energy gradient operator. Again applying Lemma 4.12.8, we may consider the gradient 266 induced by a single jump operator by focusing on its neighborhood. The relevant neighborhood Hamiltonian is 𝑯∋ 𝑗 = −𝒁 𝑗−1 𝒁 𝑗 − 𝒁 𝑗 𝒁 𝑗+1 − ℎ𝒁 𝑗 . Turning the crank, we see that the negative Bohr-frequencies and the associated jumps are 𝑗 𝜈1 = −4 − 2ℎ, 𝑗 𝜈2 = −4 + 2ℎ, 𝑗 𝜈3 = −2ℎ. 𝑨𝜈1 = |000⟩⟨010| 𝑗−1, 𝑗, 𝑗+1 , 𝑨𝜈2 = |111⟩⟨101| 𝑗−1, 𝑗, 𝑗+1 , 𝑨𝜈3 = (|001⟩⟨011| + |100⟩⟨110|) 𝑗−1, 𝑗, 𝑗+1 , (4.50) Then the energy gradient operator associated with jump 𝑿 𝑗 is †𝛽,∞,𝑯 D𝑗 [𝑯] = Í 𝑗† 𝑗 𝜈∈𝐵(𝑯 ∋ 𝑗 ) 𝜈𝛾 𝛽 (𝜈) 𝑨𝜈 𝑨𝜈 = (𝜃 1 |010⟩⟨010| + 𝜃 2 |101⟩⟨101| + 𝜃 3 |011⟩⟨011| + 𝜃 3 |110⟩⟨110|) 𝑗−1, 𝑗, 𝑗+1 + O (𝑒 −2𝛽ℎ ), where 𝜃 𝑗 = 𝜈 𝑗 𝛾 𝛽 (𝜈 𝑗 ). As we can see, any configuration with a domain wall now has a significant gradient from at least one of the jumps. The only configurations without a significant energy gradient are |0𝑛 ⟩, the ground state, and |1𝑛 ⟩, a metastable local minimum. Case 3: strong external field (ℎ > 2). In this case, the external field is sufficiently strong that the state |1𝑛 ⟩ is no longer a local minimum, and 𝑯 has no suboptimal local minima. To see this, we note that ℎ > 2 implies that 𝜈2 > 0 in Eq. (4.50), which 𝑗 𝑗† means the energetically favored jump operator is actually 𝑨−𝜈2 = 𝑨𝜈2 . This implies the energy gradient operator induced by the jump 𝑿 𝑗 in this case is †𝛽,∞,𝑯 D𝑗 [𝑯] = (𝜃 1 |010⟩⟨010| + 𝜃 ′2 |111⟩⟨111| + 𝜃 3 |011⟩⟨011| + 𝜃 3 |110⟩⟨110|) 𝑗−1, 𝑗, 𝑗+1 + O (𝑒 −2𝛽(ℎ−2) ), where 𝜃 ′2 = −𝜈2 𝛾 𝛽 (−𝜈2 ). This gives the state |1𝑛 ⟩ a significant energy gradient, and thus the ground state |0𝑛 ⟩ is the only local minimum of 𝑯. 267 depth to space circuit to Ham. UC <latexit 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HC <latexit 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<latexit 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T (C2 )⌦n system qubits <latexit 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SWAP (C2 )⌦T <latexit 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clock qubits Figure 4.3: The circuit-to-Hamiltonian mapping. For any 1D circuit, we can turn the circuit depth into an additional spatial dimension by introducing layers of swap gates; this particularly structured 2D circuit (now on 𝑛 qubits) is what we call 𝑼𝐶 . Then, we map the circuit 𝑼𝐶 to a 2D-Hamiltonian by introducing the clock qubits, which weave between the layers of system qubits in a zigzag order. This layout ensures that each clock qubit only controls nearby system qubits (in particular, the odd clock layers control only blue gates, and the even layers control only pink gates.) 4.14 Appendix:All local minima are global in BQP-hard Hamiltonians (Proof of Theorem 4.9.2) A main result of our work is that the task of finding a local minimum for 𝑯𝐶 under thermal perturbation is universal for quantum computation and hence classically hard. As we have seen in the main text and Appendix 4.9, this main result follows from Theorem 4.9.2, which we prove in this appendix. We start by defining 𝑯𝐶 in detail (visualized in Fig. 4.3). Given a 2D 𝑛-qubit circuit 𝑼𝐶 = 𝑼𝑇 · · · 𝑼2𝑼1 with 𝑇 = 2𝑡0 + 𝐿 = poly(𝑛) gates as constructed in Fig. 1 of Ref. [142], where the first and last 𝑡0 gates are identity gates and each gate of the 2D circuit 𝑼𝐶 is geometrically adjacent to the subsequent gate. We consider a geometrically local Hamiltonian on a 2D lattice with 𝑛 + 𝑇 qubits defined as follows. Definition 4.14.1 (Modified circuit-to-Hamiltonian construction). Consider a 2D circuit 𝑼𝐶 = 𝑼𝑇 · · · 𝑼2𝑼1 on 𝑛 qubits with 𝑇 = 2𝑡0 + 𝐿 gates, where the first and last 𝑡0 = 𝑐𝐿 2 gates are identity gates with 𝑐 = O (1), and each consecutive gates are geometrically adjacent. We define a geometrically-local Hamiltonian 𝑯𝐶 on a 2D lattice with 𝑛 + 𝑇 qubits as 268 follows: 𝑯𝐶 := 𝑯clock + 𝑯in + 𝑯prop acting on (C2 ) ⊗𝑛 ⊗ (C2 ) ⊗𝑇 , (4.51) where each individual term is given by 𝑯clock := 𝐽clock 𝑇−1 ∑︁ 𝑓𝑡 𝑰 ⊗ |01⟩⟨01| 𝑡,𝑡+1 , 𝑡=1 𝑯in := 𝐽in 𝑛 ∑︁ 𝑔 𝑗 |1⟩⟨1| 𝑗 ⊗ |10⟩⟨10| 𝑡 𝑗 −1,𝑡 𝑗 , 𝑗=1 𝑇 ∑︁ 1 𝑯prop (𝑡), 𝑯prop := 𝐽prop 2 𝑡=1 𝑯prop (1) := 𝑰 − ℎ1 (𝑼1 ⊗ |10⟩⟨00| 1,2 + 𝑼1† ⊗ |00⟩⟨10| 1,2 ), 𝑯prop (𝑡) := 𝑰 − ℎ𝑡 (𝑼𝑡 ⊗ |110⟩⟨100| 𝑡−1,𝑡,𝑡+1 + 𝑼𝑡† ⊗ |100⟩⟨110| 𝑡−1,𝑡,𝑡+1 ) 𝑯prop (𝑇) := 𝑰 − ℎ𝑇 (𝑼𝑇 ⊗ |11⟩⟨10|𝑇−1,𝑇 + 𝑼𝑇† ⊗ |10⟩⟨11|𝑇−1,𝑇 ) for each 1 < 𝑡 < 𝑇, . The 𝑇 qubits correspond to the 𝑇 geometrically-local gates and are placed next to each gate to ensure 𝑯𝐶 is geometrically local. The couplings are chosen as √︁ 𝐽clock = 1, 𝑓𝑡 = (𝑇 − 𝑡)/𝑇, 𝑔 𝑗 = 1/𝜉𝑡 𝑗 −1 , ℎ𝑡 = 𝑡 (𝑇 − 𝑡 + 1). We will set the other parameters 𝐽in , 𝐽prop later. The time 𝑡 𝑗 is the first time qubit 𝑗 is acted on. We will show later in Appendix 4.14 that 𝑯𝐶 has a unique ground state given by |𝜂0 ⟩ = 𝑇 √︁ ∑︁ 𝑡=0 𝑛
𝑡 𝑇−𝑡 𝜉𝑡 𝑼𝑡 · · · 𝑼1 |0 ⟩ ⊗ 1 0 where   1 𝑇 𝜉𝑡 := 𝑇 . 2 𝑡 Note this state encodes the computational history of the circuit 𝑼𝐶 . By choosing 𝑡0 = 𝐿 2 , we ensure that each time in the interesting part of the computational history (i.e., the intermediate 𝐿 gates) can be observed with Ω(1/𝑇) probability as we will show later in Proposition 4.14.1. This also implies 𝑔 𝑗 = O (𝑇). We now state a detailed version of Theorem 4.9.2 based on the definition of 𝑯𝐶 in the following. Theorem 4.14.1 (All local minima are global in 𝑯𝐶 ). Let 𝑷𝐺 be the ground-space projector for the Hamiltonian 𝑯𝐶 in Eq. (4.51). For any failure probability 0 < 𝛿 < 1, 269 there is a parameter choice 𝐽in , 𝐽prop = Poly(𝑛, 𝑇, 𝛿−1 ) and a choice of 𝑚 two-qubit jump operators 𝑇 𝑛 𝑆0 = { 𝑨𝑎 } 𝑚 𝑎=1 := {𝑰 ⊗ 𝑿𝑡 , 𝑰 ⊗ 𝒁 𝑡 } 𝑡=1 ∪ { 𝑿 𝑗 ⊗ |0⟩⟨0| 𝑡 𝑗 } 𝑗=1 (4.52) with 𝑚 = 2𝑇 + 𝑛 satisfying the following: For a sufficiently small 𝜖 = 1/Poly(𝑛, 𝑇, 𝛿−1 ), any 𝜖-approximate local minimum 𝝆 of 𝑯𝐶 under thermal perturbations with sufficiently large 𝛽 = Poly(𝑛, 𝑇, 𝛿−1 ), 𝜏 = Poly(𝑛, 𝑇, 𝛿−1 ), and system-bath interactions generated by 𝑆0 is an exact global minimum with probability tr(𝑷𝐺 (𝑯𝐶 ) 𝝆) ≥ 1 − 𝛿. We remind the reader that the thermal Lindbladians that generate the perturbations are defined in Eq. (4.18). The transition weight 𝛾 𝛽 (𝜔) is chosen to be Glauber dynamics as defined in Eq. (4.21), with energy cut-off Λ0 = 1 as a convenient choice so that ∥𝜔𝛾 𝛽 (𝜔)∥ ∞ ≤ 1. We do not expect our result to change with other reasonable choices of 𝛾 𝛽 (𝜔). Remark 4.14.1.1. Our 𝑯𝐶 is similar to previous circuit-Hamiltonian constructions (see, e.g., [3, 100, 142]), but there are some significant differences. One key change is that 𝑯prop is no longer frustration-free, and its couplings ℎ𝑡 are not uniform; consequently, this revised 𝑯prop has better spectral properties that enable us to lower bound its Bohr-frequency gap. Furthermore, 𝑯clock is given nonuniform couplings 𝑓𝑡 so that any local excitation has an incentive to move rightwards (e.g., |0011⟩ → |0001⟩), ensuring 𝑯clock has no local minima except its ground states. These modifications allow us to prove that all excited states of 𝑯𝐶 have significant negative gradients, so that they will all flow to the ground state under thermal perturbations. Characterizing low energy states of 𝑯𝐶 We will start by characterizing the low energy states of the circuit Hamiltonian. We define the following sequence of Hamiltonians: 𝑯I = 𝑯clock 𝑯II = 𝑯clock + 𝑯prop 𝑯III = 𝑯clock + 𝑯prop + 𝑯in = 𝑯𝐶 with the ground space projectors 𝑷 𝑗 such that 𝑷I ⊃ 𝑷II ⊃ 𝑷III . 270 Equivalently, we have 𝑷I 𝑷II = 𝑷II and 𝑷I 𝑷III = 𝑷II 𝑷III = 𝑷III . Our approach to calculating the gradient for 𝑯III = 𝑯𝐶 is perturbative: we start with the simple Hamiltonian 𝑯I and gradually add perturbations (which will split the spectrum, Figure 4.4). Remarkably, the gradient is stable as long as the perturbation is weak enough. That is, it suffices to analyze the gradient of the simpler, unperturbed Hamiltonians on suitable subspaces. Now we describe explicitly the ground subspaces 𝑷I , 𝑷II and 𝑷III . Let |𝐶𝑡 ⟩ = 1𝑡 0𝑇−𝑡 for each 𝑡 = 0, 1, . . . , 𝑇 and
𝜂 𝒙,𝑡 = 𝑼𝑡 · · · 𝑼1 |𝒙⟩ ⊗ |𝐶𝑡 ⟩ for each 𝒙 ∈ {0, 1}𝑛 and 0 ≤ 𝑡 ≤ 𝑇 . The set of 𝜂 𝒙,𝑡 forms an orthonormal basis for the ground space of 𝑯I = 𝑯clock , with energy 0 and a spectral gap of 𝐽clock /𝑇. The ground space projector is 𝑷I = 𝑇 ∑︁ ∑︁ |𝜂 𝒙,𝑡 ⟩⟨𝜂 𝒙,𝑡 | ⟨𝜂 𝒚,𝑡 ′ |𝜂 𝒙,𝑡 ⟩ = 𝛿 𝒚𝒙 𝛿𝑡𝑡 ′ . and 𝒙∈{0,1} 𝑛 𝑡=0 Observe that [𝑯prop , 𝑷I ] = 0, so the ground states of 𝑯II = 𝑯clock + 𝑯prop are given as the ground states of 𝑇 𝐽prop ∑︁ 𝑷I 𝑯prop 𝑷I = [𝑰 − ℎ𝑡 (𝑼𝑡 ⊗ |𝐶𝑡 ⟩⟨𝐶𝑡−1 | + 𝑼𝑡† ⊗ |𝐶𝑡−1 ⟩⟨𝐶𝑡 |)]. 2 𝑡=1 Furthermore, observe the orthogonality relations ⟨𝜂 𝒙,𝑡 |𝑯prop |𝜂 𝒚,𝑡 ′ ⟩ = 0 when 𝒙≠𝒚 for each 𝑡, 𝑡 ′ . That is, 𝑷I 𝑯prop 𝑷I is block diagonal with blocks labeled by 𝒙. Moreover, for any 𝒙, in the basis of 𝜂 𝒙,0 , 𝜂 𝒙,1 , . . . , 𝜂 𝒙,𝑇 , we can explicitly write down the effective (𝑇 + 1) × (𝑇 + 1) Hamiltonian 𝑇 −ℎ1 © ª ­−ℎ1 𝑇 −ℎ2 ® ­ ® ­ ® −ℎ2 𝑇 −ℎ3 ®  𝑇 𝐽prop ­­ ® 𝑷I 𝑯prop 𝑷I = ­ ® = 𝐽prop 𝑰 − 𝑳 𝑥 , .. .. ® 2 ­ 2 . . −ℎ3 ­ ® ­ ® .. ­ . 𝑇 −ℎ𝑇 ®® ­ −ℎ𝑇 𝑇 ¬ « 271 HIII = Hclock + Hprop + Hin HII = Hclock + Hprop <latexit 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HI = Hclock <latexit 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E E <latexit 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Figure 4.4: The degenerate levels of 𝑯clock split under perturbations 𝑯prop and 𝑯in . In particular, the ground state splitting is tracked in blue shades. The careful choice of energy scales ensures that the levels can be identified with the original degenerate blocks. where 𝑳 𝑥 is the matrix representation of the spin-𝑇/2 angular momentum operator whose spectrum is well known. In particular, the unique ground state is |𝜂 𝒙 ⟩ := 𝑇 √︁ ∑︁ 𝜉𝑡 𝜂 𝒙,𝑡 with energy 0 and spectral gap 𝐽prop . 𝑡=0 We will call |𝜂 𝒙 ⟩ the history state with respect to input |𝒙⟩. The ground space projector of 𝑯II is then given as ∑︁ 𝑷II = |𝜂 𝒙 ⟩⟨𝜂 𝒙 |. 