Discrete Harmonic Analysis and its Applications to Testing, Learning, and Complexity Citation Slote, Joseph Alfred (2025) Discrete Harmonic Analysis and its Applications to Testing, Learning, and Complexity. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/grjv-rz74. https://resolver.caltech.edu/CaltechTHESIS:06082025-235351698 Abstract This thesis consists of two parts. In Part I we present a new class of norm discretization inequalities suited for low-degree polynomials in many dimensions, with applications to discrete harmonic analysis and to quantum and classical learning theory. Discretization inequalities (of Bernstein type) control the supremum norm of polynomials f by their supremum norms over certain finite subsets T of the domain. Unlike earlier multivariate Bernstein-type discretization inequalities we establish dimension-free comparisons for simple and generic T, such as product sets T=S₁ × ⋅⋅⋅ × Sₙ for Sⱼ's consisting of well-spread points in R or C, in exchange for a constant that grows with deg(f). Our results also introduce the notion of "individual degree"—the maximum degree of f in any one variable—as a fundamental parameter for discretization inequalities: we show for the first time that dimension-free discretizations of the uniform norm are possible for T with cardinality independent of deg(f), provided f has bounded individual degree. Our work offers a new, high-dimensional perspective on discretization inequalities and yields several new results in analysis on the hypergrid (i.e., products of cyclic groups), including Bohnenblust–Hille-type inequalities, dimension-free supremum norm bounds on level-k Fourier projections, and junta theorems. These estimates in turn provide the key analytic tools for extending recent breakthroughs in learning low-degree functions to the hypergrid and to its quantum analogue, local observables on K-level qudit systems. In Part II we apply ideas from analysis of Boolean functions to study other aspects of (quantum) computation: circuit complexity and property testing. First, we introduce and study a deceptively simple model of constant-depth quantum circuits and begin the project of proving bounds on its capabilities, ultimately drawing on connections to nonlocal games and notions of approximate degree. Second, we introduce a new access model for property testing, quantum data, which allows for ultrafast testing algorithms where classical data provably yields no fast testers—such as for monotonicity, symmetry, and triangle-freeness. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Functional analysis; Approximation theory; Discretization inequalities; Learning theory; Property testing; Circuit theory Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Computing and Mathematical Sciences Awards: Bhansali Family Doctoral Prize in Computer Science, 2025. Thesis Availability: Public (worldwide access) Research Advisor(s): Umans, Christopher M. Thesis Committee: Schulman, Leonard J. (chair) Tamuz, Omer Tropp, Joel A. Umans, Christopher M. Defense Date: 29 May 2025 Non-Caltech Author Email: joseph.slote (AT) gmail.com Funders: Funding Agency Grant Number Simons Foundation Investigator Award (Chris Umans) UNSPECIFIED Record Number: CaltechTHESIS:06082025-235351698 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06082025-235351698 DOI: 10.7907/grjv-rz74 Related URLs: URL URL Type Description https://doi.org/10.19086/da.137968 DOI A dimension-free Remez-type inequality on the polytorus https://doi.org/10.1016/j.aim.2024.109824 DOI Bohnenblust–Hille inequality for cyclic groups https://doi.org/10.4230/LIPIcs.ITCS.2024.69 DOI Quantum and Classical Low-Degree Learning via a Dimension-Free Remez Inequality https://arxiv.org/abs/2411.12730 arXiv Testing classical properties from quantum data https://doi.org/10.1007/s00222-024-01306-9 DOI Dimension-free discretizations of the uniform norm by small product sets https://doi.org/10.4230/LIPIcs.ITCS.2024.92 DOI Parity vs. AC0 with Simple Quantum Preprocessing https://arxiv.org/abs/2406.08509 arXiv Noncommutative Bohnenblust--Hille Inequality for qudit systems ORCID: Author ORCID Slote, Joseph Alfred 0000-0002-6363-7821 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17423 Collection: CaltechTHESIS Deposited By: Joseph Slote Deposited On: 09 Jun 2025 21:07 Last Modified: 14 Nov 2025 21:12 Thesis Files PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