𝒙∈{0,1} 𝑛 Finally, note |𝜂0 ⟩ for 0 = (0, 0, · · · , 0) is the unique ground state of 𝑯𝐶 = 𝑯III and so 𝑷III = |𝜂0 ⟩⟨𝜂0 |. This is because 𝑯in is positive semi-definite, and |𝜂0 ⟩ is the only state in 𝑷II with zero eigenvalue with respect to 𝑯in . Proof of Theorem 4.14.1 To prove Theorem 4.14.1, we show that all excited states in 𝑰 − 𝑷III have significant gradient relative to 𝑯III . Our analysis for the gradient will be carried out in three 272 subspaces: 𝑰 − 𝑷III = ( 𝑰 − 𝑷I ) + 𝑷I ( 𝑰 − 𝑷II ) + 𝑷II ( 𝑰 − 𝑷III ) . | {z } | {z } | {z } studying 𝑯I studying 𝑯II 𝑷I studying 𝑯III 𝑷II Let L 𝑗 := L 𝛽,𝜏,𝑯 𝑗 be the thermal Lindbladian with uniform weights as in Eq. (4.30), defined with respect to 𝑯 𝑗 and the jump operators in Eq. (4.52) . Case 1: Gradients for 𝑰 − 𝑷I from 𝑯I . We first show excited states of 𝑯I have good energy gradient: −LI† [𝑯I ] ⪰ 𝑟 1 ( 𝑰 − 𝑷I ) − 𝜖 1𝑎 𝑰. (4.53) Because 𝑯I = 𝑯clock is a commuting Hamiltonian, the global gradient can be lower bounded by checking the local gradient from individual local jumps. We carry out this computation in section 4.14, where we show 𝑟 1 = Ω(1/𝑇 ln 𝛽) and 𝜖1𝑎 = O (𝑇 7/4 /𝜏 1/4 + 𝑇/𝛽 + 𝑇 (1 + 𝛽) ln 𝜏/𝜏) in Lemma 4.14.2. We then apply Theorem 4.12.1 (with 𝑯 = 𝑯I and 𝑯′ = 𝑯III ) to show excited states of 𝑯I have large gradient with respect to 𝑯III . In other words, † −LIII [𝑯III ] ⪰ 𝑟 1 ( 𝑰 − 𝑷I ) − 𝜖1 𝑰. (4.54) To do this, we only need to check that the conditions of Theorem 4.12.1 are satisfied. Note 𝑷I also projects onto eigenstates of 𝑯III since [𝑷I , 𝑯III ] = 0. Note 𝑯I has a discrete spectrum with a minimum Bohr-frequency gap of at least Δ𝜈 ≥ 1/𝑇. We can choose sufficiently small 𝐽prop , 𝐽in such that ∥𝑽I ∥ := ∥𝑯prop + 𝑯in ∥ ≤ 𝐽prop𝑇 2 + 𝐽in 𝑛𝑔max ≪ Δ𝜈 (𝑯I ) = 1/𝑇, where 𝑔𝑚𝑎𝑥 := max1≤ 𝑗 ≤𝑛 𝑔 𝑗 = O (𝑇). And plugging in Λ0 = 1 and other parameters into the error bound (4.48)   1 ∥𝑯I ∥ 3/4 1 1 1 + 𝑟1  e−𝛽Δ𝜈 /4  𝜖1 = 𝜖1𝑎 + |𝑆0 |O + + 1/3 + √ + 1+ + ∥𝑽I ∥ . 𝜏 𝛽 Δ𝜈 𝜏 1/4 𝜏 Δ𝜈 𝜏 Noting that |𝑆0 |, ∥𝑯I ∥, 𝑔max = O (𝑇), we can make 𝜖1 /𝑟 1 ≤ 𝛿/6 by choosing appropriate powers 𝜏 ≥ Ω̃(𝑇 11 /𝛿4 ), 𝛽 ≥ Ω̃(𝑇 2 /𝛿), 𝐽prop ≤ Õ (𝛿/𝑇 5 ), and 𝐽in ≤ Õ (𝛿/𝑛𝑇 4 ). (4.55) 273 Case 2: Gradients for 𝑷I ( 𝑰 − 𝑷II ) from 𝑯II . We next restrict our attention † to the action of LII [𝑯II ] inside the 𝑷I subspace, which conveniently is also an eigensubspace of both 𝑯II and 𝑯III since [𝑷I , 𝑯II ] = [𝑷I , 𝑯III ] = 0. Explicit computation in section 4.14 shows that † −𝑷I LII [𝑯II ] 𝑷I ⪰ 𝑟 2 𝑷I ( 𝑰 − 𝑷II ) − 𝜖2𝑎 𝑰, (4.56) with the bounds from Lemma 4.14.4 promising 𝑟 2 = Ω( 𝐽prop ) 𝑇 ln 𝛽 and 𝜖2𝑎 = |𝑆0 | · O 1 + 𝜏 ∥𝑯II ∥ 3/4 1 1  + + . √︁ 𝛽 𝜏 1/4 𝜏𝐽prop We then invoke Corollary 4.12.1 with 𝑸 = 𝑸 ′ = 𝑷I , 𝑯 = 𝑯II and 𝑯′ = 𝑯III to show monotonicity of energy gradient on a subspace under perturbation † ′ −𝑷I LIII [𝑯III ] 𝑷I ⪰ 𝑟 2 𝑷I ( 𝑰 − 𝑷II ) − 𝜖2𝑏 𝑰, ′ is the perturbed eigensubspace of 𝑯 where 𝑷II III that is identified with 𝑷II . To justify the application of Corollary 4.12.1, we note in 𝑯II , the eigensubspace 𝑷I has an excitation gap of Δ𝑸 ≥ 1/𝑇 − 2∥𝑯prop ∥ = Ω(1/𝑇), where extra 2∥𝑯prop ∥ term is due to shifts in eigenvalues of 𝑯I bounded by Weyl’s inequality (see Proposition 4.16.3). The perturbation on 𝑯II has strength ∥𝑽II ∥ := ∥𝑯in ∥ ≤ 𝐽in 𝑛𝑔max , and Δ𝜈 (𝑯II | 𝑷I ) = 𝐽prop . Then noting Δ𝜈 ≪ Δ𝑸 , we keep the dominant terms in the error bound (4.49) and get 𝜖2𝑏 ≤ 𝜖 2𝑎 + |𝑆0 | · O  ∥𝑽 ∥∥𝑯 ∥ ∥𝑽II ∥  II II + 𝑟2 . Δ 𝜈 Δ𝑸 Δ𝜈 Furthermore, the eigensubspace 𝑷I in 𝑯III is separated by a spectral gap of 1/𝑇 − 2∥𝑽I ∥ = Ω(1/𝑇) from the other eigenstates, we may apply Lemma 4.12.5 to show √︂ 1 1 𝑇 † ′ −LIII [𝑯III ] ⪰ 𝑟 2 𝑷I ( 𝑰 − 𝑷II ) − 𝜖2 𝑰 where 𝜖 2 = 𝜖 2𝑏 + |𝑆0 |O ( + + ). 𝛽 𝜏 𝜏 (4.57) Since |𝑆0 |, 𝑔max , ∥𝑯II ∥, ∥𝑯III ∥ = O (𝑇), we can make 𝜖2 /𝑟 2 ≤ 𝛿/6 by choosing 𝜏 ≥ Ω̃  𝑇 11  , 2 𝛿4 𝐽prop 𝛽 ≥ Ω̃  𝑇2  , 𝐽prop 𝛿 and 𝐽in ≤ Õ 2 𝛿  𝐽prop 𝑛𝑇 5 . (4.58) 274 Case 3: Gradients for 𝑷II ( 𝑰 − 𝑷III ) from 𝑯III . Now, we restrict our attention ′ , the perturbed eigensubspace in 𝑯 to 𝑷II III that corresponds to 𝑷II . We can show by explicit computation (deferred to section 4.14) that † ′ ′ ′ −𝑷II LIII [𝑯III ] 𝑷II ⪰ 𝑟 3 𝑷II ( 𝑰 − 𝑷III ) − 𝜖 3𝑎 𝑰. (4.59) This computation shows that all valid history states |𝜂 𝒙 ⟩ except for 𝒙 = 0 have nonzero gradient with respect to LIII . The derivation uses a more fine-grained version of subspace gradient monotonicity (Corollary 4.12.2) since the standard version yields insufficient bounds. Roughly, we need to capture the fact that off-diagonal perturbations induce only second-order perturbation on the eigenvalues. The final calculated bounds in Eqs. (4.14) and (4.14) give us  3/4   𝐽  (𝑛𝑇 𝐽in ) 2 1 1 𝑇 in −𝛽𝐽in 𝑟3 = Ω 2 +𝑛 and 𝜖3𝑎 ≤ 𝑇 O 1/4 + + √ +e . 𝛽 𝐽prop 𝑇 ln 𝛽 𝜏 𝐽in 𝜏 ′ is separated by energy of at least 𝐽 Using the fact that 𝑷II prop − 2∥𝑯in ∥ from the other eigenstates in 𝑯III , we can apply Lemma 4.12.5 to get † ′ −LIII [𝑯III ] ⪰ 𝑟 3 𝑷II ( 𝑰 − 𝑷III ) −𝜖3 𝑰 𝜖 3 = 𝜖3𝑎 + |𝑆0 |O where 1 1 1  + + √︁ . 𝛽 𝜏 𝐽prop 𝜏 (4.60) We may ensure 𝜖 3 /𝑟 3 ≤ 𝛿/6 by choosing 𝜏 ≥ Ω̃  𝑇 15  4 𝛿4 𝐽in 𝛽 ≥ Ω̃ ,  𝑇3  , and 𝐽in 𝛿 𝐽in ≤ Õ  𝐽prop 𝛿  𝑛 3𝑇 5 (4.61) . Altogether. Based on the conditions in Eqs. (4.55) (4.58) (4.61) and the fact that 𝑇 = Ω(𝑛), a consistent choice of parameters that satisfies all the bounds and ensures 𝜖 𝑗 /𝑟 𝑗 ≤ 𝛿/6 are 𝜏 = Θ̃  𝑇 79  𝛿 , 16 𝛽 = Θ̃  𝑇 19  𝛿4 , 𝐽prop = Θ̃ 𝛿  𝑇 , 5 and Then combining Eqs. (4.54), (4.57), and (4.60) implies that 𝜖1 1 † 𝑰 − LIII [𝑯III ], 𝑟1 𝑟1 𝜖2 1 † ′ 𝑷I − 𝑷II ⪯ 𝑰 − LIII [𝑯III ], 𝑟2 𝑟2 𝜖3 1 † ′ 𝑷II − 𝑷III ⪯ 𝑰 − LIII [𝑯III ]. 𝑟3 𝑟3 𝑰 − 𝑷I ⪯ 𝐽in = Θ̃  𝛿3  𝑇 16 . 275 ′ ′ and 𝑷′ 𝑷 Note we have used 𝑷I 𝑷II = 𝑷II II III = 𝑷III . Recall that LIII = Í𝑚 𝛽,𝜏,𝑯𝐶 . Adding all three inequalities together and normalizing suitably 𝑎=1 L 𝑎 by the number of jumps 𝑚 = |𝑆0 |, we have ! 3 𝑚 3 ©∑︁ 𝑚 ª 1 ∑︁ †𝛽,𝜏,𝑯𝐶 ©∑︁ 𝜖 𝑗 ª 𝑰 − 𝑷III ⪯ −­ L𝑎 [𝑯III ] + ­ ® ® 𝑰. 𝑟 𝑚 𝑟 𝑗 𝑗 𝑎=1 « 𝑗=1 ¬ « 𝑗=1 ¬ The above provides the desired negative gradient condition on the full Hamiltonian 𝑯III = 𝑯𝐶 . From Lemma 4.8.2, any 𝜖-approximate local minimum 𝝆 of 𝑯𝐶 under thermal perturbation satisfies 1 − tr(𝑷III 𝝆) ≤ 3 ∑︁ 1 𝛿 ≤ + 𝜖𝑚 . 2 𝑟 𝑗=1 𝑗 3 ∑︁ 𝜖 𝑗 + 𝑚𝜖 𝑟𝑗 𝑗=1
−1 𝛿 Í3 By choosing 𝜖 ≤ 2𝑚 = 1/Poly(𝑛, 𝑇, 𝛿−1 ), we guarantee that 1 − 𝑗=1 1/𝑟 𝑗 tr(𝑷III 𝝆) ≤ 𝛿. This concludes our proof of Theorem 4.14.1. Explicit calculations for energy gradients In this section of the appendix, we provide the missing calculations supporting the claims that were asserted in Eqs. (4.53), (4.56), and (4.59) in the above proof of Theorem 4.14.1. Gradient from 𝑯clock Note that 𝑯I = 𝑯clock is a commuting Hamiltonian 𝑯I = 𝑯clock = 𝐽clock 𝑇−1 ∑︁ 𝑓𝑡 𝒉𝑡,𝑡+1 , where 𝒉𝑡,𝑡+1 = 𝑰 ⊗ |01⟩⟨01| 𝑡,𝑡+1 , 𝑡=1 where we set 𝑓𝑡 = (𝑇 − 𝑡)/𝑇 and 𝐽clock = 1. We start by computing the gradient from a single jump operator, using the simplification from Lemma 4.12.8: Lemma 4.14.1. Let the jump operator 𝑨𝑡 = 𝑰 ⊗ 𝑿𝑡 for each 𝑡 ∈ [𝑇]. For all 𝑡 = 2, . . . , 𝑇, we have †𝛽,𝜏,𝑯I −D𝑡 [𝑯I ] ⪰ 𝑟 1 𝒉𝑡−1,𝑡 − 𝜖0 𝑰, √︁ where 𝑟 1 = Ω( 𝑇 ln1 𝛽 ) and 𝜖0 = O ( 𝑇/𝜏 + [(1 + 𝛽) ln 𝜏]/𝜏 + 1/𝛽). †𝛽,𝜏,𝑯I Proof. By Lemma 4.12.8, we have D𝑡 proceed in two cases. †𝛽,𝜏,𝑯 ∋𝑡 [𝑯I ] = D𝑡 [𝑯∋𝑡 ]. We then 276 Case 1: 2 ≤ 𝑡 ≤ 𝑇 − 1. In this case, the relevant part of 𝑯I that does not commute with 𝑨𝑡 is 𝑯∋𝑡 = 𝑓𝑡−1 |01⟩⟨01| 𝑡−1,𝑡 + 𝑓𝑡 |01⟩⟨01| 𝑡,𝑡+1 . Observe that 𝑯∋𝑡 has three degenerate eigenspaces with corresponding energies as follows: ∑︁ 𝑷𝑡0 = |𝒔⟩⟨𝒔| 𝑡−1,𝑡,𝑡+1 , 𝐸 0 = 0, 𝒔∈{000,100,110,111} ∑︁ 𝑷𝑡𝐿 = |𝒔⟩⟨𝒔| 𝑡−1,𝑡,𝑡+1 , 𝐸 𝐿 = 𝑓𝑡−1 , |𝒔⟩⟨𝒔| 𝑡−1,𝑡,𝑡+1 , 𝐸 𝑅 = 𝑓𝑡 . 𝒔∈{010,011} ∑︁ 𝑷𝑡𝑅 = 𝒔∈{001,101} The possible negative Bohr frequencies and the associated jumps are 𝑨𝑡𝜈1 = 𝑷𝑡0 𝑿𝑡 𝑷𝑡𝐿 = |000⟩⟨010| 𝑡−1,𝑡,𝑡+1 , 𝜈1 = − 𝑓𝑡−1 , 𝑨𝑡𝜈2 = 𝑷𝑡0 𝑿𝑡 𝑷𝑡𝑅 = |111⟩⟨101| 𝑡−1,𝑡,𝑡+1 , 𝜈 2 = − 𝑓𝑡 , 𝑨𝑡𝜈3 = 𝑷𝑡𝑅 𝑿𝑡 𝑷𝑡𝐿 = |001⟩⟨011| 𝑡−1,𝑡,𝑡+1 , 𝜈3 = 𝑓𝑡 − 𝑓𝑡−1 = −1/𝑇 . Furthermore, observe that the Bohr frequencies are exactly integer multiples of 1/𝑇, so we can lower bound the Bohr-frequency gap Δ𝜈 (𝑯∋𝑡 ) ≥ 1/𝑇. Then †𝛽,𝜏,𝑯 ∋𝑡 †𝛽,∞,𝑯 ∋𝑡 by Lemma 4.12.3 and 4.12.4, we can replace D𝑡 with D𝑡 up to an √︁ O ( 𝑇/𝜏 + [(1 + 𝛽) ln 𝜏]/𝜏) error. Letting 𝜃 𝑗 = 𝜈 𝑗 𝛾 𝛽 (𝜈 𝑗 ) for 𝑗 = 1, 2, 3 (recall 𝛾 𝛽 is given in Eq. (4.21) with Λ0 = 1), we have ∑︁ †𝛽,∞,𝑯 ∋𝑡 D𝑡 [𝑯∋𝑡 ] = 𝜈𝛾 𝛽 (𝜈) 𝑨𝜈𝑎† 𝑨𝜈𝑎 𝜈∈𝐵(𝑯 ∋𝑡 )
= 𝜃 1 |010⟩⟨010| + 𝜃 2 |101⟩⟨101| + 𝜃 3 |011⟩⟨011| 𝑡−1,𝑡,𝑡+1 + O (1/𝛽), where the last error term is due to heating transitions (positive Bohr frequencies), which incur errors of at most ∥𝜔𝛾 𝛽 (𝜔) 1 (𝜔 > 0)∥ ∞ = O (1/𝛽). Note that 𝜃 1 , 𝜃 2 , 𝜃 3 < 0, and furthermore we have min{|𝜃 1 |, |𝜃 3 |} ≥ min 𝜔∈[−1,−1/𝑇] |𝜔|𝛾 𝛽 (𝜔) =: 𝑟 1 = Ω 1
. 𝑇 ln 𝛽 Hence, †𝛽,∞,𝑯 ∋𝑡 −D𝑡