Discrete harmonic analysis and its applications to testing, learning, and complexity Thesis by Joseph Slote In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Computing and Mathematical Sciences CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2025 Defended May 29 ii © 2025 Joseph Slote ORCID: 0000-0002-6363-7821 All rights reserved except where otherwise noted iii Acknowledgements My PhD experience has felt like a sort of arrival—vocationally and scientifically, but also socially. It has been an unexpected and deeply cherished gift to meet and work with the friends and colleagues in theoretical computer science that I have. I am so grateful to my advisor, Chris Umans, for admitting me into this world, for encouraging my blue-sky questioning, for supporting my travel, and for modeling for me such a clear-eyed approach to phrasing and attacking computer science questions. My interactions with the quantum community at Caltech have been crucial, especially with Thomas Vidick and his postdocs Atul Singh Arora, Saeed Mehreban, and Ulysse Chaubad. I am also very grateful to Urmila Mehedev and her quantum seminars, which gave me an opportunity to dive deeply into literature that ended up being a large focus of my work. I am also grateful for the constellation of friends and colleagues that I have been blessed with in the larger TCS universe: many thanks especially to Henry Yuen for the opportunity to visit Columbia multiple times, and for the friendships that formed there and at Berkeley. My life is greatly enriched by my friends Shivam Nadimpalli, Ansh Nagda, Sam Gunn, Adam Bene Watts, Fermi Ma, and so many others. I did not expect to develop enough inside jokes as to form a crossword puzzle, but here we are. Thanks Sam. The other major vein of my PhD has been in harmonic analysis, and I cannot overstate my gratitude for that community’s welcome and mentorship. When I emailed Paata Ivanisvili about attending an AIM workshop on the analysis of Boolean functions and its relation to quantum computation, little did I know that the connections I formed there would lead to the main body of work in my PhD as well as deep, yearslong friendships. I have learned so much from my collaborators Alexander Volberg, Haonan Zhang, Ohad Klein, and Lars Becker, and look forward to making many more discoveries together in the years to come. And I am especially grateful to Sasha, who embodies the spirit of mathematics with such inspiring force and who has shown me so much about how to think. And also of course to Olga Volberg—I will cherish my memories of the many rich days of math, borscht, and walks around Berkeley with the Volbergs. Finally, I am deeply grateful to Irina Holmes Fay, who originally connected me with Sasha and who has remained a colleague, mentor, and friend ever since. Without Irina’s support and encouragement my PhD would have surely looked very different. iv And thanks to my longterm friends Jake and Allie, whom I met many years before while we were all greenhorn masters’ students. It has been deeply meaningful to stay connected as we’ve all found our ways through our 20s, and such a joy to discuss science and meaning when we have found ourselves in the same city. And finally, I thank my dear family, Audrey, Ben, and Susan, who have patiently listened to oh so many breathless and often dubious explanations of theoretical computer science and math, and who have been constant bulwarks of support and care during the difficult periods. Your ability to deeply appreciate people and ideas of all stripes is a rare thing, and I hope to live and work by that code as best I can in the next chapter. v Abstract This thesis consists of two parts. In Part I we present a new class of norm discretization inequalities suited for low-degree polynomials in many dimensions, with applications to discrete harmonic analysis and to quantum and classical learning theory. Discretization inequalities (of Bernstein type) control the supremum norm of polynomials f by their supremum norms over