[𝑯∋𝑡 ] ⪰ 𝑟 1 |010⟩⟨010| + |011⟩⟨011| 𝑡−1,𝑡,𝑡+1 − O (1/𝛽) 𝑰. 277 Note the first term combines to make 𝒉𝑡−1,𝑡 . We then return to the finite-𝜏 Lindbladian up to the aforementioned error: √︂ 𝑇 (1 + 𝛽) ln 𝜏 1  †𝛽,𝜏,𝑯 ∋𝑡 −D𝑡 [𝑯∋𝑡 ] ⪰ 𝑟 1 𝒉𝑡−1,𝑡 − O + + 𝑰 𝜏 𝜏 𝛽 Case 2: 𝑡 = 𝑇. The relevant part of 𝑯I in this case is 𝑯∋𝑇 = 𝑓𝑇−1 |01⟩⟨01|𝑇−1,𝑇 , which has two eigenspaces. There is only one negative Bohr frequency with a corresponding jump operator filtered at 𝜈 𝑨𝑇𝜈 = ( 𝑰−|01⟩⟨01|𝑇−1,𝑇 ) 𝑿𝑇 |01⟩⟨01|𝑇−1,𝑇 = |00⟩⟨01|𝑇−1,𝑇 where 1 𝜈 = − 𝑓𝑇−1 = − . 𝑇 Then, †𝛽,∞,𝑯 ∋𝑇 D𝑡 [𝑯∋𝑇 ] = − 𝑇1 𝛾 𝛽 (− 𝑇1 )|01⟩⟨01|𝑇−1,𝑇 + O ( 1𝛽 ) 𝑰. Note that 𝑇1 𝛾 𝛽 (− 𝑇1 ) ≥ 𝑟 1 . Applying Lemma 4.12.3 and 4.12.4 to return to the finite 𝜏 expression, √︂ 𝑇 (1 + 𝛽) ln 𝜏 1  †𝛽,𝜏,𝑯 ∋𝑇 −D𝑡 [𝑯∋𝑇 ] ⪰ 𝑟 1 𝒉𝑇−1,𝑇 − O + + 𝑰, 𝜏 𝜏 𝛽 which is the advertised result. ■ We are now ready to prove Eq. (4.53), which we state as the following lemma: Lemma 4.14.2. Assume 1 ≤ 𝑇 ≤ 𝜏. We have −LI† [𝑯I ] ⪰ 𝑟 1 ( 𝑰 − 𝑷I ) − 𝜖1𝑎 𝑰, where 𝑟1 = Ω  1  𝑇 ln 𝛽 and 𝜖 1𝑎 = O  𝑇 7/4 𝜏 + 1/4 𝑇 𝑇 (1 + 𝛽) ln 𝜏  + . 𝛽 𝜏 Proof. Note by linearity, we have LI† [𝑯I ] = ∑︁ †𝛽,𝜏,𝑯I L𝑎 [𝑯I ]. 𝑎∈𝑆 Let 𝑆I = {𝑰 ⊗ 𝑿𝑡 : 2 ≤ 𝑡 ≤ 𝑇 } be a subset of the jump operators. Then   ∥𝑯 ∥ 3/4 1 1   ∑︁ I †𝛽,𝜏,𝑯I † −LI [𝑯I ] ⪰ − D𝑎 [𝑯I ] − O |𝑆0 | + + 𝑰, 𝜏 𝛽 𝜏 1/4 𝑎∈𝑆 I 278 where the error contribution from neglecting the Lamb-shift term and the other jump operators in 𝑆0 \ 𝑆I are bounded by Proposition 4.10.3 and Lemma 4.12.1. Applying Lemma 4.14.1 to the sum on the right hand side above, we get − ∑︁ †𝛽,𝜏,𝑯I D𝑎 [𝑯I ] ⪰ 𝑟 1 𝑎∈𝑆I 𝑇−1 ∑︁ 𝒉𝑡,𝑡+1 − 𝑇𝜖 0 𝑰. 𝑡=1 It is not difficult to see that 𝑇−1 ∑︁ 𝒉𝑡,𝑡+1 ⪰ 𝑰 − 𝑷I , 𝑡=1 that is, the smallest excitation has energy 1. Hence, −LI† [𝑯I ] ⪰ 𝑟 1 ( 𝑰 − 𝑷I ) − 𝜖1𝑎 𝑰, where  ∥𝑯 ∥ 3/4   𝑇 3/2 𝑇 (1 + 𝛽) ln 𝜏 𝑇  1 1 𝜖1𝑎 = O |𝑆0 | + + + + O 1/2 + 𝜏 𝛽 𝜏 𝛽 𝜏  𝑇 7/4 𝑇 𝑇 𝑇 3/2 𝑇 (1 + 𝛽) ln 𝜏  . = O 1/4 + + + 1/2 + 𝜏 𝛽 𝜏 𝜏 𝜏  I 𝜏 1/4 The last equality uses that |𝑆0 |, ∥𝑯I ∥ = O (𝑇). Since 1 ≤ 𝑇 ≤ 𝜏, we have 𝑇 7/4 /𝜏 1/4 ≥ 𝑇/𝜏 and 𝑇 7/4 /𝜏 1/4 ≥ 𝑇 3/2 /𝜏 1/2 , so we drop the latter two terms for the final error estimate in the lemma statement. ■ Gradient from 𝑯prop In this subsection, we prove † −𝑷I LII [𝑯II ] 𝑷I ⪰ 𝑟 2 𝑷I ( 𝑰 − 𝑷II ) − 𝜖2𝑎 𝑰. Denote |𝑡⟩ := 𝜂 𝒙,𝑡 in what follows. Let 𝑳 + be the raising operator whose only nontrivial action is √︁ 𝑳 + |𝑡⟩ = (𝑡 + 1)(𝑇 − 𝑡)|𝑡 + 1⟩ for each 0 ≤ 𝑡 ≤ 𝑇 − 1 and 𝑳 − =: 𝑳 †+ . Í Furthermore, let 𝑳 𝑥 = 21 ( 𝑳 + + 𝑳 − ), 𝑳 𝑦 = 2i1 ( 𝑳 + − 𝑳 − ), 𝑳 𝑧 = 𝑇𝑡=0 (𝑡 − 𝑇/2)|𝑡⟩⟨𝑡|. These operators form a set of angular momentum operators as [𝑳 𝑎 , 𝑳 𝑏 ] = i𝜖 𝑎𝑏𝑐 𝑳 𝑐 for 𝑎, 𝑏, 𝑐 ∈ {𝑥, 𝑦, 𝑧}. As noted earlier, we have 𝑷I 𝑯prop 𝑷I = 𝐽prop ( 𝑇 − 𝑳 𝑥 ). 2 279 The eigenstates are known to be |𝑣 𝑘 ⟩ = ei𝜋 𝑳 𝑦 /2 |𝑘⟩ with eigenvalues 𝜆 𝑘 = 𝑘 𝐽prop for 𝑘 = 0, 1, . . . , 𝑇 . This integer spectrum means the minimum Bohr-frequency gap in the subspace 𝑷I is Δ𝜈 (𝑯prop | 𝑷I ) = 𝐽prop . Next, we give jump operators with nontrivial gradient on any excited state of 𝑷I 𝑯prop 𝑷I . These will be the 1-local jumps acting on the clock register 𝑰 ⊗ 𝒁ℓ for each 1 ≤ ℓ ≤ 𝑇 which nicely respects the block-diagonal structure of 𝑯prop such that ⟨𝜂 𝒙,𝑡 | 𝑰 ⊗ 𝒁ℓ |𝜂 𝒚,𝑡 ′ ⟩ = 0 if 𝒚 ≠ 𝒙. (4.62) Thus, fixing 𝒙, we merely need to consider effective jump operators 𝑷I ( 𝑰 ⊗ 𝒁ℓ )𝑷I ≡ 𝝈ℓ such that 𝝈ℓ |𝑡⟩ = (−1) 1𝑡 ≥ℓ |𝑡⟩. (4.63) Lemma 4.14.3 (Good transition rates). For the operators 𝝈ℓ in Eq. (4.63) and any 0 ≤ 𝑘 < 𝑇, we have that 1 max |⟨𝑣 𝑘 |𝝈ℓ |𝑣 𝑘+1 ⟩| ≥ √ . ℓ∈[𝑇] 𝑇 Proof. Fix any 0 ≤ 𝑘 < 𝑇. Observe that ⟨𝑣 𝑘 | 𝑳 𝑧 |𝑣 𝑘+1 ⟩ = ⟨𝑘 |e−i𝜋 𝑳 𝑦 /2 𝑳 𝑧 ei𝜋 𝑳 𝑦 /2 |𝑘 + 1⟩ = ⟨𝑘 | 𝑳 𝑥 |𝑘 + 1⟩ = 1 √︁ (𝑘 + 1)(𝑇 − 𝑘). 2 Then, 𝑇 1 ∑︁ |⟨𝑣 𝑘 |𝝈𝑡 |𝑣 𝑘+1 ⟩| max |⟨𝑣 𝑘 |𝝈𝑡 |𝑣 𝑘+1 ⟩| ≥ 𝑡 𝑇 𝑡=1 2 |⟨𝑣 𝑘 | 𝑳 𝑧 |𝑣 𝑘+1 ⟩| 𝑇 1 ≥ √ 𝑇 ≥ as advertised. ■ Now that we understand the connectivity between the 𝑡 labels, we may restore the 2𝑛 many labels 𝒙 |𝑣 𝑘 ⟩ → 𝑣 𝑘,𝒙 such that ⟨𝑣 𝑘 ′ ,𝒚 |𝑣 𝑘,𝒙 ⟩ = 𝛿 𝑘 𝑘 ′ 𝛿 𝒙 𝒚 . Fortunately, we do not need to address the explicit labels 𝒙 due to the orthogonality † properties in Eq. (4.62). We may now calculate the gradient operator 𝑷I LII [𝑯II ] 𝑷I . 280 Lemma 4.14.4. Consider the thermal Lindbladian LII = Í 𝛽,𝜏,𝑯II . Then, 𝑎∈𝑆0 L 𝑎 † [𝑯II ] 𝑷I ⪰ 𝑟 2 𝑷I ( 𝑰 − 𝑷II ) − 𝜖2 𝑰 −𝑷I LII for 𝐽prop 𝑟 2 = Ω( ) 𝑇 ln 𝛽 and 𝜖2 = |𝑆0 | · O ∥ 3/4 1 ∥𝑯II + 𝜏 𝜏 1/4 ! + 1 1 + √︁ . 𝛽 𝜏𝐽prop Proof. Observe that 𝑷I projects onto a low-energy subspace of 𝑯II with an excitation gap of at least 𝐽clock /𝑇 −2∥𝑯prop ∥ from Weyl’s inequality. Furthermore, 𝑯II restricted to 𝑷I has eigenvalues that are integer multiples of 𝐽prop , so the Bohr-frequency gap in the subspace is Δ𝜈 (𝑯II | 𝑷I ) = 𝐽prop . Assuming 𝐽prop /2 < 𝐽clock /𝑇 − 2∥𝑯prop ∥, we may apply Lemma 4.12.6 with 𝑯 = 𝑯II , 𝑸 = 𝑷I to get 𝐸 † 𝑷I LII [𝑯II ] 𝑷I ≈ ∑︁ ∫ 0 ∑︁ 𝑷I 𝑨𝜈𝑎† 𝑷I 𝑨𝜈𝑎 𝑷I −∞ 𝑎∈𝑆0 𝜈∈𝐵(𝑯II | 𝑷I ) ⪯ ∑︁ 2 𝛾 𝛽 (𝜔)𝜔 𝑓ˆ𝜇 (𝜔 − 𝜈) d𝜔 ∫ 0 ∑︁ 𝑷I 𝑨𝜈𝑎† 𝑷I 𝑨𝜈𝑎 𝑷I 2 𝛾 𝛽 (𝜔)𝜔 𝑓ˆ𝜇 (𝜔 − 𝜈) d𝜔, −∞ 𝑎∈𝑆II 𝜈∈𝐵(𝑯II | 𝑷I ) (4.68) where 𝜇 = Δ𝜈 /2 and  ∥𝑯∥ 3/4 1 1 1 𝐸 = |𝑆0 |O + + +√ . 𝜏 𝛽 𝜏 1/4 Δ𝜈 𝜏  The second line uses the negativity of the half-integral to reduce to the following subset of jump operators from Eq. (4.52) 𝑆II = {𝑰 ⊗ 𝒁ℓ : ℓ ∈ [𝑇]}. Let us now explicitly display the matrix elements of the above jump operators in the 𝑣 𝑘,𝒙 basis: ∑︁ 𝑷I ( 𝑰 ⊗ 𝒁ℓ )𝑷I = 𝑣 𝑘 ′ ,𝒚 𝑣 𝑘 ′ ,𝒚 𝑰 ⊗ 𝒁ℓ 𝑣 𝑘,𝒙 𝑣 𝑘,𝒙 𝑘,𝑘 ′ ,𝒙,𝒚 = ∑︁ 𝑘,𝑘 ′ ,𝒙 𝑣 𝑘 ′ ,𝒙 𝑣 𝑘 ′ ,𝒙 𝑰 ⊗ 𝒁ℓ 𝑣 𝑘,𝒙 𝑣 𝑘,𝒙 , 281 where we applied Eq. (4.62) to drop the sum on 𝒚. Thus we can rewrite the RHS of Eq. (4.68) as ∫ 0 ∑︁ (𝑐𝑜𝑛𝑡.) = |𝑣 𝑘,𝒙 ⟩⟨𝑣 𝑘,𝒙 |𝝈ℓ |𝑣 𝑘 ′ ,𝒙 ⟩⟨𝑣 𝑘 ′ ,𝒙 |𝝈ℓ |𝑣 𝑘,𝒙 ⟩⟨𝑣 𝑘,𝒙 | −∞ ℓ,𝑘 ′ ,𝑘,𝒙  ⪯ −Ω  ⪯ −Ω 2 𝛾 𝛽 (𝜔)𝜔 𝑓ˆ𝜇 (𝜔 − 𝐽prop (𝑘 ′ − 𝑘)) d𝜔 𝐽prop ln 𝛽  ∑︁ 𝐽prop 𝑇 ln 𝛽 𝑣 𝑘,𝒙 𝑣 𝑘,𝒙 max 𝑣 𝑘,𝒙 𝝈ℓ 𝑣 𝑘−1,𝒙 2 ℓ∈[𝑇] 𝑘,𝒙  ∑︁ 𝑣 𝑘,𝒙 𝑣 𝑘,𝒙 𝑘 ≥1,𝒙  ∑︁ 𝐽prop  ⪯ −Ω 𝑷I − 𝑣 0,𝒙 𝑇 ln 𝛽 𝒙   𝑣 0,𝒙 = −Ω( 𝐽prop ) · 𝑷I ( 𝑰 − 𝑷II ). 𝑇 ln 𝛽 The first line uses the orthogonality condition ⟨𝑣 𝑘 ′ ,𝒚 |𝑣 𝑘,𝒙 ⟩ = 𝛿 𝑘 𝑘 ′ 𝛿 𝒙 𝒚 , and the fact that the identical 𝜈 labels on 𝑨𝜈𝑎† and 𝑨𝜈𝑎 enforce that transitions from 𝑘 ′ need to be to the same 𝑘 on both sides. The second line uses the negativity of the half-integral to focus on cooling transitions (which includes 𝑘 → 𝑘 − 1) and evaluates the integral (which concentrates near 𝜔 ≈ 𝐽prop ). Lastly, we combine the above with the error bound on 𝐸 to conclude the proof. ■ Gradient from 𝑯in The goal of this subsection is to prove † ′ ′ ′ −𝑷II LIII [𝑯III ] 𝑷II ⪰ 𝑟 3 𝑷II ( 𝑰 − 𝑷III ) − 𝜖 3𝑎 𝑰. ′ is the projector onto perturbed low-energy eigenstates of 𝑯 Here 𝑷II III = 𝑯II + 𝑯in , corresponding to ∑︁ 𝑷II = |𝜂 𝒙 ⟩⟨𝜂 𝒙 |, 𝒙∈{0,1} 𝑛 where |𝜂 𝒙 ⟩ = 𝑇 √︁ ∑︁ 𝑡=0 𝜉𝑡 𝜂 𝒙,𝑡 = 𝑇 √︁ ∑︁ 𝑡=0 𝜉𝑡 𝑈𝑡 · · · 𝑈1 |𝒙⟩ ⊗ |𝐶𝑡 ⟩ and   1 𝑇 𝜉𝑡 = 𝑇 . 2 𝑡 Recall our Definition 4.14.1 where given a circuit with 𝐿 computational gates, we pad it in the beginning and the end with 𝑡0 identity gates to make a total 𝑇 = 2𝑡0 + 𝐿 gates. We can understand 𝜉𝑡 as the probability from a symmetric binomial distribution Binom(𝑇, 12 ), which has substantial weight near the center where the interesting computation takes place. 282 Proposition 4.14.1 (Lower bound 𝜉𝑡 in the center). Suppose 𝑇 = 2𝑡0 + 𝐿 and 𝑡0 = 𝑐𝐿 2 are positive integers. Then we have e−𝑐/4 𝜉𝑡 ≥ 𝑇 +1 for each 𝑡 ∈ [𝑡0 , 𝑇 − 𝑡0 ]. Proof. As a property of the binomial distribution Binom(𝑇, 12 ), we have 𝜉𝑡 ≥ 𝜉𝑡0 for all 𝑡 ∈ [𝑡0 , 𝑇 − 𝑡0 ]. Observe that   −1 ∫ 1  𝑡 𝑡  𝑇 𝑡  𝑇−𝑡 𝑡 𝑇−𝑡 = (𝑇 + 1) 𝑥 (1 − 𝑥) d𝑥 ≤ (𝑇 + 1) 1− , 𝑇 𝑇 𝑡 0 where the inequality comes from the fact that arg max𝑥∈[0,1] 𝑥 𝑡 (1 − 𝑥)𝑇−𝑡 = 𝑡/𝑇. Then   1 𝑇 𝑓 (𝐿) 𝜉𝑡0 = 𝑇 , ≥ 𝑇 +1 2 𝑡0  𝑐𝐿 2 +𝐿 1  𝑇  𝑡0  𝑇  𝑇−𝑡0  1  𝑐𝐿 2  1 where 𝑓 (𝐿) = 𝑇 = 1+ 1− . 𝑇 − 𝑡0 2𝑐𝐿 2(𝑐𝐿 + 1) 2 𝑡0 The last equality is obtained after plugging in 𝑇 = 2𝑡0 + 𝐿, 𝑡0 = 𝑐𝐿 2 and simplifying. We can use the first-derivative test to check that 𝑓 (𝐿) is monotonically decreasing, and so 𝑓 (𝐿) ≥ lim 𝐿→∞ 𝑓 (𝐿) = e−𝑐/4 . Hence, 𝜉𝑡 ≥ 𝜉𝑡0 ≥ 𝑓 (𝐿)/(𝑇 + 1) ≥ e−𝑐/4 /(𝑇 + 1). ■ Using the fact that 𝑈𝑡 𝑗 −1 · · · 𝑈1 acts trivially on the 𝑗-th qubit (by definition of 𝑡 𝑗 ), we see that 𝑷II 𝑯in 𝑷II is diagonal in the |𝜂 𝒙 ⟩ basis: ⟨𝜂 𝒙 |𝑯in |𝜂 𝒚 ⟩ = 𝐽in 𝑇 √︁ ∑︁ 𝜉𝑡 𝜉𝑡 ′ ⟨𝜂 𝒙,𝑡 | 𝑛  ∑︁ 𝑡,𝑡 ′ =0 = 𝛿 𝒙,𝒚 · 𝐽in  𝑔 𝑗 |1⟩⟨1| 𝑗 ⊗ |𝐶𝑡 𝑗 −1 ⟩⟨𝐶𝑡 𝑗 −1 | |𝜂 𝒚,𝑡 ′ ⟩ 𝑗=1 𝑛 ∑︁ 𝑥 𝑗 𝑔 𝑗 𝜉𝑡 𝑗 −1 . 𝑗=1 Since 𝑔 𝑗 = 1/𝜉𝑡 𝑗 −1 (see Definition 4.14.1), then the above implies that that |𝜂 𝒙 ⟩ are eigenstates of 𝑷II 𝑯in 𝑷II with eigenvalue 𝐽in · wt(𝒙), where wt(𝒙) is the Hamming weight of bit string 𝒙. While |𝜂 𝒙 ⟩ are eigenstates of 𝑷II 𝑯in 𝑷II , unfortunately, only |𝜂0 ⟩ is an eigenstate of 𝑯in ; this will require an additional perturbation step to handle this off-block-diagonal effect. Let 𝑯III = 𝑯˜ III + 𝑽III , where ⊥ ⊥ 𝑯˜ III = 𝑯II + 𝑷II 𝑯in 𝑷II + 𝑷II 𝑯in 𝑷II ⊥ ⊥ and 𝑽III = 𝑷II 𝑯in 𝑷II + 𝑷II 𝑯in 𝑷II . 283 We will also denote L̃III as the thermal Lindbladian with respect to 𝑯˜ III . We start by studying the gradient on 𝑯˜ III and showing all its excited states have negative energy gradients. It suffices to focus on the states in 𝑷II , which is a lowenergy subspace of eigenstates of 𝑯˜ III with excitation gap of at least 𝐽prop − 2∥𝑯in ∥ from Weyl’s inequality. The effective Hamiltonian in this subspace has a simple form of decoupled qubits, which we write as 𝑯˜ eff := 𝑯˜ III | 𝑷II = 𝑷II 𝑯in 𝑷II = 𝐽in ∑︁ wt(𝒙)|𝜂 𝒙 ⟩⟨𝜂 𝒙 | ≡ 𝐽in 𝒙 𝑛 ∑︁ ( 𝑰 − 𝒁 eff 𝑗 )/2, 𝑗=1 eff 𝑥𝑗 where 𝒁 eff 𝑗 is the Pauli Z operator of a virtual qubit defined as 𝒁 𝑗 |𝜂 𝒙 ⟩ = (−1) |𝜂 𝒙 ⟩. Observe that 𝑯˜ eff has eigenvalues that are integer multiples of 𝐽in , and thus its Bohr-frequency gap is Δ𝜈 ( 𝑯˜ eff ) = 𝐽in . Hence, assuming 𝐽in /2 < 𝐽prop − 2∥𝑯in ∥, we can apply Lemma 4.12.6 to see that the gradient operator sandwiched by 𝑷II can be understood by fully restricting to the subspace: ∫ 0 ∑︁ 𝐸˜ 𝑎 ∑︁ † 𝑎† 𝑎 ˜ 𝑷II L̃III [ 𝑯III ] 𝑷II ≈ 𝑷II 𝑨𝜈 𝑷II 𝑨𝜈 𝑷II 𝛾 𝛽 (𝜔)𝜔| 𝑓ˆ𝜇 (𝜔 − 𝜈)| 2 d𝜔, −∞ 𝑎∈𝑆0 𝜈∈𝐵( 𝑯˜ eff ) (4.77) where 𝜇 = 𝐽in /2 and  ˜ III ∥ 3/4 1 1 1 ∥ 𝑯 + + +√ . 𝐸˜ 𝑎 = |𝑆0 |O 𝜏 𝛽 𝜏 1/4 𝐽in 𝜏  To show that this has good gradients for all states in 𝑷II ( 𝑰 − 𝑷III ), it is sufficient to consider the following subset of the jump operators from Eq. (4.52):  𝑛 𝑆III = 𝑿 𝑗 ⊗ |0⟩⟨0| 𝑡 𝑗 𝑗=1 . These jump operators from 𝑆III effectively flip the individual virtual qubits by ⟨𝜂 𝒚 |( 𝑿 𝑗 ⊗ |0⟩⟨0| 𝑡 𝑗 )|𝜂 𝒙 ⟩ = ⟨𝒚| 𝑿 𝑗 |𝒙⟩ where we have denoted 𝛼 𝑗 = ( Í𝑡 𝑗 −1 𝑡=0 √ 𝜉𝑡 =: ⟨𝒚| 𝑿 𝑗 |𝒙⟩ 𝛼 𝑗 , 2 𝑡<𝑡 𝑗 𝜉𝑡 ) . Let Í √ 𝑿 eff 𝑗 = 𝑷II ( 𝑿 𝑗 ⊗ |0⟩⟨0| 𝑡 𝑗 )𝑷II / 𝛼 𝑗 such that ∥ 𝑿 eff 𝑗 ∥ = 1 for each 𝑗 = 1, . . . , 𝑛. Note 𝑿 eff 𝑗 is effectively the Pauli X operator for the 𝑗-th virtual qubit. Furthermore, note since 𝑡 𝑗 ∈ [𝑡0 , 𝑇 − 𝑡0 ] by our circuit construction in Definition 4.14.1, we have 𝛼 𝑗 ≥ 𝜉𝑡20 ≥ Ω(1/𝑇 2 ) by Proposition 4.14.1. 284 Next, we replace 𝑯˜ III by 𝑯˜ eff so we only need to talk about the virtual qubits. We first restore the RHS of Eq. (4.77) to the thermal Lindbladian form by undoing the approximation in Lemma 4.12.6, which incurs another error bounded by 𝐸˜ 𝑎 : 2𝐸˜ 𝑎 † 𝑷II L̃III [ 𝑯˜ III ] 𝑷II ≈ L †𝛽,𝜏,𝑯eff [𝑯eff ]. We then focus on the subset 𝑆III of jump operators and write 𝐸˜ 𝑏 Í𝑛 † ˜ 𝑗=1 𝛼 𝑗 L̃ 𝑗 [ 𝑯eff ], L †𝛽,𝜏,𝑯eff [𝑯eff ] ≈ (4.78) 𝛽,𝜏, 𝑯˜ eff where we denoted L̃ 𝑗 := L 𝑗 to be the thermal Lindbladian associated with eff effective jump operator 𝑿 𝑗 and pulled out the normalization factor 𝛼 𝑗 . The error   ∥ 𝑯˜ eff ∥ 3/4 1 1 ˜ 𝐸 𝑏 = |𝑆0 |O + + 𝜏 𝛽 𝜏 1/4 comes from neglecting the other jump operators 𝑆0 \ 𝑆III , and is bounded using Lemma 4.12.1. Since the effective operators on different virtual qubits commute, we can treat them †𝛽,𝜏,𝒉 𝑗 independently. More formally, by Lemma 4.12.8 we have L̃ †𝑗 [ 𝑯˜ eff ] = L 𝑗 [𝒉 𝑗 ], 𝑗 where 𝒉 𝑗 = 𝐽in ( 𝑰 − 𝒁eff )/2. To bound the global gradient in Eq. (4.78), we first consider cooling a single qubit. Lemma 4.14.5 (Cooling a qubit). On a qubit, consider the thermal Lindbladian 𝛽,𝜏,𝑯 L = L𝑎 with the Hamiltonian 𝑯 = 𝐽in ( 𝑰 − 𝒁)/2 and one jump operator 𝑨 = 𝑿. Then, −L † [𝑯] ⪰ 𝑟 in ( 𝑰 − 𝒁) − 𝜖in 𝑰, where 𝑟 in = Ω 𝐽  in ln 𝛽 and 𝜖 in = O  𝐽 3/4 1 1 in + + √ 𝜏 1/4 𝜏 𝜏𝐽  + e−𝛽𝐽in . in Proof. Again, we invoke a series of approximations: L † [𝑯] ≈ D † [𝑯] ∫ ∞ ≈ 𝛾 𝛽 (𝜔)𝜔 𝑨(𝜔) † 𝑨(𝜔)d𝜔 ∫−∞∞ ≈ 𝛾 𝛽 (𝜔)𝜔𝑺(𝜔) † 𝑺(𝜔)d𝜔 −∞ ∫ ∞ ∑︁ 2 † = 𝑨𝜈 𝑨𝜈 𝛾 𝛽 (𝜔)𝜔 𝑓ˆ𝜇 (𝜔 − 𝜈) d𝜔 𝜈∈𝐵(𝑯) −∞ ⪯ −Ω(𝐽in /ln 𝛽)|1⟩⟨1| + O (e−𝛽𝐽in )|0⟩⟨0|. 285 The last line uses the transition matrix elements for the two Bohr frequencies 𝜈 = ±𝐽in : 𝑨+1 = |1⟩⟨0|, and 𝑨−1 = |0⟩⟨1|. Combine the error bound to conclude the proof. ■ Summing up over the contributions from the individual qubits, the global gradient satisfies 𝑛 h i ∑︁ † ˜ −𝑷II L̃III [ 𝑯III ] 𝑷II ⪰ 𝛼 𝑗 𝑷II 𝑟 in ( 𝑰 − 𝒁 eff ) − 𝜖 𝑰 𝑷II − (2𝐸˜ 𝑎 + 𝐸˜ 𝑏 ) 𝑰 in 𝑗 𝑗=1 ⪰ 𝑟 3 𝑷II ( 𝑰 − 𝑷III ) − 𝜖˜3 𝑰, Í where we used the fact that 𝑛𝑗=1 ( 𝑰 − 𝒁 eff 𝑗 ) ⪰ 𝑷III , and denoted 𝑟 3 = 𝑟 in min 𝑗 𝛼 𝑗 , Í𝑛 ˜ ˜ and 𝜖˜3 = 2𝐸 𝑎 + 𝐸 𝑏 + 𝜖in 𝑗=1 𝛼 𝑗 . Since 𝛼 𝑗 = Ω(1/𝑇 2 ) and 𝛼 𝑗 ≤ 1, we have that 𝑟3 = Ω  𝐽  in 𝑇 2 ln 𝛽 and 𝜖˜3 ≤ 2𝐸˜ 𝑎 +𝐸˜ 𝑏 +𝑛𝜖in = |𝑆0 |O  ∥ 𝑯˜ III ∥ 𝜏 1/4 3/4  1 1 1 + + +√ +e−𝛽𝐽in . 𝜏 𝛽 𝐽in 𝜏 (4.85) Lastly, to obtain the gradient for the final Hamiltonian 𝑯III , we need to add the off-block-diagonal perturbation and show that the gradient persists on the subspace. Directly applying subspace monotonicity (Corollary 4.12.1) yields loose bounds; we will need to invoke a finer-grained subspace monotonicity (Corollary 4.12.2) that exploits the fact that 𝑽 is off-block-diagonal and contribute to eigenvalue change at second order O (∥𝑽 ∥ 2 ). We apply Corollary 4.12.2 with 𝑯 = 𝑯˜ III and 𝑯′ = 𝑯III , and parameters 𝑸 = 𝑷II Δ𝑸 = 𝐽prop − 2∥𝑯in ∥ 𝑽 = 𝑷II 𝑯in ( 𝑰 − 𝑷II ) + ( 𝑰 − 𝑷II )𝑯in 𝑷II Δ𝜈 = 𝐽in Therefore, by Corollary 4.12.2, † ′ ′ ′ LIII [𝑯III ] 𝑷II ⪰ 𝑟 3 𝑷II ( 𝑰 − 𝑷III ) − 𝜖3𝑎 𝑰, −𝑷II where 2/3 1 ∥ 𝑯˜ III ∥ 3/4 Λ0 Λ0 Λ0 e−𝛽Δ𝑸 /4 𝜖3𝑎 ≤ 𝜖˜3 + |𝑆0 | · O + + + + + √ √︁ 𝜏 𝛽 𝜏 1/4 𝜏 1/3 Δ𝜈 𝜏 Δ𝑸 𝜏  ∥𝑯 ∥∥𝑽 ∥ ∥𝑽 ∥ 2   ∥𝑽 ∥ ∥𝑽 ∥ 2   ∥𝑽 ∥ 2 𝑸 + ∥𝑯𝑸 ∥ · + + 𝑟3 + + . Δ𝑸 Δ𝑸 Δ 𝜈 Δ𝑸 Δ 𝜈 Δ𝑸 Δ 𝜈 Δ𝑸  286 Noting that |𝑆0 |, ∥ 𝑯˜ III ∥ = O (𝑇), ∥𝑽 ∥ = O (𝑛𝑇 𝐽in ), ∥𝑯𝑸 ∥ = 𝑛𝐽in , and 𝐽in ≪ 𝐽prop , we simplify the error bound above by keeping the dominant term to get  3/4  1 𝑇 1 (𝑛𝑇 𝐽in ) 2 −𝛽𝐽in 𝜖3𝑎 ≤ 𝑇 O 1/4 + + √ +e +𝑛 (4.91) 𝛽 𝐽prop 𝜏 𝐽in 𝜏 as advertised earlier in Eq. (4.59). 4.15 Appendix:Operator Fourier Transform Recall that the exact form of thermal Lindbladians in Appendix 4.10 involves the operator Fourier transform (OFT) [39] for a set of jump operators 𝑨𝑎 . In this appendix, we provide the key properties of OFT which are used in the proofs of many statements in Appendices 4.12 and 4.16. For any operator 𝑨 (the jump operator), and Hermitian operator 𝑯 (the Hamiltonian), and a weight function 𝑓 , the operator Fourier Transform (OFT) is an integral over time-evolution of the operator 𝑨, ∫ ∞ 1 ˆ 𝑨 𝑓ˆ (𝜔) := √ ei𝑯𝑡 𝑨e−i𝑯𝑡 e−i𝜔𝑡 𝑓 (𝑡)d𝑡. 2𝜋 −∞ ˆ Often, we will also write 𝑨(𝜔) when we choose 𝑓 to be the default normalized window function    1 if 𝑡 ∈ [−𝜏/2, 𝜏/2] 1  𝑓𝜏 (𝑡) = √ · (4.92) 𝜏   0 else.  Í It is usually helpful to consider the energy eigenspaces 𝑯 = 𝑖 𝐸𝑖 |𝜓𝑖 ⟩⟨𝜓𝑖 | = Í 𝐸 ∈Spec(𝑯) 𝐸 𝑷 𝐸 and and write 𝑨 as the following decomposition ∑︁ ∑︁ ∑︁ 𝑨= 𝑷 𝐸2 𝑨𝑷 𝐸1 = 𝑨𝜈 where 𝑨𝜈 := 𝑷 𝐸2 𝑨𝑷 𝐸1 . 𝐸 2 ,𝐸 1 ∈Spec(𝑯) 𝐸 2 −𝐸 1 =𝜈 𝜈∈𝐵(𝑯) Formally, these energy differences 𝜈 ∈ 𝐵(𝑯) := {𝐸𝑖 − 𝐸 𝑗 | 𝐸𝑖 , 𝐸 𝑗 ∈ Spec(𝑯)} are called the Bohr frequencies (Figure 4.2), and 𝑨𝜈 collects the matrix elements ⟨𝜓𝑖 | 𝑨 𝜓 𝑗 that changes the energy by 𝜈 = 𝐸𝑖 − 𝐸 𝑗 . The Bohr frequencies are natural for the Heisenberg-picture evolution of 𝑨 since ∑︁ ei𝑯𝑡 𝑨e−i𝑯𝑡 = ei𝜈𝑡 𝑨𝜈 . 𝜈∈𝐵(𝑯) Then, executing the Fourier integral yields the frequency domain representation ∫ ∞ ∑︁ 1 ˆ ˆ ˆ 𝑨 𝑓 (𝜔) = 𝑨𝜈 𝑓 (𝜔 − 𝜈) where 𝑓 (𝜔) := √ 𝑓 (𝑡)e−i𝜔𝑡 d𝑡, 2𝜋 −∞ 𝜈∈𝐵 which contains a collection of Bohr frequencies 𝜈 near 𝜔. Conceptually, we can think of the operator Fourier transform 𝑨ˆ 𝑓 (𝜔) as the smooth probe of 𝑨𝜈 with exact Bohr frequency 𝜈, which generally requires resolving arbitrarily close eigenvalues. 287 Useful properties We instantiate some useful properties of the Operator Fourier Transform. Proposition 4.15.1 (Operator Parseval’s identity [39]). Consider a set of operators { 𝑨𝑎 } 𝑎 and its operator Fourier transform with weight 𝑓 ∈ L2 (R) and Hamiltonian 𝑯. Then, we have a certain symmetry ( 𝑨𝑎𝑓 (𝜔)) † = 𝑨𝑎† 𝑓 ∗ (−𝜔) and certain Parseval-type identity ∑︁ ∫ ∞ ∑︁ ∫ ∞ 𝑎 † ˆ𝑎 ˆ 𝑨 𝑓 (𝜔) 𝑨 𝑓 (𝜔)d𝜔 = ei𝑯𝑡 𝑨𝑎† 𝑨𝑎 e−i𝑯𝑡 | 𝑓 (𝑡)| 2 d𝑡 −∞ 𝑎∈𝑆 𝑎∈𝑆 ⪯∥ ∑︁ −∞ 𝑨𝑎† 𝑨𝑎 ∥∥ 𝑓 ∥ 22 · 𝑰 𝑎∈𝑆 ∑︁ ∫ ∞ 𝑨ˆ 𝑎𝑓 (𝜔) 𝑨ˆ 𝑎𝑓 (𝜔) † d𝜔 = −∞ 𝑎∈𝑆 ∑︁ ∫ ∞ 𝑎∈𝑆 ⪯∥ ∑︁ ei𝑯𝑡 𝑨𝑎 𝑨𝑎† e−i𝑯𝑡 | 𝑓 (𝑡)| 2 d𝑡 −∞ 𝑨𝑎 𝑨𝑎† ∥∥ 𝑓 ∥ 22 · 𝑰. 𝑎∈𝑆 Intuitively, the above tells us that the Fourier Transforms 𝑨ˆ 𝑎𝑓 (𝜔) with different frequencies 𝜔 are “orthogonal” to each other and that the average of squares of strengths is bounded (reminiscent of a probability). Without using this norm sum constraint from Fourier transforms, one easily gets loose bounds. An alternative view of the above (as a natural purification) will prove useful for manipulating norms of expression involving 𝑨𝑎𝑓 (𝜔). Corollary 4.15.1 (Purification of Operator Fourier Transform). In the prevailing notation, the abstract operator Fourier Transform has a norm bound, √︄ ∑︁ ∑︁ ∫ ∞ 𝑎 ∥ 𝑨ˆ 𝑓 (𝜔) ⊗ |𝑎⟩ ⊗ |𝜔⟩d𝜔∥ ≤ ∥ 𝑓 ∥ 2 ∥ 𝑨𝑎† 𝑨𝑎 ∥, 𝑎∈𝑆 −∞ 𝑎∈𝑆 where the continuous basis vectors satisfy the normalization ⟨𝜔′ |𝜔⟩ = 𝛿(𝜔′ − 𝜔). Proof. We multiply the conjugate ! ! ∑︁ ∫ ∞ ′ ∑︁ ∫ ∞ 𝑨ˆ 𝑎𝑓 (𝜔′) † ⊗ ⟨𝑎′ | ⊗ ⟨𝜔′ |d𝜔′ · 𝑨ˆ 𝑎𝑓 (𝜔) ⊗ |𝑎⟩ ⊗ |𝜔⟩d𝜔 𝑎 ′ ∈𝑆 = −∞ ∑︁ ∫ ∞ 𝑎∈𝑆 𝑎∈𝑆 −∞ 𝑨ˆ 𝑎𝑓 (𝜔) † 𝑨ˆ 𝑎𝑓 (𝜔)d𝜔. −∞ Take the operator norm and use Proposition 4.15.1 to conclude the proof. ■ 288 The above “purification” trick applies to other quantities with a liberal choice of summation indices, whether they are 𝑎, 𝜔, or 𝜈. Lemma 4.15.1 (Norm inequalities from operator purification). For any operator 𝑶 and any set of operators 𝑨𝑖 , 𝑨′𝑗 acting on the same Hilbert space, we have that † ′ 𝑖, 𝑗 𝑨𝑖 𝑶 𝑨 𝑗 𝐺 𝑖 𝑗 ≤ ∥𝑮 ∥∥𝑶 ∥ Í v u t ∑︁ ∑︁ 𝑨𝑖† 𝑨𝑖 𝑖 ∑︁ 𝑨𝑖† 𝑨′𝑗 𝐺 𝑖 𝑗 ≤ ∥𝑮 ∥ v u t ∑︁ 𝑖, 𝑗 ′ 𝑨 𝑗† 𝑨′𝑗 𝑗 ∑︁ 𝑨𝑖† 𝑨𝑖 𝑖 ′ 𝑨 𝑗† 𝑨′𝑗 𝑗 Proof. By homogeneity, it suffices to set normalization to be ∥ Í ∥ 𝑖 𝑨𝑖† 𝑨𝑖 ∥ = ∥𝑶 ∥ = ∥𝑮 ∥ = 1. Introduce purifications ∑︁ 𝑮 ′ := 𝑰 ⊗ 𝐺 𝑖 𝑗 |𝑖⟩⟨ 𝑗 |, 𝑶 ′ := 𝑶 ⊗ 𝑰, Í ′† ′ 𝑗 𝑨𝑗 𝑨𝑗 ∥ = 𝑖, 𝑗 𝑽 ′ := ∑︁ 𝑨′𝑗 ⊗ | 𝑗⟩, 𝑽 := 𝑗 ∑︁ 𝑨𝑖 ⊗ |𝑖⟩, 𝑖 which are all bounded by ∥𝑮 ′ ∥, ∥𝑶 ′ ∥, ∥𝑽 ′ ∥, ∥𝑽 ∥ ≤ 1. Then, ∑︁ 𝑨𝑖† 𝑶 𝑨′𝑗 𝐺 𝑖 𝑗 = ∥𝑽 ′𝑶 ′ 𝑮 ′𝑽 ∥ ≤ 1 𝑖, 𝑗 ∑︁ 𝑨𝑖† 𝑨′𝑗 𝐺 𝑖 𝑗 = ∥𝑽 ′ 𝑮 ′𝑽 ∥ ≤ 1. 𝑖, 𝑗 Rescale to obtain the advertised result. ■ Secular approximation Due to the energy-time uncertainty principle, the energies 𝜔 that are accessed by finite-time quantum algorithms always inherit an uncertainty. Indeed, when we choose our weight function 𝑓𝜏 (𝑡) in Eq. (4.92) for our operator Fourier transform in the frequency domain, we have 𝑨ˆ 𝑓 (𝜔) = ∑︁ 𝜈∈𝐵 𝑨𝜈 𝑓ˆ(𝜔 − 𝜈) where i𝜔𝜏/2 − e−i𝜔𝜏/2 e 𝑓ˆ(𝜔) = √ i𝜔 2𝜋𝜏 when 𝑓 (𝑡) = 𝑓𝜏 (𝑡). Note with this choice, 𝑓ˆ(𝜔) has a heavy tail ∼ 1/𝜔, which is reminiscent of unamplified phase estimation. Therefore, even when restricting to jumps with 𝜔 < 0, 289 there is a decent chance that 𝑨ˆ 𝑓 (𝜔) mistakenly activates a heating transition (𝜈 > 0) instead of a cooling transition (𝜈 < 0), unintentionally heating up the system instead of cooling it. To control the resulting error, in this section, we introduce the secular approximation [39] of the Fourier transformed operators 𝑨 𝑓 (𝜔). The secular approximation applies truncation to the Fourier-transformed operators in the frequency domain by truncating Bohr frequencies 𝐸 ∈ 𝐵 that deviate substantially from the frequency label 𝜔. Truncation at energy difference 𝜇 can be achieved by setting a step function and defining the following secular-approximated operators as follows: ∑︁ 𝑺ˆ 𝑓 ,𝜇 (𝜔) := 𝑨𝜈 𝑓ˆ(𝜔 − 𝜈) · 𝑠ˆ 𝜇 (𝜔 − 𝜈) where 𝑠ˆ 𝜇 (𝜔) := 1 (|𝜔| < 𝜇). 𝜈∈𝐵 We often drop subscript 𝑓 , 𝜇 for simplicity. The truncation error is an operator whose norm can be bounded by the following (as a variant of Corollary 4.15.1). Corollary 4.15.2 (Secular approximation). In the prevailing notation, √︄ ∑︁ ∑︁ ∫ ∞ 𝑎 𝑎 𝑨𝑎† 𝑨𝑎 ∥. ∥ ( 𝑺ˆ 𝑓 (𝜔) − 𝑨ˆ 𝑓 (𝜔)) ⊗ |𝑎⟩ ⊗ |𝜔⟩d𝜔∥ ≤ ∥ 𝑓ˆ · (1 − 𝑠ˆ 𝜇 )∥ 2 ∥ 𝑎∈𝑆 −∞ 𝑎∈𝑆 The error is controlled by the 2-norm for the truncated tail ∫ 2 ˆ ∥ 𝑓 · (1 − 𝑠ˆ 𝜇 )∥ 2 = | 𝑓ˆ(𝜔)| 2 d𝜔 |𝜔|≥𝜇 noting that conveniently, the Fourier transform preserves the 2-norm of functions. For our bump function (4.92) in particular, we can integrate the tail-bound ∥ 𝑓ˆ𝜏 · (1 − 𝑠ˆ 𝜇 )∥ 22 ≤ 4 . 𝜋𝜇𝜏 (4.93) This conveniently leads to bounds on other quantities involving operator Fourier Transforms 𝑨ˆ 𝑎 (𝜔). Corollary 4.15.3 (Error from secular approximation). For any real function 𝜃 : R → R, ∑︁ ∫ ∞ 𝑎∈𝑆 −∞ ′ 𝜃 (𝜔 ) 𝑨ˆ 𝑎𝑓 (𝜔′) † 𝑨ˆ 𝑎𝑓 (𝜔′)d𝜔′ − ∑︁ ∫ ∞ 𝑎∈𝑆 𝜃 (𝜔′) 𝑺ˆ 𝑎𝑓 ,𝜇 (𝜔′) † 𝑺ˆ 𝑎𝑓 ,𝜇 (𝜔′)d𝜔′ −∞ √︄ ∑︁ ≤ 2∥𝜃 ∥ ∞ ∥ 𝑓ˆ · (1 − 𝑠ˆ 𝜇 )∥ 2 ∥ 𝑓 ∥ 2 ∥ 𝑨𝑎† 𝑨𝑎 ∥. 𝑎∈𝑆 290 Í Proof. It suffices to set normalization ∥ 𝑎∈𝑆 𝑨𝑎† 𝑨𝑎 ∥ = 1. Introduce purifications ∫ ∞ 𝑭 := 𝑰 ⊗ 𝑰 ⊗ 𝜃 (𝜔)|𝜔⟩⟨𝜔|d𝜔, −∞ ! ∑︁ ∫ ∞ 𝑨ˆ 𝑎 (𝜔) ⊗ |𝑎⟩ ⊗ |𝜔⟩d𝜔 , 𝑽 := 𝑓 𝑎∈𝑆 ′ 𝑽 := −∞ ∑︁ ∫ ∞ 𝑎∈𝑆 ! 