certain finite subsets T of the domain. Unlike earlier multivariate Bernstein-type discretization inequalities we establish dimension-free comparisons for simple and generic T , such as product sets T = S1 × · · · × Sn for Sj ’s consisting of well-spread points in R or C, in exchange for a constant that grows with deg(f ). Our results also introduce the notion of individual degree—the maximum degree of f in any one variable—as a fundamental parameter for discretization inequalities: we show for the first time that dimension-free discretizations of the uniform norm are possible for T with cardinality independent of deg(f ), provided f has bounded individual degree. Our work offers a new, high-dimensional perspective on discretization inequalities and yields several new results in analysis on the hypergrid (i.e., products of cyclic groups), including Bohnenblust–Hille-type inequalities, dimensionfree supremum norm bounds on level-k Fourier projections, and junta theorems. These estimates in turn provide the key analytic tools for extending recent breakthroughs in learning low-degree functions to the hypergrid and to its quantum analogue, local observables on K-level qudit systems. In Part II we apply ideas from analysis of Boolean functions to study other aspects of (quantum) computation: circuit complexity and property testing. First, we introduce and study a deceptively simple model of constant-depth quantum circuits and begin the project of proving bounds on its capabilities, ultimately drawing on connections to nonlocal games and notions of approximate degree. Second, we introduce a new access model for property testing, quantum data, which allows for ultrafast testing algorithms where classical data provably yields no fast testers—such as for monotonicity, symmetry, and triangle-freeness. vi Published Content and Contributions In order of announcement Part I [SVZ24a] Joseph Slote, Alexander Volberg, and Haonan Zhang. “BohnenblustHille inequality for cyclic groups”. In: Adv. Math. 452 (2024), Paper No. 109824, 35. doi: 10.1016/j.aim.2024.109824. I Equal contribution. [SVZ25] Joseph Slote, Alexander Volberg, and Haonan Zhang. “A dimensionfree Remez-type inequality on the polytorus”. In: Discrete Analysis (to appear) (2025). arXiv: 2305.10828 [math.CA]. I J.S. proved the main theorem, building on tools from [SVZ24a]. [KSVZ24] Ohad Klein, Joseph Slote, Alexander Volberg, and Haonan Zhang. “Quantum and Classical Low-Degree Learning via a DimensionFree Remez Inequality”. In: 15th Innovations in Theoretical Computer Science Conference, ITCS 2024. Ed. by Venkatesan Guruswami. Vol. 287. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, 69:1–69:22. doi: 10.4230/LIPICS.ITCS. 2024.69. – Also appeared at TQC 2024. I Equal contribution. [SVZ24b] Joseph Slote, Alexander Volberg, and Haonan Zhang. Noncommutative Bohnenblust–Hille Inequality for qudit systems. 2024. arXiv: 2406.08509 [math.FA]. I Equal contribution. [Bec+25] Lars Becker, Ohad Klein, Joseph Slote, Alexander Volberg, and Haonan Zhang. “Dimension-free discretizations of the uniform norm by small product sets”. In: Invent. Math. 239.2 (2025), pp. 469–503. doi: 10.1007/s00222-024-01306-9. I Equal contribution on all matters except (i) J.S. did not participate in the proof of the “subsampling theorem” [Bec+25, Theorem 3], appearing here as Theorem 17; and (ii) J.S. independently proved the Lp comparison, [Bec+25, Theorem 5], appearing here as Theorem 7. vii Part II [Slo24] Joseph Slote. “Parity vs. AC0 with Simple Quantum Preprocessing”. In: 15th Innovations in Theoretical Computer Science Conference, ITCS 2024. Ed. by Venkatesan Guruswami. Vol. 287. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, 92:1–92:21. doi: 10.4230/LIPICS.ITCS.2024.92. – Also appeared at TQC 2024. [CNS25] Matthias Caro, Preksha Naik, and Joseph Slote. “Testing classical properties from quantum data”. In: TQC 2025 (to appear) (2025). arXiv: 2411.12730 [quant-ph]. I J.S. was main contributor to “Fourier sampling does not suffice” and “Hardness of 3-fold intersection detection” theorems. J.S. and P.N. made equal contribution to analysis of monotone ensembles. viii Table of Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Published Content and Contributions . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii v vi vii x x x xi I Discretization inequalities and learning 1 Chapter I: Mathematical overview . . . . . . . . . . . . . . . . . . . . 1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Applications in analysis . . . . . . . . . . . . . . . . . . . . . . 1.3 Applications in learning theory . . . . . . . . . . . . . . . . . . Chapter II: From learning theory to discretization inequalities . . . . . Chapter III: The discretization inequality and its accoutrements . . . . 3.1 Theorem 1, sketch of Proof I: via Fourier multipliers . . . . . . 3.2 Theorem 1, sketch of Proof II: via interpolation . . . . . . . . . 3.3 Relationships to other literatures . . . . . . . . . . . . . . . . . Chapter IV: Proof I: Discretization by Fourier Multipliers . . . . . . . 4.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Aside: characterizing inseparable parts for prime K . . . . . . Chapter V: Proof II: Discretization by interpolation . . . . . . . . . . . 5.1 The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Nonexistence of subexponential-cardinality meshes . . . . . . . Chapter VI: Applications . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Lp discretization inequalities . . . . . . . . . . . . . . . . . . . 6.2 Junta theorem for functions on the hypergrid . . . . . . . . . . 6.3 Qudit Bohnenblust–Hille in the Heisenberg–Weyl basis . . . . . 6.4 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7 8 10 12 22 26 29 31 35 35 38 47 50 50 57 59 59 60 61 72 II Other applications 77 Chapter VII: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VIII: Parity vs. AC0 with simple quantum preprocessing . . . 8.1 Lower bounds when QNC0 is ancilla-free . . . . . . . . . . . . . 8.2 Lower bounds against AC0 ◦ QNC0 via nonlocal games . . . . . 79 83 90 94 ix 8.3 Towards a switching lemma for AC0 ◦ QNC0 . . . . . . . . . . . 8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . Chapter IX: Testing classical properties from quantum data . . . . . . 9.1 Passive quantum testing upper bounds . . . . . . . . . . . . . 9.2 Fourier sampling does not suffice . . . . . . . . . . . . . . . . . 9.3 Separating passive quantum from query-based classical property testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 A challenge: lower bounds for monotonicity testing . . . . . . . 9.5 Appendix: Useful facts . . . . . . . . . . . . . . . . . . . . . . 100 107 108 109 123 135 Bibliography 162 138 143 158 x List of Illustrations Number Page 1.1 Failure of the “discrete maximum modulus principle” . . . . . . 5 3.1 A visual depiction of Theorem 14 . . . . . . . . . . . . . . . . . 23 9.1 Property testing resource inequalities . . . . . . . . . . . . . . . 116 List of Tables Number Page 9.1 Upper and lower bounds for testing and learning . . . . . . . . 110 9.2 Our bounds in context . . . . . . . . . . . . . . . . . . . . . . . 118 List of Symbols ZK cyclic group of order K R real numbers C complex numbers Tn unit n-torus (or polytorus) {z ∈ C : |z| = 1}n Dn unit n-disk (or polydisk) {z ∈ C : |z| ≤ 1}n ΩK K th roots of unity, multiplicative representation of ZK k · kX supremum norm over domain X D maximum support pseudoprojection, see Definition 1 xi Conventions Computer Science • [n] := {1, 2, . . . , n} • x ∼ X for X a finite set means denotes a sample from the uniform probability measure over X. e (n)) means there exists a k ∈ N for • The “soft-O” notation g(n) ≤ O(f which   g(n) ≤ O f (n) · logk n .  d • O attaches first in Landau notation: O(x)d := O(x) Analysis • Integral norms. For functions f : X → C, X finite, the notation kf kp will denote Lp norm of f w.r.t. the uniform measure. The notation kfbkp denotes the `p norm of the Fourier transform of f (with respect to counting measure). • Vinogradov notation. For two quantities X and Y , the notation X . Y means there exists a universal constant C > 0 such that X ≤ C · Y . The notation X .