𝑺ˆ 𝑎𝑓 (𝜔) ⊗ |𝑎⟩ ⊗ |𝜔⟩d𝜔 . −∞ Then, by a telescoping sum, ′ ′ ′ (𝐿𝐻𝑆) = ∥𝑽 † 𝑭𝑽 − 𝑽 † 𝑭𝑽 ′ ∥ ≤ ∥(𝑽 † − 𝑽 † )𝑭𝑽 ∥ + ∥𝑽 † 𝑭(𝑽 − 𝑽 ′)∥. We conclude the proof using the individual bounds on the operator norm ∥𝑽 ∥ ≤ ∥ 𝑓 ∥ 2 ∥𝑽 ′ ∥ ≤ ∥ 𝑓ˆ · 𝑠ˆ 𝜇 ∥ 2 ≤ ∥ 𝑓 ∥ 2 ∥𝑭∥ ≤ ∥𝜃 ∥ ∞ ∥𝑽 ′ − 𝑽 ∥ ≤ ∥ 𝑓ˆ · (1 − 𝑠ˆ 𝜇 )∥ 2 . This concludes the proof. ■ 4.16 Appendix:Proving monotonicity of energy gradient under level splitting It will often be helpful to understand how the energy gradients of a Hamiltonian 𝑯 change when a perturbation 𝑽 is added to yield 𝑯′ = 𝑯 + 𝑽. This allows us to characterize the energy gradient of 𝑯′ by analyzing the unperturbed spectrum of 𝑯, which is usually much simpler than that of 𝑯′. Indeed, this is an important part of our proof strategy for showing our key result that 𝑯𝐶 has no suboptimal local minima (Theorem 4.14.1 in section 4.14). The relationship we can prove, which was previously stated in Theorem 4.12.1 in section 4.12, takes the form of monotonicity. As the name implies, the result only holds in one direction; it fails when the 𝑯′ and 𝑯 are switched. It is imperative that 𝑯 have a highly degenerate spectrum with the Bohr-frequency gap Δ𝜈 (𝑯) := min 𝜈1 ≠𝜈2 ∈𝐵(𝑯) |𝜈1 − 𝜈2 |, 291 which sets an energy scale for which 𝑽 is perturbative. Note that the Bohr-frequency gap is upper bounded by the gap in the spectrum10 Δ𝜈 ≤ Δ 𝐸 where Δ𝐸 = min 𝐸 1 ≠𝐸 2 ∈Spec(𝑯) |𝐸 1 − 𝐸 2 |. We now state a more general version of Theorem 4.12.1, which we prove in the remainder of this appendix. Theorem 4.16.1 (Monotonicity of gradient under level splitting, expanded version). Consider a Hamiltonian 𝑯 with a highly degenerate spectrum and Bohr-frequency gap Δ𝜈 := min𝜈1 ≠𝜈2 ∈𝐵(𝑯) |𝜈1 − 𝜈2 |, and a perturbed Hamiltonian 𝑯′ = 𝑯 + 𝑽. Suppose the perturbation is weaker than the Bohr-frequency gap, ∥𝑽 ∥ ≤ 18 Δ𝜈 . For Í Í 𝛽,𝜏,𝑯 ′ 𝛽,𝜏,𝑯 any 𝛽, 𝜏 > 0, let L = 𝑎∈𝑆 L𝑎 , L ′ = 𝑎∈𝑆 L𝑎 be thermal Lindbladians 𝑎 𝑎 with jumps { 𝑨 } 𝑎∈𝑆 , where ∥ 𝑨 ∥ ≤ 1 and the transition weight 𝛾 𝛽 (𝜔) is given by Eq. (4.21). Let 𝛿𝜆 = max 𝑗 |𝜆 𝑗 (𝑯) − 𝜆 𝑗 (𝑯′)|, where 𝜆 𝑗 ( 𝑿) is the 𝑗-th largest eigenvalue of 𝑿, and let 𝜃 max = max𝜈∈𝐵(𝑯) |𝜈𝛾 𝛽 (𝜈) 1 (𝜈 ≤ Δ𝜈 /2)|. For any two operators 𝑶 and 𝑶 ′, where [𝑶 ′, 𝑯′] = 0, we have the monotone property that −L † [𝑯] ⪰ 𝑟𝑶 − 𝜖 𝑰 implies ′ − L † [𝑯′] ⪰ 𝑟𝑶 ′ − 𝜖 ′ 𝑰, where ! 2/3 3/4 −𝛽Δ𝜈 /4 Λ ∥𝑽 ∥ 1 ∥𝑯∥ e Λ 0 0 + 𝛿𝜆 + 𝜃 max + 𝑟 ∥𝑶 − 𝑶 ′ ∥ . + 𝜖 ′ ≤ 𝜖+|𝑆|·O + + 1/3 +√ 1/4 𝜏 𝛽 Δ 𝜏 𝜏 Δ𝜈 𝜏 𝜈 For the special case of 𝑶 = 𝑰 − 𝑷 and 𝑶 ′ = 𝑰 − 𝑷′, where 𝑷 projects onto an eigensubspace of 𝑯, and 𝑷′ projects onto the corresponding perturbed eigensubspace in 𝑯′, then we have the following simpler error bound: ! 2/3 −𝛽Δ /4 3/4 𝜈 Λ
1 ∥𝑯∥ Λ0 e Λ0 + 𝑟 0 + 𝜖 ′ ≤ 𝜖 + |𝑆| · O + + 1/3 +√ + 1+ ∥𝑽 ∥ . 1/4 𝜏 𝛽 Δ𝜈 𝜏 𝜏 Δ𝜈 𝜏 The above result is nontrivial because naïve perturbation theory fails: the Lindbladian depends sensitively on the perturbation 𝑽 (as it uses a long Hamiltonian simulation time 𝜏∥𝑽 ∥ ≫ 1). In fact, it drastically fails if the energy spectrum of 𝑯 has a (nearly) continuous spectrum (as the opposite of the premise of gapped degenerate subspaces). 10The Bohr-frequency gap can be much smaller than the eigenvalue gap. For example, consider the energies {−0.99, 0, 1}, which has an eigenvalue gap of 0.99 and a Bohr-frequency gap of 0.01. 292 We can understand the energy scale associated with the minimum Bohr-frequency gap Δ𝜈 as the meaningful quantity for which 𝑽 is a perturbation 1 ≪ ∥𝑽 ∥ ≪ Δ𝜈 . 𝜏 Otherwise, the 1/𝜏 energy resolution is too small compared to the intended perturbation ∥𝑽 ∥. The proof of Theorem 4.16.1 will be quite involved. Technically, we heavily utilize the manipulations using the operator Fourier Transform (Appendix 4.15). The key subroutines of the proof are discussed separately as follows. First, in Appendix 4.16, we will simplify the intimidating expression for energy gradient L † [𝑯]. Secondly, in Appendix 4.16, we isolate the key nonperturbative argument, which roughly says level splitting only improves the gradient. We then provide some results from perturbation theory in section 4.16. The altogether proof is presented in Appendix 4.16, with minor supporting calculations in section 4.16. We also prove two corollaries of Theorem 4.16.1 that apply to subspace gradients in Appendix 4.16. Since this appendix only consider thermal Lindbladians, in what follows we will ′ drop the superscripts 𝛽, 𝜏, 𝑯, i.e. L ≡ L 𝛽,𝜏,𝑯 , L ′ ≡ L 𝛽,𝜏,𝑯 , D ≡ D 𝛽,𝜏,𝑯 , etc. Expressing the energy gradient The thermal Lindbladian is quite cumbersome to manipulate. Nicely, the energy gradient operator associated with the dissipative part D † [𝑯] permits a much simpler approximate form up to a controllable error. Combining with error bounds on the Lamb-shift term [𝑯 𝐿𝑆 , 𝑯] (Proposition 4.10.3) allows us to approximate the full gradient operator L † [𝑯]. Lemma 4.16.1 (Expression for energy gradient). Consider the operator Fourier Transforms 𝑨ˆ 𝑎 (𝜔) weighted by the bump function 𝑓𝜏 in Eq. (4.92) with Hamiltonian 𝑯. Then, for any Fourier transform pairs 𝛾(𝜔) and 𝑐(𝑡), the energy gradient associated with the purely dissipative Lindbladian   ∑︁ ∫ ∞ 1 ˆ𝑎 † ˆ𝑎 † 𝑎 † 𝑎 ˆ ˆ D [𝑯] = 𝛾(𝜔) 𝑨 (𝜔) 𝑯 𝑨 (𝜔) − { 𝑨 (𝜔) 𝑨 (𝜔), 𝑯} d𝜔 2 −∞ 𝑎∈𝑆 can be well approximated as a simpler form using ∑︁ ∫ ∞ ∑︁ 2 † ∥D [𝑯] − 𝛾(𝜔)𝜔 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔)d𝜔∥ ≤ √ · ∥𝑐∥ 1 · ∥ 𝑨𝑎† 𝑨𝑎 ∥. 2𝜋𝜏 𝑎∈𝑆 −∞ 𝑎∈𝑆 293 Intuitively, the following expression is the simplest proxy one can write down to capture the rate of energy change ∑︁ ∫ ∞ 𝜔 × 𝛾(𝜔) 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔)d𝜔 ∼ (energy difference) × (rate). 𝑎∈𝑆 −∞ Indeed, the Bohr frequency 𝜔 is essentially the energy difference after jump operator 𝑨𝑎 (𝜔); but because of the energy uncertainty in the operator Fourier transform (i.e., because 𝑓ˆ(𝜔) is not a delta function), this interpretation must be corrected by an error scaling as the energy resolution ∼ 1/𝜏. The starting point of the calculation is a certain integration-by-part trick that relates the Hamiltonian operator 𝑯 to the scalar 𝜔. Proposition 4.16.1 (Integration by parts). In the setting of Lemma 4.16.1,  1  ˆ ˆ [𝑯, 𝑨(𝜔)] = 𝜔 𝑨(𝜔) + √ 𝑨(𝜏/2)e−i𝜔𝜏/2 − 𝑨(−𝜏/2)ei𝜔𝜏/2 . i 2𝜋𝜏 Proof. Integration of the derivative can be expanded by the product rule 1  −i𝜔𝜏/2 i𝜔𝜏/2  𝑨(𝜏/2)e − 𝑨(−𝜏/2)e √ 2𝜋𝜏 ∫ 𝜏/2   d 1 𝑨(𝑡)e−i𝜔𝑡 d𝑡 =√ 2𝜋𝜏 −𝜏/2 d𝑡 ∫ 𝜏/2   1 =√ i[𝑯, 𝑨(𝑡)]e−i𝜔𝑡 − i𝜔 𝑨(𝑡)e−i𝜔𝑡 d𝑡 2𝜋𝜏 −𝜏/2 ˆ ˆ = i[𝑯, 𝑨(𝜔)] − i𝜔 𝑨(𝜔). Rearrange to conclude the proof. ■ Observe that taking the infinite time limit 𝜏 → ∞ (i.e., perfect energy resolution) in the above proposition recovers the relation for the true Bohr frequencies 𝜈 [𝑯, 𝑨𝜈 ] = 𝜈 𝑨𝜈 for each 𝜈 ∈ 𝐵(𝑯). At finite 𝜏, the above leads to simple bounds on the correction term. We now present the proof of Lemma 4.16.1. 294 Proof of Lemma 4.16.1. We calculate   ∑︁ ∫ ∞ 1 † 𝑎 † 𝑎 𝑎 † 𝑎 D [𝑯] = 𝛾(𝜔) 𝑨ˆ (𝜔) 𝑯 𝑨ˆ (𝜔) − { 𝑨ˆ (𝜔) 𝑨ˆ (𝜔), 𝑯} d𝜔 2 −∞ 𝑎∈𝑆   ∑︁ ∫ ∞ 1 ˆ𝑎 † 𝑎 𝑎 † ˆ𝑎 ˆ ˆ = 𝛾(𝜔) 𝑨 (𝜔) [𝑯, 𝑨 (𝜔)] − [𝑯, 𝑨 (𝜔) ] 𝑨 (𝜔) d𝜔 2 𝑎∈𝑆 −∞ ∫ ∞ ∑︁ = 𝛾(𝜔)𝜔 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔)d𝜔 + 𝑬, −∞ 𝑎∈𝑆 where the error term 𝑬 is given by Proposition 4.16.1 as ∫   −i ∑︁ ∞ 𝛾(𝜔) 𝑨ˆ 𝑎 (𝜔) † 𝑨𝑎 (𝜏/2)e−i𝜔𝜏/2 − 𝑨𝑎 (−𝜏/2)ei𝜔𝜏/2 d𝜔 𝑬 := √ 2 2𝜋𝜏 𝑎∈𝑆 −∞   ∑︁ ∫ ∞ i 𝛾(𝜔) 𝑨𝑎 (𝜏/2) † ei𝜔𝜏/2 − 𝑨𝑎 (−𝜏/2) † e−i𝜔𝜏/2 𝑨ˆ 𝑎 (𝜔)d𝜔. + √ 2 2𝜋𝜏 𝑎∈𝑆 −∞ To bound this error term, let us calculate one of the four individual terms as an example ∑︁ ∫ ∞ 𝛾(𝜔) 𝑨𝑎 (𝜏/2) † 𝑨ˆ 𝑎 (𝜔)ei𝜔𝜏/2 d𝜔 𝑎∈𝑆 −∞ ∫ ∞∫ ∞ ∫ 𝜏/2 1 −i𝜔𝑡1 = 𝑐(𝑡1 )e d𝑡1 ei𝜔𝜏/2 𝑨𝑎 (𝜏/2) † 𝑨𝑎 (𝑡2 )e−i𝜔𝑡2 d𝑡 2 d𝜔 √ 2𝜋 𝜏 −∞ −∞ −𝜏/2 𝑎∈𝑆 ∫ 𝜏/2 1 ∑︁ 𝑐(𝜏/2 − 𝑡2 ) 𝑨𝑎 (𝜏/2) † 𝑨𝑎 (𝑡 2 )d𝑡2 . =√ 𝜏 𝑎∈𝑆 −𝜏/2 ∑︁ √ We can bound the operator norm of this term by ∥𝑐∥ 1 / 𝜏 after applying the triangle inequality and using the fact that ∑︁ ∑︁ ∥ 𝑨𝑎 (𝜏/2) † 𝑨𝑎 (𝑡2 )∥ = ∥ 𝑨𝑎† (𝜏/2) 𝑨𝑎 (𝑡2 )∥ 𝑎∈𝑆 ≤∥ ∑︁ 𝑨 𝑎† ∑︁ 𝑎∈𝑆 ∑︁ ⊗ ⟨𝑎|∥ · ∥ 𝑨𝑎 ⊗ |𝑎⟩∥ = ∥ 𝑨𝑎† 𝑨𝑎 ∥. 𝑎∈𝑆 𝑎∈𝑆 𝑎∈𝑆 Repeat a similar argument for the other three terms to conclude the proof. ■ Monotonicty of rates Thermal Lindbladians generally depend sensitively on the Hamiltonian as it uses Hamiltonian simulation for a long time 𝜏 ei𝑯𝜏 for 𝜏 ≫ 1. 295 Therefore, even adding a small perturbation to the Hamiltonian 𝑯′ = 𝑯 + 𝑽 may have a nonperturbative effect on L since 𝜏∥𝑽 ∥ ≫ 1 implies ∥L − L ′ ∥ 1−1 ≫ 0. In other words, at a large 𝜏, it is not obvious at all why the Lindbladians L, L ′ are related. Indeed, the original Davies’ generator (𝜏 → ∞) is unstable against arbitrarily small perturbations to the Hamiltonian; whenever energy degeneracy is broken, the Lindbladian can change substantially. Nevertheless, what we can show as a compromise is that the rate only increases if the perturbation only introduces level splitting; this amounts to the assumption that the original Hamiltonian has highly degenerate subspaces with a certain Bohr frequencies gap Δ𝜈 as another large energy scale 1 ≪ ∥𝑽 ∥ ≪ Δ𝜈 . 𝜏 Intuitively, level splitting causes decoherence (and only decoherence) in the Bohr frequencies; for large 𝜏, the Lindblidan can indeed tell the transitions 𝜔, 𝜔′ apart if the Bohr frequencies are sufficiently different. Fortunately, even though decoherence can change the Lindbladian by a lot, we establish certain monotonicity of transition rates. That is, L ′ must have as good transition rates as L. A good example of 𝑶 would be an energy subspace projector. However, the argument works for general 𝑶, which makes it more flexible to use. Lemma 4.16.2 (Decoherence increases the rates). For any set of operators { 𝑨𝑎 } 𝑎∈𝑆 , suppose there exists an operator 𝑶 such that ∑︁ 𝑨𝑎† 𝑨𝑎 ⪰ 𝑶 where [𝑶, 𝑯] = 0. 𝑎∈𝑆 Then, the operator Fourier Transforms 𝑨ˆ 𝑎 (𝜔) (subscript 𝑓 omitted) for some ∫∞ normalized weight −∞ | 𝑓 (𝑡)| 2 d𝑡 = 1 and Hamiltonian 𝑯 satisfies ∑︁ ∫ ∞ 𝑎∈𝑆 −∞ 𝑨ˆ 𝑎 (𝜔) † 𝑨ˆ 𝑎 (𝜔)d𝜔 ⪰ 𝑶. 296 Proof. By Proposition 4.15.1, ∑︁ ∫ ∞ 𝑎∈𝑆 ˆ𝑎 ∫ ∞ † ˆ𝑎 𝑨 (𝜔) 𝑨 (𝜔)d𝜔 − 𝑶 = −∞ ! i𝑯𝑡 e ∑︁ −∞ 𝑎∈𝑆 ∫ ∞ ∑︁ = −∞ ∫ ∞ = 𝑨 𝑨 e−i𝑯𝑡 | 𝑓 (𝑡)| 2 d𝑡 − 𝑶 𝑎† 𝑎 ! ei𝑯𝑡 𝑨𝑎† 𝑨𝑎 − 𝑶 e−i𝑯𝑡 | 𝑓 (𝑡)| 2 d𝑡 𝑎∈𝑆 ei𝑯𝑡 𝑿 † · 𝑿e−i𝑯𝑡 | 𝑓 (𝑡)| 2 d𝑡 −∞ ⪰ 0. ∫∞ The second equality uses that −∞ | 𝑓 (𝑡)| 2 d𝑡 = 1 and that ei𝑯𝑡 𝑶e−i𝑯𝑡 = 𝑶. The last line establishes PSD order using the assumption that there exists operator 𝑿 such Í that 𝑎∈𝑆 𝑨𝑎† 𝑨𝑎 − 𝑶 = 𝑿 † 𝑿. Together, we establish the desired statement. ■ Perturbation theory of eigenstates and eigenvalues We state a few useful facts about perturbed eigenspace and eigenvalues that would be useful in the proofs. Proposition 4.16.2 (Davis-Kahan sin Θ theorem (see also Theorem VII.3.1 of [22])). Let 𝑯 and 𝑯˜ be two equal-sized Hermitian matrices. Let 𝑷 be the projector onto eigenstates of 𝑯 with eigenvalue in an interval [𝑎, 𝑏]. Let 𝑷˜ ⊥ be the projector onto eigenstates of 𝑯˜ with eigenvalues outside the interval [𝑎 − 𝛿, 𝑏 + 𝛿]. Then ˜ ∥𝑷 𝑷˜ ⊥ ∥ ≤ ∥𝑯 − 𝑯∥/𝛿. Here ∥ · ∥ is the spectral norm (or any unitarily invariant norm). Furthermore, the following fact bounds errors on perturbed eigenvalues: Proposition 4.16.3 (Weyl’s inequality). For any two equal-sized Hermitian matrices ˜ we have |𝜆 𝑗 (𝑯) − 𝜆 𝑗 ( 𝑯)| ˜ ≤ ∥𝑯 − 𝑯∥ ˜ for all 𝑗, where 𝜆 𝑗 ( 𝑿) is the 𝑗-th 𝑯 and 𝑯, largest eigenvalue of matrix 𝑿. Together, these facts imply that Lemma 4.16.3. Let 𝑯 and 𝑯˜ = 𝑯 + 𝑽 be Hamiltonians. Let 𝑷 be the projector onto eigenstates of 𝑯 with eigenvalues in some interval [𝑎, 𝑏], which are separated from the other eigenvalues by a gap of at least Δ. If ∥𝑽 ∥ ≤ Δ/4, then 297 1. There exists a spectral projector 𝑷˜ onto eigenstates of 𝑯˜ with eigenvalues in [𝑎 − Δ/4, 𝑏 + Δ/4], which are separated from the other eigenvalues by a gap of at least Δ/2. ˜ ≤ 8∥𝑽 ∥/Δ. 2. ∥𝑷 − 𝑷∥ Proof. The existence of 𝑷˜ (item 1) holds because of Proposition 4.16.3 above. Then observe that ˜ = ∥𝑷 − 𝑷 𝑷˜ + 𝑷 𝑷˜ − 𝑷∥ ˜ ≤ ∥𝑷 𝑷˜ ⊥ ∥ + ∥𝑷⊥ 𝑷∥ ˜ ≤ 8∥𝑽 ∥/Δ, ∥𝑷 − 𝑷∥ where the last inequality is obtained by applying Proposition 4.16.2 with 𝛿 = Δ/4 to ˜ bound ∥𝑷 𝑷˜ ⊥ ∥ and ∥𝑷⊥ 𝑷∥. ■ Lemma 4.16.4 (Off-block-diagonal perturbation). Consider a block diagonal Hermitian matrix 𝑫 = 𝑫 1 + 𝑫 2 , where the two blocks correspond to orthogonal subspace projectors 𝑷1 and 𝑰 − 𝑷1 = 𝑷2 and are separated by eigenvalue gap at least Δ. Add an off-block-diagonal Hermitian perturbation 𝑽 = 𝑽12 + 𝑽21 such that ∥𝑽 ∥ ≤ Δ/4. Then, there is an anti-Hermitian operator 𝑩 and an absolute constant 𝐶0 such that 𝑫 + 𝑽 = e−𝑩 𝑫e 𝑩 + (𝑫 + 𝑽 − e−𝑩 𝑫e 𝑩 ) where ∥ 𝑩∥ ≤ 𝐶0 ∥𝑽 ∥ , Δ and ∥e 𝑩 (𝑫 + 𝑽)e−𝑩 − 𝑫 ∥ ≤ 𝐶0 ∥𝑽 ∥ 2 . Δ This implies the sorted eigenvalues of 𝑫 are perturbed by 𝐶0 ∥𝑽 ∥ 2 /Δ. We remark that the scaling with respect to ∥𝑽 ∥ is consistent with perturbation theory: the angle change is first-order ∼ ∥𝑽Δ ∥ , and the eigenvalue change is second-order 2 ∼ ∥𝑽Δ∥ . Note that for diagonal perturbation, the eigenvalue change is only bounded by ∼ ∥𝑽 ∥. Proof. Observe that 𝑩 −𝑩 e (𝑫 + 𝑽)e = 𝑫 + (𝑽 + [𝑩, 𝑫]) + [𝑩, 𝑽] + ∞ ∑︁ 1 𝑘! 𝑘=2 ad 𝑘𝑩 (𝑫 + 𝑽). Let us choose 𝑩 to cancel the first order term in 𝑽, i.e., 𝑽 = −[𝑩, 𝑫]. (4.113) 298 We can solve for 𝑩 by working in the eigenbasis of 𝑫 = 𝑂 𝑖 𝑗 = ⟨𝜓𝑖 |𝑶|𝜓 𝑗 ⟩, we can rewrite Eq. (4.16) as 𝑉𝑖 𝑗 = 𝐵𝑖 𝑗 (𝐷 𝑖 − 𝐷 𝑗 ) or Í 𝐵𝑖 𝑗 = 𝑖 𝐷 𝑖 |𝜓𝑖 ⟩⟨𝜓𝑖 |. Then denoting 𝑉𝑖 𝑗 . 𝐷𝑖 − 𝐷 𝑗 Note 𝐵𝑖 𝑗 = 𝑉𝑖 𝑗 = 0 whenever |𝐷 𝑖 − 𝐷 𝑗 | ≥ Δ by assumption. Hence, we can solve for 𝑩 using the Heisenberg picture Fourier transform: ∫ ∞ 1 𝑓 (𝑡)ei(𝐷 𝑖 −𝐷 𝑗 )𝑡 𝑉𝑖 𝑗 d𝑡 = 𝑉𝑖 𝑗 𝑓ˆ(𝐷 𝑗 − 𝐷 𝑖 ) for each 𝑖, 𝑗 𝐵𝑖 𝑗 = √ 2𝜋 ∫−∞ ∞ 1 𝑓 (𝑡)ei𝑫𝑡 𝑽e−i𝑫𝑡 d𝑡, or 𝑩 = √ | {z } 2𝜋 −∞ =:𝑽 (𝑡) where we choose the function 𝑓 (𝑡) whose Fourier transform matches the reciprocal at sufficiently large values, 1 𝑓ˆ(𝜔) = −𝜔 when |𝜔| ≥ Δ, but remain “nice” near 𝜔 = 0. One example is to use a smooth bump function 1 𝜔 − · 𝑏( ) 𝜔 Δ where 𝑏(𝑥) =   1  if |𝑥| ≥ 1   O (𝑥 2 )  if |𝑥| ≈ 0. For concreteness, we take 𝑏(𝑥) =    1 − exp( 1−1 1 )  if  1  else |𝑥| < 1 𝑥2 . Then, taking triangle inequality and using the unitary invariance of the operator norm, 1 ∥𝑽 ∥ ∥ 𝑩∥ ≤ √ ∥ 𝑓 ∥ 1 · ∥𝑽 ∥ ≤ 𝐶0 . Δ 2𝜋 The last inequality bounds the Fourier transform by change-of-variable 𝑥 = 𝜔/Δ and leaves a constant 𝐶0 that depends on the inverse Fourier transform of the “dimensionless function” 𝑏(𝑥)/𝑥. This bound on 𝑩 then allows us to control the higher-order errors 𝑩 −𝑩 e (𝑫+𝑽)e ∫ 1 ∫ 1 𝑩𝑠 −𝑩𝑠 = 𝑫+𝑽+ e [𝑩, 𝑽]e d𝑠+[𝑩, 𝑫]+ e 𝑩𝑠 [𝑩, [𝑩, 𝑫]]e−𝑩𝑠 (1−𝑠)d𝑠. 0 0 299 Substitute [𝑩, 𝑫] = −𝑽 and rearranging, we get ∫ 1 𝑩 −𝑩 e (𝑫 + 𝑽)e − 𝑫 = e 𝑩𝑠 [𝑩, 𝑽]e−𝑩𝑠 𝑠d𝑠. 0 Apply the triangle inequality, we get 𝑩 ∥e (𝑫 + 𝑽)e −𝑩 ∫ 1 − 𝑫 ∥ ≤ ∥ [𝑩, 𝑽] ∥ 𝑠d𝑠 ≤ 𝐶0 0 ∥𝑽 ∥ 2 . Δ Finally, to obtain sorted eigenvalues of 𝑫, use the fact that e−𝑩 is unitary and apply Weyl’s inequality for e−𝑩 𝑫e 𝑩 . ■ fˆ ŝµ (! 0 <latexit 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⌫0) <latexit 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O(⌧ 1 <latexit 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Spec(H) ) µ <latexit 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!0 <latexit 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2 <latexit 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B(H 0 ) <latexit 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⌫ 0 ⌫ 2 B(H) <latexit 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B(H) ⌫ <latexit 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sha1_base64="vetgvobRuP6Wi6IgFBOnC1ecE8E=">AAAOKHichZfLbuM2FIY109s0rdtJu+xGqBGgKNrADqaXzQCTKM79Yie240lkBKRM24SpCyiKHcNQX6Db9iX6NN0Vs+2TlBLtiOFxO9qYPN9/eDniL1k4YTQVjcbbJ0/fe/+DDz969vHGJ5/WPvv8+eYX/TTOeEB6QcxiPsAoJYxGpCeoYGSQcIJCzMgNnnkFv5GEpzSOumKekGGIJhEd0wAJFeq07p/XG9uN8nJho7ls1J3l1b7frG36ozjIQhKJgKE0vWs2EjFcIC5owEi+seFnKUlQMEMTskBhms5DnLtbIRLT1GZFcB27y8T45+GCRkkmSBQoiWLjjLkidotduCPKSSDYXDVQwKma2g2miKNAqL2uneXxunCoAluuOWVZS1XGWfpS8Ix85yLGylj6ErOMDBdTNTHnZKwyrcE4mhGRb2yZwQgJTHExyxbCmBO54XMSkV+COAxRNFr4kgT5wi8W4pthb1dFixUHiC12c5vuGXQPUMQ8g3s231tRjCH09o3UfUBbBm2BgVvVwAB6B0bqAaCHBj0E9MigR4AeG/QY0BODngB6atBTQM8MegbouUHPAb0w6AUo1UVVKgC9SyP1EtC2Qdtg4HY1MIBex0jtgNROlQqgd2WkXoHUqyoVQO/aSL0GtGvQLqA9g/YA7Ru0D+iNQW8AHRh0AOhr18CqY/NbA98+pt8Wrk5SJZCIqwZlcVQ8LR4NIHdL20NvS6QBAmBPA+B3iTXAAHgaAJPLfQ2Av2VLA+BeeaABMK481AB4Vh5pAOwqjzUATpUnGgCTylMNgD/lmQbAmvJcA+BKeaEB8Jy81ADYTbY1AGaSHQ2AVeSVBsAI8loD4AHZ1QAcf9nTAJx82dcAHHp5owE473KgATjqsjzqisBTLm81uQUgpZNweVBdX3csCa40iCuRXKuSfBqvhimauf1+LO4wI2Nh5RUl5nQyFcBYOMqWc/qqZaVNSzhFYh3Eq0RIwocxQwjLt0QJveIgbj2mrRUEnsJxSCarAvm6Y0lGaDLJ9Q/hubVTVpwB1ifcLg4vAF8D1F9AQSnN75rDhS5rGfJnhEffN7Z3fiBv3hV1601XVx6K3hm2txfFPCwX4xeLdRf1Zu76a9adaOHOUKcs6jv5fSFeM1xiypqFbMeWMUPH/k84rtZXycqnPi+fhOCGMIRTrcd0MtH10pt66NtrecioKqwzjKqtWb6R07dz1hSQ5A/rJvYuaRZRUXFqc07/E6o3neAV7XKAk5jNK0G76IER5kmlKDq2oGsKumsEiKsny5ulJk7UkkNXxX4tgj6jIRWpnUJHeSlTvxa5zhJjurJn32cajYovp3i1daweX+owqo+opv3JBBv9ne3mj9svOi/qr/aWn1PPnK+cr51vnKbzk/PKOXLaTs8JHOL85vzu/FH7s/ZX7e/aWy19+mSZ86Xz6Kr98y+jbGZs</latexit> ⌫ Figure 4.5: The energy scales 1𝜏 ≪ ∥𝑽 ∥ ≪ Δ𝜈 in one plot. The Hamiltonian perturbation 𝑽 causes eigenvalues of 𝑯 to change by at most 𝛿𝜆 ≤ ∥𝑽 ∥, which splits the Bohr frequencies such that |𝜈 − 𝜈′ | ≤ 2𝛿𝜆 . Here 𝜇 is the cut-off frequency for Í the( secular approximation 𝑺ˆ 𝑎𝑓 ,𝜇 (𝜔′) = 𝜈′ 𝑨𝜈𝑎′ 𝑓ˆ(𝜔′ − 𝜈′) 𝑠ˆ 𝜇 (𝜔′ − 𝜈′). As long as we ) 𝜆 choose 𝜇 < Δ𝜈 −4𝛿 small enough, the secular approximation 𝑺ˆ 𝑎 (𝜔′) can only contain 2 Bohr frequencies 𝑨𝜈′ from at most one block 𝜈, i.e., different blocks decohere. Proof of Theorem 4.16.1 We combine the above ingredients for the proof of Theorem 4.16.1. In what follows, let 𝐸 𝑗 , 𝑷 𝐸 𝑗 , 𝜈 be the eigenvalues, eigenspace projectors, and Bohr frequencies of the unperturbed Hamiltonian 𝑯; furthermore let 𝐸 ′𝑗 , 𝑷 𝐸 ′𝑗 𝜈′, be their counterpart for the perturbed Hamiltonian 𝑯′. It will be helpful to display the structure of the Bohr frequencies and the energy eigenspaces under perturbation by ∑︁ ∑︁ ∑︁ 𝑨𝑎 = 𝑷 𝐸1 𝑨𝑎 𝑷 𝐸2 = 𝑨𝜈𝑎 𝜈∈𝐵(𝑯) 𝐸 1 −𝐸 2 =𝜈 = ∑︁ ∑︁ 𝜈 ′ ∈𝐵(𝑯 ′ ) 𝐸 1′ −𝐸 2′ =𝜈 ′ 𝜈∈𝐵(𝑯) 𝑷 𝐸1′ 𝑨𝑎 𝑷 𝐸2′ = ∑︁ 𝜈 ′ ∈𝐵(𝑯 ′ ) 𝑨𝜈𝑎′ = ∑︁ 𝜈∈𝐵(𝑯) 𝑎 𝑨≈𝜈 , 300 where we defined ∑︁ 𝑎 𝑨≈𝜈 := 𝑨𝜈𝑎′ and 𝜈′ ≈ 𝜈 ⇐⇒ |𝜈′ − 𝜈| ≤ 2𝛿𝜆 . 𝜈 ′ ∈𝐵(𝑯 ′ ), 𝜈 ′ ≈𝜈 In other words, the perturbed set of Bohr frequencies can be identified with the original degenerate blocks according to eigenvalue perturbation11 bounded by 𝛿𝜆 , under the assumption that the perturbation is weaker than the Bohr frequency differences Δ𝜈 > 4𝛿𝜆 . This structure is crucial for proving the monotonicity of 𝑎 (𝜔′) to be the operator gradients; see Figure 4.5. For later use, we also define 𝑨ˆ ≈𝜈 𝑎 with respect to the perturbed Hamiltonian 𝑯′, and consider Fourier transform of 𝑨≈𝜈 𝑎 (𝜔′) at truncation scale 𝜇, i.e., its secular approximation 𝑺ˆ ≈𝜈 ∑︁ ∑︁ 𝑎 𝑎 𝑨ˆ ≈𝜈 (𝜔′) = 𝑨𝜈𝑎′ 𝑓ˆ𝜏 (𝜔′ − 𝜈′), 𝑺ˆ ≈𝜈 (𝜔′) = 𝑨𝜈𝑎′ 𝑓ˆ𝜏 (𝜔′ − 𝜈′) 1 (|𝜔′ − 𝜈′ | < 𝜇). 𝜈 ′ ≈𝜈 𝜈 ′ ≈𝜈 (4.114) In what follows, we will denote 𝜃 (𝜔) = 𝛾(𝜔)𝜔. It is worth recalling the following bounds: ∥𝑐 𝛽 ∥ 1 ∥ 𝑨𝑎 ∥, ∥ 𝑓𝜏 ∥ 2 , √ ≤ 1, 2𝜋 ∥𝜃 ∥ ∞ = O (Λ0 ), Λ0 = Θ(1). Our strategy for proving Theorem 4.12.1 is to rewrite the energy gradients L † [𝑯] ′ and L † [𝑯′] in a form amenable to Lemma 4.16.2 between a set of operators and their Fourier Transforms. Step 1. For the perturbed Hamiltonian 𝑯′, we apply a sequence of approximations to establish ∫ ∞ ∑︁ ∑︁ ′ 𝑎 𝑎 𝜃 − (𝜈) 𝑨ˆ ≈𝜈 (𝜔′) † 𝑨ˆ ≈𝜈 (𝜔′)d𝜔′ − L † [𝑯′] ≤ 𝜖 𝐴 (4.115) 𝑎∈𝑆 𝜈∈𝐵(𝑯) −∞ 𝐸 for some 𝜖 𝐴 > 0 and a function 𝜃 − to be soon specified. Recall we write 𝑨 ≈ 𝑩 if 11 Note that we always have 𝛿 𝜆 ≤ ∥𝑽 ∥ by Proposition 4.16.3, but we keep 𝛿𝜆 as an separate parameter which helps yield better bounds when 𝛿𝜆 ≪ ∥𝑽 ∥, such as the case when the 𝑽 is an off-diagonal perturbation (see Lemma 4.16.4). 