` Y allows the implicit constant C = C(`) to depend on ` only. We do not allow an additive constant (e.g., the D in X ≤ C Ẏ + D) in Vinogradov notation, which distinguishes it from the Landau notation X ≤ O(Y ). • Quantities X and Y are comparable, X ≃ Y , if that X . Y and Y . X. Part I Dimension-free discretization inequalities with applications to learning theory 1 2 Published as In order of announcement: [SVZ24a] Joseph Slote, Alexander Volberg, and Haonan Zhang. “BohnenblustHille inequality for cyclic groups”. In: Adv. Math. 452 (2024), Paper No. 109824, 35. doi: 10.1016/j.aim.2024.109824. [SVZ25] Joseph Slote, Alexander Volberg, and Haonan Zhang. “A dimensionfree Remez-type inequality on the polytorus”. In: Discrete Analysis (to appear) (2025). arXiv: 2305.10828 [math.CA]. [KSVZ24] Ohad Klein, Joseph Slote, Alexander Volberg, and Haonan Zhang. “Quantum and Classical Low-Degree Learning via a DimensionFree Remez Inequality”. In: 15th Innovations in Theoretical Computer Science Conference, ITCS 2024, January 30 to February 2, 2024, Berkeley, CA, USA. Ed. by Venkatesan Guruswami. Vol. 287. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, 69:1–69:22. doi: 10.4230/LIPICS.ITCS.2024.69. [SVZ24b] Joseph Slote, Alexander Volberg, and Haonan Zhang. Noncommutative Bohnenblust–Hille Inequality for qudit systems. 2024. arXiv: 2406.08509 [math.FA]. [Bec+25] Lars Becker, Ohad Klein, Joseph Slote, Alexander Volberg, and Haonan Zhang. “Dimension-free discretizations of the uniform norm by small product sets”. In: Invent. Math. 239.2 (2025), pp. 469–503. doi: 10.1007/s00222-024-01306-9. 3 Chapter 1 Mathematical overview A summary of the mathematical contributions in Part I. For motivations from computer science, one may skip to Chapter 2. The analysis of Boolean functions is by now a well-established theory and an essential fiber in the fabric of theoretical computer science and discrete mathematics more generally. Any function f : {−1, 1}n → R is uniquely expressed as a multilinear (or multi-affine) polynomial f (x) = P S⊆[n] f (S)χS b with χS := Q j∈S xj , and various measures and aspects of the Fourier coefficients {fb(S)}S⊆[n] provide significant insight into the structure of f . This perspective has led to many discoveries in computational complexity theory, learning theory, voting theory, coding theory, and combinatorics, among other areas [ODo14]. But not every function encountered in theoretical computer science is on the hypercube. A key generalization of {±1}n is the so-called hypergrid, [K]n , conveniently represented as the product of multiplicative cyclic groups ΩnK := {exp(2πik/K) : k = 0, 1, . . . , K − 1}n . Harmonic and functional analysis on ΩnK is important in combinatorics [ALM91; Mes95], number theory [BS80], and graph theory [ADFS04], and naturally models many-candidate social choice functions and even certain aspects of K-level qudit systems in quantum computing. Coming from the hypercube {±1}n = Ωn2 , some aspects of functional and harmonic analysis on ΩnK for general K are familiar. For example, the basics of influence and hypercontractivity are well-understood in this setting [Wei80; JPPP17]. But the analysis of functions on ΩnK is not just a retelling of the Boolean story; new and formidable barriers appear already at K = 3. One example is the well-known Plurality is stablest conjecture from hardness of approximation [KKMO07], which remains open even though the Boolean case (Majority is stablest) was resolved 20 years ago [MOO10] (announced in 2005). In Part I we present a sequence of works surmounting another barrier in 4 the analysis over ΩnK appearing only for K ≥ 3, motivated by a desire to extend a recent breakthrough in learning theory by Eskenazis and Ivanisvili [EI22] to more-general product spaces and their quantum analogues. Ultimately we reach beyond this goal, obtaining results with consequences for approximation theory and discrete analysis more broadly. A challenge Analysis over ΩnK is hard in part because Fourier expansions of functions on ΩnK , K ≥ 3 are no longer multilinear. As we elaborate in the next chapter, for our target applications an essential technical step is to compare the supremum norm of f : ΩnK → C to the supremum norm of its extension to the product of convex hulls of ΩK , conv(ΩK )n . Specifically, for f of degree at most d, we seek a comparison independent of dimension n of the form ? kf kconv(ΩK )n ≤ C(K, d)kf kΩnK . (1.0.1) Here and throughout, k · kX will denote the