301 ∥ 𝑨 − 𝑩∥ ≤ 𝐸. ′ ′ 𝐸1 L † [𝑯′] ≈ D † [𝑯′] ∫ ∞ 𝐸 2 ∑︁ ≈ 𝜃 (𝜔′) 𝑨ˆ 𝑎 (𝜔′) † 𝑨ˆ 𝑎 (𝜔′)d𝜔′ 𝑎∈𝑆 𝐸3 ≈ 𝑎∈𝑆 = −∞ ∑︁ ∫ ∞ 𝜃 (𝜔′) 𝑺ˆ 𝑎 (𝜔′) † 𝑺ˆ 𝑎 (𝜔′)d𝜔′ −∞ ∑︁ ∑︁ ∫ ∞ 𝑎∈𝑆 𝜈∈𝐵(𝑯) 𝑎 𝑎 𝜃 (𝜔′) 𝑺ˆ ≈𝜈 (𝜔′) † 𝑺ˆ ≈𝜈 (𝜔′)d𝜔′ −∞ ∫ ∞ 𝐸 4 ∑︁ ∑︁ 𝑎 𝑎 ≈ 𝜃 (𝜔′) 𝑨ˆ ≈𝜈 (𝜔′) † 𝑨ˆ ≈𝜈 (𝜔′)d𝜔′ 𝑎∈𝑆 𝜈∈𝐵(𝑯) 𝐸 5 ∑︁ ≈ ∑︁ −∞ ∫ ∞ 𝜃 (𝜈) −∞ 𝑎∈𝑆 𝜈∈𝐵(𝑯) 𝐸 6 ∑︁ ≈ ∑︁ 𝑎 𝑎 𝑨ˆ ≈𝜈 (𝜔′) † 𝑨ˆ ≈𝜈 (𝜔′)d𝜔′ ∫ ∞ 𝜃 − (𝜈) 𝑎∈𝑆 𝜈∈𝐵(𝑯) 𝑎 𝑎 𝑨ˆ ≈𝜈 (𝜔′) † 𝑨ˆ ≈𝜈 (𝜔′)d𝜔′ . −∞ The approximations 𝐸 1 , 𝐸 2 are bounded by ! ∑︁ ∥𝑯∥ 3/4 ∥𝑯∥ 3/4 𝑎† 𝑎 |𝑆|O ∥𝑐 ∥ ∥ 𝑨 𝑨 ∥ = ( ) 𝐸1 ≤ O 𝛽 1 𝜏 1/4 𝜏 1/4 𝑎∈𝑆 Í 2∥ 𝑎∈𝑆 𝑨𝑎† 𝑨𝑎 ∥ 1 𝐸2 ≤ ∥𝑐 𝛽 ∥ 1 = |𝑆|O ( ). √ 𝜏 2𝜋𝜏 In the approximations 𝐸 3 and 𝐸 4 , we choose the secular approximation parameter 𝜇 such that 𝜇 < (Δ𝜈 − 4𝛿𝜆 )/2 (4.124) with associated errors given by Corollary 4.15.3 as √︄ 1 ), 𝜇𝜏 𝑎∈𝑆 √︄ ∑︁ 1 𝑎† 𝑎 𝐸 4 ≤ 2∥𝜃 ∥ ∞ ∥ 𝑨≈𝜈 𝑨≈𝜈 ∥∥ 𝑓ˆ𝜏 · (1 − 𝑠ˆ 𝜇 )∥ 2 ∥ 𝑓𝜏 ∥ 2 = |𝑆|O (Λ0 ), 𝜇𝜏 𝑎,𝜈 𝐸 3 ≤ 2∥𝜃 ∥ ∞ ∥ ∑︁ 𝑨𝑎† 𝑨𝑎 ∥∥ 𝑓ˆ𝜏 · (1 − 𝑠ˆ 𝜇 )∥ 2 ∥ 𝑓𝜏 ∥ 2 = |𝑆|O (Λ0 √︁ where we applied Eq. (4.93) to bound ∥ 𝑓ˆ𝜏 · (1 − 𝑠ˆ 𝜇 )∥ 2 ≤ 4/𝜋𝜇𝜏, and used Proposition 4.16.5 to bound the spectral norm of the sum of jump operators in the second line. 302 To justify the equality on the fourth line (different blocks 𝜈s decohere), observe the choice of the parameter 𝜇 in Eq. (4.16) implies ∑︁ ∑︁ 𝑎 𝑎 𝑺ˆ ≈𝜈 (𝜔′) 1 (|𝜔′ − 𝜈| < 𝜇 + 2𝛿𝜆 ) = 𝑺ˆ ≈𝜈 (𝜔′) 1 (|𝜔′ − 𝜈| < Δ𝜈 /2), 𝑺ˆ 𝑎 (𝜔′) = 𝜈∈𝐵(𝑯) 𝜈∈𝐵(𝑯) (4.125) 𝑎 (𝜔′) is given in Eq. (4.114). This ensures that for any given 𝜔′, 𝑺ˆ 𝑎 (𝜔′) where 𝑺ˆ ≈𝜈 can activate at most one block of transitions with Bohr frequencies closest to 𝜈 (see Figure 4.5). Consequently, ∑︁ 𝑎† 𝑎 𝑺ˆ 𝑎† (𝜔′) 𝑺ˆ 𝑎 (𝜔′) = 𝑺ˆ ≈𝜈 (𝜔′) 𝑺ˆ ≈𝜈 (𝜔′). 𝜈∈𝐵(𝑯) Next, for the approximation 𝐸 5 , we define the following “rounded” function 𝜃¯ (𝜔′) where an input 𝜔′ close to 𝜈 ∈ 𝐵(𝑯) is assigned the same value 𝜃 (𝜈), with uniqueness of 𝜈 guaranteed by Eq. (4.125),    𝜃 (𝜈)  if |𝜔′ − 𝜈| ≤ 𝜇 + 2𝛿𝜆 for 𝜈 ∈ 𝐵(𝑯) ′ 𝜃¯ (𝜔 ) := .   𝜃 (𝜔′) else  This lets us formally pull 𝜃 (𝜔′) out of the integral. Of course, this rounding introduces an error scaling with the energy spread multiplied with the derivative ∥ 𝜃¯ − 𝜃 ∥ ∞ ≤ (2𝜇 + 4𝛿𝜆 ) · ∥d𝜃/d𝜔∥ ∞ . Roughly, this error quantifies how the energy gradient (i.e., 𝜃 (𝜔) = 𝛾 𝛽 (𝜔)𝜔) changes due to perturbation in Bohr frequency. Thus, ∑︁ 𝑎† 𝑎 𝐸 5 ≤ ∥𝜃 − 𝜃¯ ∥ ∞ 𝑨≈𝜈 𝑨≈𝜈 ∥ 𝑓ˆ𝜏 · 𝑠ˆ 𝜇 ∥ 22 = |𝑆|O (2𝜇 + 4𝛿𝜆 ), 𝑎,𝜈 where we applied Propositions 4.16.4 and 4.16.5 (deferred to section 4.16), and used the fact that ∥ 𝑓ˆ𝜏 𝑠ˆ 𝜇 ∥ 2 ≤ ∥ 𝑓ˆ𝜏 ∥ 2 ≤ 1. Finally, in the last approximation 𝐸 6 , we define the truncated weight 𝜃 − (𝜈) = 𝜃 (𝜈) 1 (𝜈 ≤ Δ𝜈 /2). This truncation has the property that 𝜃 − (𝜈) ≤ 0 for each 𝜈 ∈ 𝐵(𝑯), which ensures the last line is negative semidefinite. Thus, ∑︁ 𝑎† 𝑎 𝐸 6 ≤ ∥𝜃 − 𝜃 − ∥ ∞ ∥ 𝑨≈𝜈 𝑨≈𝜈 ∥∥ 𝑓𝜏 ∥ 22 ≤ |𝑆|∥𝜃 − 𝜃 − ∥ ∞ , 𝑎,𝜈 where ∥𝜃 − 𝜃 − ∥ ∞ ≤ max 𝜔𝛾 𝛽 (𝜔) ≤ 𝜔≥Δ𝜈 /2 e−𝛽Δ𝜈 /4 , 𝛽 303 using the tail bound in Eq. (4.20). Altogether, 𝜖 𝐴 = 𝐸1 + 𝐸2 + 𝐸3 + 𝐸4 + 𝐸5 + 𝐸6    Λ0 1 ∥𝑯∥ 3/4 e−𝛽Δ𝜈 /4 ≤ O |𝑆| + + 𝜇 + √ + 𝛿𝜆 + . 𝜏 𝜇𝜏 𝛽 𝜏 1/4 1/3 , (Δ − 4𝛿 )/4) so as to optimize the error We then choose 𝜇 = min(Λ2/3 𝜈 𝜆 0 /𝜏 √ O (𝜇 + Λ0 / 𝜇𝜏) while subject to the constraint that 𝜇 < (Δ𝜈 − 4𝛿𝜆 )/2. This choice √︁ √ √ 1/3 + 2Λ / (Δ − 4𝛿 )𝜏 ≤ O (Λ2/3 /𝜏 1/3 + Λ / Δ 𝜏), implies Λ0 / 𝜇𝜏 ≤ Λ2/3 0 0 𝜈 𝜆 𝜈 0 /𝜏 0 where we used 𝛿𝜆 ≤ ∥𝑽 ∥ ≤ Δ𝜈 /8 which is a combination of Proposition 4.16.3 and the assumption in the theorem statement. This yields the following error bound !! 2/3 3/4 −𝛽Δ /4 𝜈 Λ 1 ∥𝑯∥ Λ0 e 0 𝜖 𝐴 ≤ O |𝑆| + + 1/3 +√ + 𝛿𝜆 + . (4.126) 1/4 𝜏 𝛽 𝜏 𝜏 Δ𝜈 𝜏 Step 2. For the original Hamiltonian 𝑯, we may repeat the above argument with trivial perturbation (𝑽 = 0) to get ∑︁ ∑︁ 𝑎 † ∥ 𝜃 − (𝜈)( 𝑨≈𝜈 ) 𝑨≈𝜈 − L † [𝑯] ∥ ≤ 𝜖 𝐵 (4.127) 𝑎∈𝑆 𝜈∈𝐵(𝑯) for some 𝜖 𝐵 > 0. More detailedly, we have ∫ ∞ 𝐸 7 ∑︁ ∑︁ † L [𝑯] ≈ 𝜃 − (𝜈) 𝑨ˆ 𝜈𝑎 (𝜔) † 𝑨ˆ 𝜈𝑎 (𝜔)d𝜔 𝑎∈𝑆 𝜈∈𝐵(𝑯) = ∑︁ ∑︁ −∞ 𝜃 − (𝜈)( 𝑨𝜈𝑎 ) † 𝑨𝜈𝑎 𝑎∈𝑆 𝜈∈𝐵(𝑯) 𝐸 8 ∑︁ ≈ ∑︁ 𝑎 † 𝑎 𝜃 − (𝜈)( 𝑨≈𝜈 ) 𝑨≈𝜈 . 𝑎∈𝑆 𝜈∈𝐵(𝑯) The second line is operator Parseval’s identity, where the time evolution simplifies due to commutativity [ 𝑨𝜈𝑎† 𝑨𝜈𝑎 , 𝑯] = 0. The error 𝐸 7 can be bounded by the same 𝑎 → 𝑨𝑎 ). bounds for 𝜖 𝐴 in Eq. (4.126) by setting 𝑽 = 0 (i.e., 𝑨≈𝜈 𝜈 𝑎 , which acts on eigenstates The last line is a brute-force rewriting of 𝑨𝜈𝑎 into 𝑨≈𝜈 of 𝑯′ instead of 𝑯, with error 𝐸 8 bounded by perturbation theory. This rewriting allows us to prove Theorem 4.16.1 by directly applying Lemma 4.16.2 between the following set of operators and their Fourier transforms: √︁ 𝑎 { |𝜃 − (𝜈)| 𝑨≈𝜈 } 𝑎,𝜈 and √︁ 𝑎 { |𝜃 − (𝜈)| 𝑨ˆ ≈𝜈 (𝜔′)} 𝑎,𝜈,𝜔′ for the perturbed Hamiltonian 𝑯′ . 304 We give an explicit error bound on 𝐸 8 in Proposition 4.16.6 (deferred to section 4.16), which yields !   ∑︁ ∥𝑽 ∥ ∥𝑽 ∥ 𝑎† 𝑎 𝐸8 = O 𝑨 𝑨 𝜃 max = O |𝑆|𝜃 max , Δ𝜈 Δ𝜈 𝑎∈𝑆 where we used the bound max𝜈∈𝐵(𝑯) |𝜃 − (𝜈)| = 𝜃 max provided in the theorem statement. Collect the errors to bound 𝜖 𝐵 = 𝐸 7 + 𝐸 8 ≤ 𝜖 𝐴 + 𝐸 8 . Step 3. Now we may finish the proof of Theorem 4.16.1 by applying Lemma 4.16.2. First, Eq. (4.127) implies ∑︁ ∑︁ 𝑎 † − 𝜃 − (𝜈)( 𝑨≈𝜈 ) 𝑨≈𝜈 + 𝜖 𝐵 𝑰 ⪰ −L † [𝑯] 𝑎∈𝑆 𝜈∈𝐵(𝑯) ⪰ 𝑟𝑶 − 𝜖 𝑰 ⪰ 𝑟𝑶 ′ − (𝑟 ∥𝑶 ′ − 𝑶 ∥ + 𝜖) 𝑰. Similarly, Eq. (4.115) implies ′† ′ −L [𝑯 ] + 𝜖 𝐴 𝑰 ⪰ − ∑︁ ∑︁ ∫ ∞ 𝜃 − (𝜈) 𝑎∈𝑆 𝜈∈𝐵(𝑯) 𝑎 𝑎 𝑨ˆ ≈𝜈 (𝜔′) † 𝑨ˆ ≈𝜈 (𝜔′)d𝜔′ −∞ ⪰ 𝑟𝑶 ′ − (𝜖 + 𝜖 𝐵 + 𝑟 ∥𝑶 − 𝑶 ′ ∥) 𝑰. Note in the last step we used the assumption that [𝑶 ′, 𝑯′] = 0 and applied ′ Lemma 4.16.2. Hence, we have shown that −L † [𝑯′] ⪰ 𝑟𝑶 ′ − 𝜖 ′ 𝑰, where 𝜖 ′ = 𝜖 + 𝜖 𝐴 + 𝜖 𝐵 + 𝑟 ∥𝑶 − 𝑶 ′ ∥ can be bounded by 2/3  ∥𝑯∥ 3/4 Λ0 ∥𝑽 ∥ Λ0 e−𝛽Δ𝜈 /4 ′ + 𝛿𝜆 + 𝜃 max + 𝑟 ∥𝑶 − 𝑶 ∥ . 𝜖 ≤ 𝜖 + O |𝑆| + + 1/3 + √ + 𝜏 𝛽 Δ𝜈 𝜏 1/4 𝜏 Δ𝜈 𝜏  ′ 1 A simpler bound. We now consider the special case of 𝑶 = 𝑰 − 𝑷, 𝑶 = 𝑰 − 𝑷′ to derive a simpler bound as in the theorem statement. Note we have ∥𝑶 − 𝑶 ′ ∥ = ∥𝑷 − 𝑷′ ∥ ≤ 8∥𝑽 ∥/Δ𝐸 ≤ 8∥𝑽 ∥/Δ𝜈 , using Lemma 4.16.3 and the fact the spectral gap is lower bounded by the Bohr-frequency gap, Δ𝐸 ≥ Δ𝜈 . Furthermore generally 𝜃 max = ∥𝜃 − ∥ ∞ = O (Λ0 ) for our choice of 𝛾 𝛽 (𝜔) in Equation 4.21. And we always have 𝛿𝜆 ≤ ∥𝑽 ∥ by Proposition 4.16.3. Plugging these into Equation 4.16, we have  2/3 ∥𝑯∥ 3/4 Λ0 Λ0 e−𝛽Δ𝜈 /4 ∥𝑽 ∥  𝜖 ≤ 𝜖 + O |𝑆| + + 1/3 + √ + + ∥𝑽 ∥ + (Λ0 + 𝑟) . 𝜏 𝛽 Δ𝜈 𝜏 1/4 𝜏 Δ𝜈 𝜏 ′  1 This concludes our proof of Theorem 4.16.1. 305 Supplementary calculations In this section, we provide some missing calculations that prove some propositions used in the proof in the previous section. Proposition 4.16.4 (Bounds on the derivative). There exists an absolute constant 𝐶 such that for any 𝛽, Λ0 , ! ! 2 2 d d e−𝜔 /2Λ0 𝜔 ∥ ∞ ≤ 𝐶. ∥ 𝜃 (𝜔)∥ ∞ = O ∥ d𝜔 d𝜔 1 + e 𝛽𝜔 Proof. By the product rule, ! 2 2 2 2 2 2 2 2 e−𝜔 /2Λ0 − e−𝜔 /2Λ0 𝜔2 /Λ20 e−𝜔 /2Λ0 𝛽𝜔e 𝛽𝜔 d 𝜔e−𝜔 /2Λ0 = − ≤ (𝑐𝑜𝑛𝑠𝑡.) d𝜔 1 + e 𝛽𝜔 1 + e 𝛽𝜔 (1 + e 𝛽𝜔 ) 2 using change of variable 𝑥 = 𝛽𝜔 and 𝑦 = 𝜔/Λ0 to obtain the absolute constant bound. ■ Proposition 4.16.5. In the prevailing notation, ∑︁ ∑︁ ∑︁ ∑︁ 𝑨𝜈𝑎† 𝑨𝜈𝑎 , 𝑎∈𝑆 𝜈∈𝐵(𝑯) 𝑎† 𝑎 𝑨≈𝜈 𝑨≈𝜈 ≤ 𝑎∈𝑆 𝜈∈𝐵(𝑯) ∑︁ 𝑨𝑎† 𝑨𝑎 . 𝑎∈𝑆 𝑎† Proof. We focus on 𝑨≈𝜈 since the case of 𝑨𝜈𝑎† is a special case. Resolve the identity by nearby energy subspaces ∑︁ ∑︁ ∑︁ ∑︁ 𝑎 𝑰= 𝑷𝐸 = 𝑷 𝐸 such that 𝑨≈𝜈 = 𝑷≈𝐸¯2 𝑨𝑎 𝑷≈𝐸¯1 . 𝐸 𝐸¯ 𝐸≈𝐸¯ 𝐸¯ 2 −𝐸¯ 1 =𝜈 | {z } =:𝑷≈𝐸¯ Now, we calculate ∑︁ ∑︁ 𝑎∈𝑆 𝜈∈𝐵(𝑯) ∑︁ 𝑎† 𝑎 𝑨≈𝜈 𝑨≈𝜈 = 𝑷≈𝐸¯1 𝑨𝑎† 𝑷≈𝐸¯2 𝑨𝑎 𝑷≈𝐸¯1 = 𝑎∈𝑆, 𝐸¯ 2 , 𝐸¯ 1 = max 𝑷≈𝐸¯1 ∑︁ 𝑷≈𝐸¯1 𝑨𝑎† 𝑨𝑎 𝑷≈𝐸¯1 𝑎∈𝑆, 𝐸¯ 1 ∑︁ 𝐸¯ 𝑎∈𝑆 𝑨𝑎† 𝑨𝑎 𝑷≈𝐸¯1 ≤ ∑︁ 𝑨𝑎† 𝑨𝑎 . 𝑎∈𝑆 The last line uses that the operator norm of block-diagonal matrices equals the maximum among the blocks. ■ 306 Proposition 4.16.6 (Jumps with perturbed Hamiltonian). In the prevailing notation, and for any function ℎ(𝜈), we have that ∑︁ ∑︁ ∑︁ ∑︁ 𝑎 † 𝑎 ∥ ℎ(𝜈)( 𝑨𝜈𝑎 ) † 𝑨𝜈𝑎 − ℎ(𝜈)( 𝑨≈𝜈 ) 𝑨≈𝜈 ∥ 𝑎∈𝑆 𝜈∈𝐵(𝑯) 𝑎∈𝑆 𝜈∈𝐵(𝑯) ≤O ∑︁ ! ∥𝑽 ∥ · max |ℎ(𝜈)| · . Δ𝜈 𝜈∈𝐵(𝑯) 𝑨𝑎† 𝑨𝑎 𝑎∈𝑆 Í Proof. It suffices to set ∥ 𝑎∈𝑆 𝑨𝑎† 𝑨𝑎 ∥ = 1. Consider the operator Fourier transform with a smooth bump weight ∥𝑔∥ 2 = 1, 𝑎 𝑨𝑔,𝑯 (𝜔) 𝑔(𝜔) ˆ ∝ for |𝜔| ≥ Δ2𝜈   0  if   O (1)  else. which automatically decohere different Bohr frequency blocks (i.e., no need to apply secular approximation). Also, extend the function locally: n ℎ(𝜔) := ℎ(𝜈) if |𝜔 − 𝜈| ≤ 2∥𝑽 ∥ . Then, ∑︁ ∫ ∞ 𝑎∈𝑆 𝑎 𝑎 (𝜔)d𝜔 ℎ(𝜔) 𝑨𝑔,𝑯 (𝜔) † 𝑨𝑔,𝑯 −∞ = ∑︁ = ∑︁ ∑︁ = ∑︁ ∑︁ ∫ ∞ ∑︁ 𝑎∈𝑆 𝜈,𝜈 ′ ∈𝐵(𝑯) ℎ(𝜔)( 𝑨𝜈𝑎′ ) † 𝑨𝜈𝑎 𝑔ˆ ∗ (𝜔 − 𝜈′) 𝑔(𝜔 ˆ − 𝜈)d𝜔 −∞ ℎ(𝜈) ( 𝑨𝜈𝑎 ) † 𝑨𝜈𝑎 ∫ ∞ | 𝑔(𝜔 ˆ − 𝜈)| 2 d𝜔 −∞ 𝑎∈𝑆 𝜈,∈𝐵(𝑯) ℎ(𝜈)( 𝑨𝜈𝑎 ) † 𝑨𝜈𝑎 . 𝑎∈𝑆 𝜈∈𝐵(𝑯) Now, we add the perturbation 𝑯 + 𝑽. The insight is that we can introduce an artificial Hamiltonian ∑︁ ∑︁ 𝑯¯ := 𝐸 𝑷 𝐸 ′ such that ∥ 𝑯¯ − 𝑯∥ ≤ 2∥𝑽 ∥ 𝐸 ∈spec(𝑯) 𝐸 ′ ≈𝐸 with exactly the same spectrum of the original Hamiltonian 𝑯, but with the basis according to the perturbed Hamiltonian 𝑯′. Then, the same argument with the artificial Hamiltonian implies ∑︁ ∫ ∞ ∑︁ ∑︁ 𝑎 † 𝑎 𝑎 † 𝑎 ℎ(𝜔) 𝑨𝑔, (𝜔) 𝑨 (𝜔)d𝜔 = ℎ(𝜈)( 𝑨≈𝜈 ) 𝑨≈𝜈 . 