supremum norm over a domain X. To make sense of (1.0.1), recall that any f : ΩnK → C admits the unique Fourier expansion f (z) = X α∈{0,1,...,K−1}n fb(α)z α with z α := n Y α zj j . j=1 In this way, f can be uniquely extended to an analytic polynomial over Cn . Throughout Part I we will conflate f : ΩnK → C with its associated polynomial and use terms like the degree of f to mean its total degree as an analytic polynomial. Note also that all polynomials arising thusly have individual degree (i.e., the largest degree of any coordinate) at most K − 1. Our results will hold for domains beyond ΩnK so we will find it most natural to state our results directly for analytic polynomials of bounded individual degree. Now that we can parse Equation (1.0.1), what does it mean? One might compare it to the maximum modulus principal, which in this context would say that kf kconv(ΩK )n ≤ kf kPKn , where PK is the boundary of conv(ΩK ), i.e., the boundary of the regular K-gon. Equation (1.0.1) asks whether a discrete grid of points can be used in place of the entire product of boundaries, possibly at the expense of introducing a dimension-free multiplicative constant. 5 Figure 1.1: Failure of the “discrete maximum modulus principle.” Here we plot the modulus of p(z) = z 2 /3 − z/3 + 1 evaluated on conv(Ω3 ). Notice that while |p(z)| is greatest on the boundary of conv(Ω3 ) (the maximum modulus principle), it is not maximized by a vertex from Ω3 itself. We conclude kf kconv(ΩK )n ≤ Ckf kΩnK cannot hold with constant 1 in general. Let us sketch the difficulty in resolving (1.0.1) for different K. When K = 2, the comparison is a trivial equality with constant C = 1: because functions on the hypercube have multilinear extensions, we obtain kf k[−1,1]n = kf k{±1}n by convexity. But at K = 3 this doesn’t work; in fact one can prove (see Figure 1.1 and [SVZ24a, Appendix B]) that even for n = 1 we must have C(d, K) > 1 in (1.0.1). As a result, naive approaches to proving (1.0.1) for K ≥ 3 yield constants with exponential dependence on n. On the other end of the K spectrum, if we take “K = ∞” and consider analytic functions on the polytorus Tn := {z ∈ C : |z| = 1}n , the question is trivial, and again we get equality with constant 1: kf kconv(T)n = kf kDn = kf kTn (this of course is just the maximum modulus principle). But there is no such easy fact for finite K because ΩK is not the entire boundary of conv(ΩK ). There seems to be a particular difficulty for finite K ≥ 3. 6 Norm discretization inequalities In approximation theory, comparisons of the kind (1.0.1) are known as Bernstein-type discretization inequalities—or discretizations of the uniform norm—and there is a vast body of work on the subject; see for example the surveys [DPTT19; KKLT22].1 Bernstein-type discretization inequalities have the general shape of kf kX ≤ C(d, n)kf kT (1.0.2) for some domain X (usually convex) and finite sampling set T ⊂ X. The typical goal for multivariate Bernstein-type inequalities is to establish a comparison of the kind (1.0.2) for C independent of degree d, but possibly depending on dimension n, while keeping T as small as possible. While this is the most natural multivariate generalization of the univariate Bernstein discretization inequality [Ber31; Ber32] (where independence of C from d was of course paramount), dependence of C on dimension n makes much of the norm discretization literature inapplicable for our hoped-for bound (1.0.1). On the other hand, in some cases results with universal constants C are known, for example in the recent work of Dai and Prymak [DP24] resolving an important conjecture of Kroó [Kro11]. But these results also do not seem to apply either. The most salient reason is that the literature does not seem to distinguish between individual degree K − 1 and (total) degree d, and as a result the cardinality of T always depends directly on deg(f ). For our purposes, where d can easily grow much larger than K, we still require T = ΩnK . And even if we restrict ourselves to the very special case of d ≤ K, there is the issue of the structure of T . ΩnK has a very simple but rigid product structure, whereas Bernstein-type discretization inequalities in the literature with C