𝑯¯ 𝑔, 𝑯¯ 𝑎∈𝑆 −∞ ¯ 𝑎∈𝑆 𝜈∈𝐵( 𝑯) 307 Lastly, we may bound the difference by the purification ∑︁ ∫ ∞ ∑︁ ∫ ∞ 𝑎 𝑎 ˆ ∥ 𝑨𝑔,𝑯 (𝜔) ⊗ |𝑎⟩ ⊗ |𝜔⟩d𝜔 − 𝑨ˆ 𝑔, (𝜔) ⊗ |𝑎⟩ ⊗ |𝜔⟩d𝜔∥ 𝑯¯ 𝑎∈𝑆 =∥ −∞ 𝑎∈𝑆 ∑︁ ∫ ∞ 𝑎∈𝑆 −∞ √︄∫ ∞ ≤ 𝑨𝑎𝑯 (𝑡) ⊗ |𝑎⟩ ⊗ 𝑔(𝑡)|𝑡⟩d𝑡 − −∞ ∑︁ ∫ ∞ 𝑎∈𝑆 2 ¯ 2∥ei𝑯𝑡 − ei 𝑯𝑡 ∥𝑔(𝑡) d𝑡 = O ( −∞ −∞ 𝑨𝑎𝑯¯ (𝑡) ⊗ |𝑎⟩ ⊗ 𝑔(𝑡)|𝑡⟩d𝑡 ∥ ∥𝑽 ∥ ). Δ𝜈 The factor of 2 is due to left and right Hamiltonian evolution ¯ ¯ ¯ ¯ ¯ ei𝑯𝑡 𝑨𝑎 e−i𝑯𝑡 − ei 𝑯𝑡 𝑨𝑎 e−i 𝑯𝑡 = ei𝑯𝑡 𝑨𝑎 (e−i𝑯𝑡 − e−i 𝑯𝑡 ) + (ei𝑯𝑡 − ei 𝑯𝑡 ) 𝑨𝑎 e−i 𝑯𝑡 . ¯ ¯ ≤ 2∥𝑽 ∥𝑡 and that To evaluate the integral, we use that ∥ei𝑯𝑡 − ei 𝑯𝑡 ∥ ≤ ∥𝑯 − 𝑯∥𝑡 𝑔(𝑡) is rapidly decaying for large |𝑡| ≥ Δ1𝜈 . To conclude the proof, use the purification tricks (Lemma 4.15.1). ■ Monotonicity of gradient on a subspace For our proof that BQP-hard Hamiltonians has no suboptimal local minima in section 4.14, we will need the following refinements of Theorem 4.16.1 where the gradient operator acts on a low-energy subspace with an excitation gap. Intuitively, we care only about the Bohr-frequency gap restricted to the low-energy subspace 𝑯𝑸 instead of the full Hilbert space; the gradient on that subspace should not be sensitive to the excited states above the excitation gap. Corollary 4.16.1 (Monotonicity of gradient on a subspace; Corollary 4.12.1 restated). Í Consider a Hamiltonian 𝑯 = 𝐸¯ 𝐸¯ 𝑷 𝐸¯ and its perturbation 𝑯′ := 𝑯 + 𝑽. Let 𝑷 be the ground space projector for 𝑯 and 𝑷′ be the corresponding perturbed eigensubspace of 𝑯′. Let 𝑸 be a low-energy eigensubspace projector of 𝑯 (i.e., Í 1 𝑸 = 𝐸 ≤𝐸𝑸 𝑷 𝐸 for 𝐸 𝑸 ∈ Spec(𝑯)) with excitation gap Δ𝑸 . Assume ∥𝑽Δ∥ ∥𝑯∥ ≤ 144 Δ𝜈 𝑸 where Δ𝜈 := min𝜈1 ≠𝜈2 ∈𝐵(𝑯|𝑸 ) |𝜈1 − 𝜈2 | is the Bohr-frequency gap of 𝑯 within the Í Í 𝛽,𝜏,𝑯 𝛽,𝜏,𝑯 ′ subspace 𝑸. For any 𝛽, 𝜏 > 0, let L = 𝑎∈𝑆 L𝑎 , L ′ = 𝑎∈𝑆 L𝑎 be thermal 𝑎 𝑎 Lindbladians with jumps { 𝑨 } 𝑎∈𝑆 , where ∥ 𝑨 ∥ ≤ 1 and the transition weight 𝛾 𝛽 (𝜔) is given by Eq. (4.21). Then we have the monotone property that −𝑸L † [𝑯]𝑸 ⪰ 𝑟𝑸( 𝑰 − 𝑷) − 𝜖 𝑰 implies − 𝑸 ′ L ′† [𝑯′]𝑸 ′ ⪰ 𝑟𝑸 ′ ( 𝑰 − 𝑷′) − 𝜖 ′ 𝑰 308 where 𝑸 ′ projects onto the perturbed eigensubspace of 𝑯′ identified with 𝑸, and 2/3 1 ∥𝑯∥ 3/4 Λ0 Λ0 Λ0 e−𝛽Δ𝜈 /4 e−𝛽Δ𝑸 /4 𝜖 ≤ 𝜖 + |𝑆| · O + + + + + + √ √︁ 𝜏 𝛽 𝛽 𝜏 1/4 𝜏 1/3 Δ𝜈 𝜏 Δ𝑸 𝜏     ∥𝑽 ∥ ∥𝑽 ∥  Λ0 ∥𝑽 ∥∥𝑯∥ + 1+ +𝑟 + . Δ𝜈 Δ𝑸 Δ𝑸 Δ𝜈  ′ ′ Proof. The idea is that 𝑸 ′ L † [𝑯′]𝑸 ′ essentially depends only on the low energy subspace 𝑸 ′ and the corresponding restricted transition 𝑸 ′ 𝑨𝑎 𝑸 ′. ∑︁ ∫ ∞ ′ ′† ′ ′ 𝐸1 𝑸 L [𝑯 ]𝑸 ≈ 𝜃 (𝜔′)𝑸 ′ 𝑨ˆ 𝑎 (𝜔′) † 𝑨ˆ 𝑎 (𝜔′)𝑸 ′d𝜔′ 𝑎∈𝑆 −∞ ∫ ∞ 𝐸 2 ∑︁ ≈ 𝜃 (𝜔′)𝑸 ′ 𝑨ˆ 𝑎 (𝜔′) † 𝑸 ′ 𝑨ˆ 𝑎 (𝜔′)𝑸 ′d𝜔′ 𝑎∈𝑆 = −∞ ∑︁ ∫ ∞ 𝜃 (𝜔′) 𝑹ˆ ′𝑎 (𝜔′) † 𝑹ˆ ′𝑎 (𝜔′)d𝜔′ 𝑎∈𝑆 −∞ ′ †𝛽,𝜏,𝑯𝑸 𝐸3 ′ ≈ L {𝑹′𝑎 } 𝐸4 [𝑯′𝑸 ′ ] †𝛽,𝜏,𝑯 ′ ′ ≈ L {𝑹 𝑎 } 𝑸 [𝑯′𝑸 ′ ]. The approximation 𝐸 2 inserts the low-energy projector 𝑸 ′. To do so, we resolves the identity by 𝑰 = 𝑸 ′ + ( 𝑰 − 𝑸 ′) and uses that ∑︁ ∫ ∞ 𝜃 (𝜔′)𝑸 ′ 𝑨ˆ 𝑎 (𝜔′) † ( 𝑰 − 𝑸 ′) 𝑨ˆ 𝑎 (𝜔′)𝑸 ′d𝜔′ 𝑎∈𝑆 −∞ ∫ ∞ 𝐸 21 ∑︁ ≈ 𝜃 (𝜔′) 𝑺ˆ 𝑎 (𝜔′) † 𝑺ˆ 𝑎 (𝜔′)d𝜔′ 𝑎∈𝑆 −∞ 𝐸 22 ≈ 0. That is, we need the excitation gap to be large so that ( 𝑰 − 𝑸 ′) 𝑨𝑎 𝑸 ′ have a vanishing contribution to the gradient. These error combines 𝐸 2 = 𝐸 21 + 𝐸 22 , where Λ0 𝑨𝑎† 𝑨𝑎 ∥∥𝜃 ∥ ∞ ∥ 𝑓𝜏 · (1 − 𝑠ˆ 𝜇 )∥ 2 ∥ 𝑓𝜏 ∥ 2 = |𝑆|O ( √ ) 𝜇𝜏 𝑎∈𝑆 ∑︁ e−𝛽(Δ𝑸′ −𝜇)/2 𝐸 22 ≤ ∥ 𝑨𝑎† 𝑨𝑎 ∥ ′ max |𝜃 (𝜔′)|∥ 𝑓𝜏 · 𝑠ˆ 𝜇 ∥ 22 = |𝑆| . 𝜔 ≥Δ𝑸′ −𝜇 𝛽 𝑎∈𝑆 𝐸 21 ≤ 2∥ ∑︁ Thus, we consider a safe choice of 𝜇 = Δ𝑸 ′ /2. Also, since the perturbation is Δ𝑸 1 1 small by assumption, ∥𝑽 ∥ ≤ 144 Δ𝜈 ∥𝑯∥ ≤ 144 Δ𝑸 , the excitation gap remains large 1 Δ𝑸 ′ ≥ Δ𝑸 − 2∥𝑽 ∥ ≥ Δ𝑸 − 72 Δ𝑸 ≥ Δ𝑸 /2. The third line (after approximation 309 ′ by defining a modified 𝐸 2 ) formally disposes of the excited state of 𝑯′ above 𝐸 𝑸 ′ Hamiltonian ′ ′ 𝑯′𝑸 ′ := 𝑯′𝑸 ′ + 𝐸 𝑸 ′ (𝑰 − 𝑸 ) for the Fourier transform 𝑹ˆ′𝑎 (𝜔′). This Hamiltonian is merely a proof artifact, and one can also set the energy of the excited subspace 𝑰 − 𝑸 ′ to be infinity. The error 𝐸 3 is merely the error to put it back to the form of an energy gradient. The approximation 𝐸 4 changes the jumps with norm bounded by ∑︁ ∑︁ ∥ 𝑹′𝑎 ⊗ |𝑎⟩ − 𝑹 𝑎 ⊗ |𝑎⟩∥ 𝑎∈𝑆 𝑎∈𝑆 √︄ ∑︁ ∥𝑽 ∥ ≤2 ∥ ) 𝑨𝑎† 𝑨𝑎 ∥∥𝑸 − 𝑸 ′ ∥ = |𝑆|O ( Δ 𝑸 𝑎∈𝑆 ∥ using subspace perturbation bounds ∥𝑸 − 𝑸 ′ ∥ ≤ 8 ∥𝑽 Δ𝑸 (Lemma 4.16.3). Since 1 † the (suitably normalized) gradient operator 2|𝑆|∥𝑶 ∥ · L [𝑶] can be block-encoded using O (1) block-encodings of the jumps (Theorem 4.10.1, Proposition 4.10.7, Proposition 4.10.8), perturbation to the jumps propagates to the gradient operator by !  ∑︁ ∑︁ ∥𝑽 ∥  . 𝐸4 = O 𝑹′𝑎 |𝑎⟩ − 𝑹 𝑎 |𝑎⟩ · ∥𝑯′𝑸 ′ ∥ = |𝑆|O ∥𝑯′𝑸 ′ ∥ Δ𝑸 𝑎 𝑎 To summarize, the above gives the bound ′† ′ †𝛽,𝜏,𝑯𝑸 ′ ∥𝑸 ′ L [𝑯′]𝑸 ′ − L {𝑹 𝑎 } 1 ∥𝑯∥ 3/4 Λ0 e−𝛽Δ𝑸 /4 ∥𝑽 ∥ ′ [𝑯′𝑸 ′ ] ∥ ≤ |𝑆| · O + + (4.148) + ∥𝑯𝑸 ′ ∥ + √︁ 𝜏 𝛽 Δ𝑸 𝜏 1/4 Δ𝑸 𝜏 using ∥𝑯′ ∥ ≤ ∥𝑯∥ + ∥𝑽 ∥ ≤ 2∥𝑯∥ and Δ𝑸 ′ ≥ Δ𝑸 /2. Similarly, the 𝐸5 †𝛽,𝜏,𝑯 𝑸L † [𝑯]𝑸 ≈ L {𝑹 𝑎 } 𝑸 [𝑯𝑸 ] for 𝑯𝑸 := 𝑯𝑸 + 𝐸 𝑸 ( 𝑰 − 𝑸) with 𝐸 5 also bounded by the RHS of Eq. (4.16) but with ∥𝑽 ∥ → 0. Now, we may use the monotonicity of gradient (Theorem 4.16.1) for Hamiltonian pairs 𝑯𝑸 and 𝑯′𝑸 ′ , jumps {𝑹 𝑎 } 𝑎∈𝑆 , with the characteristic Bohr-frequency gap Δ𝜈 = min𝜈1 ≠𝜈2 ∈𝐵(𝑯𝑸 ) |𝜈1 − 𝜈2 |. The modified Hamiltonian perturbation ∥𝑯𝑸 − 𝑯′𝑸 ′ ∥ is bounded by the sum of the following two errors: ∥𝑯𝑸 − 𝑯′𝑸 ′ ∥ = ∥𝑯𝑸 − 𝑯𝑸 ′ + 𝑯𝑸 ′ − 𝑯′𝑸 ′ ∥ ≤ 8 ∥𝑽 ∥ ∥𝑽 ∥ ∥𝑯∥ + ∥𝑽 ∥ ≤ 9 ∥𝑯∥, Δ𝑸 Δ𝑸 (4.149) ! 310 ′ ′ ′ ′ ′ ′ ∥𝐸 𝑸 ( 𝑰 − 𝑸) − 𝐸 𝑸 ′ ( 𝑰 − 𝑸 )∥ = ∥𝐸 𝑸 ( 𝑰 − 𝑸) − 𝐸 𝑸 ( 𝑰 − 𝑸 ) + 𝐸 𝑸 ( 𝑰 − 𝑸 ) − 𝐸 𝑸 ′ ( 𝑰 − 𝑸 )∥ ≤ 8𝐸 𝑸 ∥𝑽 ∥ ∥𝑽 ∥ + ∥𝑽 ∥ ≤ 9 ∥𝑯∥. Δ𝑸 Δ𝑸 (4.150) So ∥𝑯𝑸 − 𝑯′𝑸 ′ ∥ ≤ 18∥𝑽 ∥∥𝑯∥/Δ𝑸 , which is less than Δ𝜈 /8 by the assumption in the corollary statement so we may apply Theorem 4.16.1. Note that error due to perturbing the 𝑸( 𝑰 − 𝑷) can be bounded directly by ∥𝑸 ′ ( 𝑰 − 𝑷′) − 𝑸( 𝑰 − 𝑷)∥ ≤ ∥𝑸 ′ − 𝑸∥ + ∥𝑷 − 𝑷′ ∥ = O  ∥𝑽 ∥ Δ𝑸 + ∥𝑽 ∥  . Δ𝜈 (4.151) Apply Theorem 4.16.1 with 𝑯 = 𝑯𝑸 , 𝑯′ = 𝑯′𝑸 ′ , 𝑶 = 𝑸( 𝑰 − 𝑷), and 𝑶 ′ = 𝑸 ′ ( 𝑰 − 𝑷′), 𝛿𝜆 ≤ ∥𝑯𝑸 − 𝑯𝑸 ′ ∥, and 𝜃 max = O (Λ0 ), along with the additional approximation error in Eq. (4.16), we obtain the result as advertised in the corollary statement. ■ Corollary 4.16.2 (Monotonicity of gradient on a subspace under off-block-diagonal perturbation; Corollary 4.12.2 restated). In the setting of Corollary 4.16.1, instead ∥ ∥𝑽 ∥ assume ∥𝑽 Δ𝜈 , Δ𝑸 ≤ (𝑐𝑜𝑛𝑠𝑡.), and that the perturbation is off-block-diagonal, i.e., 𝑸𝑽𝑸 = ( 𝑰 − 𝑸)𝑽 ( 𝑰 − 𝑸) = 0. Then, −𝑸L † [𝑯]𝑸 ⪰ 𝑟𝑸( 𝑰 − 𝑷) − 𝜖 𝑰 − 𝑸 ′ L ′† [𝑯′]𝑸 ′ ⪰ 𝑟𝑸 ′ ( 𝑰 − 𝑷′) − 𝜖 ′ 𝑰, implies where 2/3 e−𝛽Δ𝜈 /4 e−𝛽Δ𝑸 /4 1 ∥𝑯∥ 3/4 Λ0 Λ0 Λ0 + + + 𝜖 ≤ 𝜖 + |𝑆| · O + + + √ √︁ 𝜏 𝛽 𝛽 𝜏 1/4 𝜏 1/3 Δ𝜈 𝜏 Δ𝑸 𝜏  ∥𝑯 ∥∥𝑽 ∥ ∥𝑽 ∥ 2   ∥𝑽 ∥ ∥𝑽 ∥ 2   ∥𝑽 ∥ 2 𝑸 + + ∥𝑯𝑸 ∥ · + + +𝑟 . Δ𝑸 Δ𝑸 Δ 𝜈 Δ𝑸 Δ 𝜈 Δ𝑸 Δ𝑸 Δ 𝜈 ′  Proof. When the perturbation 𝑽 is off-diagonal in the energy eigenbasis of 𝑯, we can use tighter bounds on the changes in eigenvalues and eigensubspaces of 𝑯′ from Lemma 4.16.4, which implies 𝛿𝜆 = max |𝐸 𝑗 − 𝐸 ′𝑗 | = O  ∥𝑽 ∥ 2  Δ𝑸 𝑗  ∥𝑽 ∥ ∥𝑽 ∥ 2 ∥𝑯𝑸 ∥ + . Δ𝑸 Δ𝑸  ∥𝑽 ∥ ∥𝑽 ∥ 2  ∥𝑷 − 𝑷′ ∥ ≤ O + . Δ𝑸 Δ𝑸 Δ 𝜈 ∥𝑯𝑸 − 𝑯′𝑸 ′ ∥ = O  311 The second line and third line are due to rotation and then subspace perturbation (Lemma 4.16.3). Essentially, this is because (1) all subspace rotations are small ∥ ∥𝑽 ∥ 2 ( ∥𝑽 Δ𝑸 ≪ 1) and (2) the energy perturbation is smaller than the level spacing ( Δ𝑸 ≪ Δ𝜈 ). We then follow the same argument in the proof of Corollary 4.16.1 above with the improved bounds for Eqs. (4.149), (4.150), and (4.151). We can also use an improve bound of 𝜃 max ≤ ∥𝑯𝑸 ∥. Together these improvements yield the better error bound on 𝜖 ′ as advertised. ■ Example where perturbation kills energy gradient We present an example where despite ∥𝑽 ∥ ≪ Δ𝜈 , the gradient is lost due to the perturbation. Therefore, the resulting change in gradient is not multiplicative ∥𝑽 ∥ ∥ (1 − ∥𝑽 Δ𝜈 ), but merely additive. If the gradient is polynomially small, we need Δ𝜈 to be also polynomially small to secure the gradient. Proposition 4.16.7 (Perturbation kills the gradient). Let 𝑯 = 𝒁 = |1⟩⟨1| − |−1⟩⟨−1|, 𝑨 = 𝒁 + 𝜖 𝑿, and 𝑽 = 𝜖 𝑿. Then, for the 𝛽, 𝜏 → ∞ heat bath Lindbladian, we have that L † [𝑯] ⪯ −2𝜖 2 ( 𝑰 − 𝑷) but L ′† [𝑯′] = 0. Proof. † L [𝑯] = 𝜖 2   1 1 |1⟩⟨−1|𝑯|−1⟩⟨1| − |1⟩⟨1|𝑯 − 𝑯|1⟩⟨1| = −2𝜖 2 |1⟩⟨1|. 2 2 But L ′† [𝑯′] = 0 because [𝑯′, 𝑨] = 0, i.e., 𝑨 is diagonal in the new energy basis. ■ Bibliography [1] Nilin Abrahamsen. A polynomial-time algorithm for ground states of spin trees. arXiv preprint arXiv:1907.04862, 2019. 204 [2] Naman Agarwal, Zeyuan Allen-Zhu, Brian Bullins, Elad Hazan, and Tengyu Ma. Finding approximate local minima faster than gradient descent. 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[38] has guarantees assuming only the mixing time, and ETH is one way of bounding the mixing time. 3 Ref. [180] did not include the algorithmic cost of quantum simulated annealing, so we fill in using our modernized version (section 2.12). Likewise, the simulated annealing cost of [186] could also be improved (still assuming perfect QPE). 4 The proof of correctness assumes each eigenstate 𝑯 can be perfectly distinguished. 5 Assuming the Markov property and clustering quantities both decay exponentially. 6 Their exactly detailed-balanced quantum channel seems to qualitatively differ from other quantum MCMC algorithms. It assumes certain efficiently implementable basis measurements with a good overlap with the energy basis. For example if measured in the computational basis, the 1D transverse field Ising model seems to exhibit an exponential mixing time at constant temperature, see [133, Page 5]. 7 A comparison of existing thermal state preparation algorithms. We focus on methods with provable guarantees for an 𝜖-approximation of the Gibbs state (in trace distance) and list their cost, assumptions, and caveats. We use Poly(·) to denote polynomials and Õ (·) to absorb logarithmic dependences. The first few algorithms are Monte Carlo-style methods, incoherent or coherent; we represent their costs by the total black-box Hamiltonian simulation time. The incoherent ones are based on semi-groups, with complexity being the cost of emulating the semi-group (which is basically dominated by the phase estimation time 𝑡 𝑄𝑃𝐸 ) multiplied by the mixing time 𝑡 𝑚𝑖 𝑥 . Instead, the coherent version implements block-encoding for discriminants and prepares the purified √︃ Gibbs state −1 ) (as the via quantum simulated annealing. The number of discriminant calls is Õ (𝛽∥𝑯∥ 𝜆 𝑔𝑎 𝑝 counterpart for the mixing time 𝑡 𝑚𝑖 𝑥 ) where 𝜆g𝑎 𝑝 refers to the minimum gap of discriminants along the adiabatic path. We calculate the costs for Theorem 2.1.3 and Theorem 2.1.4, assuming the algorithmic parameters and the mixing time 𝑡 𝑚𝑖 𝑥 or spectral gap 𝜆 𝑔𝑎 𝑝 satisfy certain self-consistency constraint. The mixing time and spectral gap can be (loosely) converted to each other, assuming approximate detailed balance. Of course, as in all classical MCMC methods, the mixing time 𝑡 𝑚𝑖𝑥 or spectral gap 𝜆 𝑔𝑎 𝑝 can be exponentially small depending on the temperature and the system; indeed, there are systems whose thermal states are expected to have high complexity, such as certain spin glass models, and we do not expect efficient quantum algorithms to exist in general. Optimistically, we often care about the Gibbs state that appears in nature, which suggests the mixing time remains reasonably small.