independent of n are either proofs of existence of T of low cardinality, or they construct T ’s that have intricate ε-net-type structures, as is the case for the sampling sets in Dai and Prymak [DP24], which are far from being product sets. In fact, as we explain in the sequel, it turns out that for T = ΩnK to be a workable sampling set at all for us (i.e., yielding C independent of n), the constant C must depend on degree d. So our hoped-for norm discretization inequality is of a distinct flavor from the traditional results of multivariate norm discretization. Indeed, if (1.0.1) were true, it would represent a new perspective in high-dimensional norm discretization—one that demands C be 1 Warning: in the norm discretization literature, d is typically used for dimension and n for degree, precisely the opposite of what is typical in analysis of Boolean functions. 7 independent of dimension while allowing degree dependence—and would show that new and interesting features appear in this regime, such as the role of individual degree. 1.1 Results The first headline result of Part I is a resolution of (1.0.1). (It turns out kf kconv(ΩK )n and kf kTn are easily comparable independent of dimension2 so we will state results using the latter as it is cleaner.) In fact, a broad generalization of (1.0.1) is proved, but we stick to the below for the purposes of this discussion. Theorem 1 (Dimension-free discretization of the uniform norm). Let f be an n-variate analytic polynomial of degree d and individual degree K − 1. Then kf kTn ≤ O(log K)d kf kΩnK . We will give two proofs of Theorem 1 which generalize it in different ways. Proof I: Fourier multipliers [SVZ25]. This is the historically-first proof and obtains only an implicit constant (still dimension-free) in place of O(log K)d . Along the way to Theorem 1 the proof develops a rich class of Fourier multipliers that are L∞ -bounded independent of dimension and may be of independent interest. For prime K this class of multipliers is characterized exactly thanks to connections to transcendental number theory and Baker’s celebrated theorem on the logarithms of algebraic numbers [Bak22]. Theorem 2 (Bounded Fourier projections, prime K ≥ 3). Suppose K is an odd prime and let S be a maximal subset of {0, 1, . . . , K − 1}n such that for all α, β ∈ S: • (Total) degrees are equal: Pn j=1 αj = j=1 βj . Pn • Individual degree symmetry: there is a bijection π : [n] → [n] such that for all j ∈ [n], αj = βπ(j) or αj = K − βπ(j) . (In particular, α and β are nonzero on the same number of indices.) Then for any n-variate analytic polynomial f of degree at most d and individual P degree at most K − 1, the S-part of f , fS (x) := α∈S fb(α)z α , satisfies kfS kΩn .d,K kf kΩnK . K 2 One direction is immediate. For the other, let cK > 0 be such that cK T ⊂ conv(ΩK ). Because f has bounded degree, kf kTn ≃ kf k(cK T)n , which is at most kf k(convΩK )n . 8 Proof II: polynomial interpolation [KSVZ24; Bec+25]. This proof is probabilistic and obtains Theorem 1 with the quoted constant. The key technique is a novel interpolation formula, which is our second headline result. Theorem 3 (Dimension-free multivariate interpolation). For any z ∈ Dn (z) there exist explicit coefficients {ax }x∈ΩnK such that for any n-variate analytic polynomial f of degree d and individual degree K − 1, f (z) = X a(z) x f (x) , x∈Ωn K and (z) d x |ax | ≤ O(log K) . P While interpolation of degree-d polynomials f can be accomplished by much smaller sets than ΩnK , Theorem 3 appears to be the first multivariate (z) interpolation formula for which the coefficients {ax }x have `1 norm independent of dimension n. Moreover, a point set with cardinality exponential in n, like ΩnK , is necessary to get ka(z) k`1 independent of dimension. This dimensionindependence is crucial for obtaining Theorem 1 from Theorem 3 and to downstream applications. We hope Theorem 3 finds other uses in approximation theory and analysis. Remark. Theorems 1 and 3 may actually be generalized from ΩnK to a very generic class of sampling sets S ⊂ Dn (though the K-dependence of the constant becomes more complicated). This is done in the work [Bec+25]; see Chapter 3 for the general statement. 1.2 Applications in analysis Theorems 1 and 3 have several consequences in analysis and approximation theory. New Bohnenblust–Hille inequalities Theorem 4 (Cyclic-group Bohnenblust–Hille inequality). Let f : ΩnK → C have degree at most d. Then kfbk 2d .K,d kf kΩnK . d+1 Here and throughout, kfbkp denotes the `p norm of the Fourier coefficients for f . This theorem extends to ΩnK the classical inequality of Bohnenblust and Hille (BH), which originally appeared in the study of Dirichlet series and has a long 9 history in harmonic analysis. The hypercube formulation was studied more recently [Ble01; DMP18] and has found surprising applications in computer science. The BH inequality for ΩnK , K ≥ 3 was not known until we gave a proof in [SVZ24a]. That publication actually predates our discretization inequality (Theorem 1), but we elected to omit [SVZ24a] from this thesis because (i) with the discretization inequality in hand, Theorem 4 becomes a one-line reduction to the original BH inequality [BH31]; and (ii) the techniques employed in [SVZ24a] are directly subsumed by those in the Fourier multiplier proof of Theorem 1, [SVZ25]. Theorem 1 is also a key ingredient in the proof of a noncommutative (or quantum) Bohnenblust–Hille inequality. Theorem 5 (Qudit Bohnenblust–Hille inequality). Let A be a d-local quantum observable on n-many K-level qudits. Then b 2d . kAk kAkop . d,K d+1 The “Fourier transform” Ab here refers to the vector of coefficients obtained by decomposing A in the Heisenberg–Weyl basis, which is a unitary generalization of the familiar Pauli basis. This is discussed in detail in Section 6.3. Theorem 5 generalizes the qubit BH inequality that was first proved in [HCP22] and then improved in [VZ23]. The proof of Theorem 5 combines Theorem 1 with a careful analysis of the commutation structure of Heisenberg– Weyl matrices. Level-` Fourier projections are bounded Theorem 6 (Boundedness of the level-` Fourier projection). Let f : ΩnK → C have degree at most d and let ` be a positive integer. Then with f` = P α b |α|=` f (α)z denoting the `-homogeneous part of f , kf` kΩnK .K,d kf kΩnK . This theorem shows that the level-` Fourier projection, when applied to lowdegree polynomials, is bounded independent of dimension. This is a common Fourier multiplier-type estimate, and dimension free-ness for polynomials on ΩnK , K ≥ 3 was not known until the discovery of Theorem 1. The K = 2 case has a short proof usually attributed to Figiel [MS86, §14.6]. When K is prime Theorem 6 can be seen as a specialization of the very fine-grained projection theorem Theorem 2, though more generally it follows easily from Theorem 1 and can be stated with the explicit constant O(log K)d . 10 Lp discretization Theorem 7 (Dimension-free discretization of Lp norms). Let d, n ≥ 1, K ≥ 2. Let 1 ≤ p ≤ ∞. Then for any polynomial f : Tn → C of degree at most d and individual degree at most K − 1, we have kf kLp (Tn ) ≤ d · O(log K)d kf kLp (ΩnK ) . This Lp discretization inequality could be called a Marcinkiewicz–Zygmund-type (MZ-type) discretization inequality, though it differs from the typical MZ-type inequalities in the literature in that the constant in Theorem 7 is independent of p but depends strongly on d. The proof combines the interpolation formula of Theorem 3 with invariance of the uniform (Haar) measure on ΩnK . One can also get a dimension-free comparison à la Theorem 7 via hypercontractivity [Wei80; JPPP17], but again the constant will depend on p. 1.3 Applications in learning theory A basic task in learning theory is to learn an L2 approximation to a degree-d function f , given access to random samples. The naive algorithm requires O(nd ) samples but we obtain an algorithm with exponentially-better n dependence. Theorem 8 (Cyclic-group low degree learning). Let f : ΩnK → D have degree d. Then with    d+1 n 1 2 (log K)O(d ) · log · δ ε independent random samples of the form (x, f (x)), x ∼ ΩnK , we may with confidence 1 − δ learn a function fe : ΩnK → C with kf − fek22 ≤ ε. This learning theorem extends a breakthrough result of Eskenazis and Ivanisvili [EI22] from functions on the hypercube to those on ΩnK , and is obtained by combining the cyclic-group Bohnenblust–Hille inequality, Theorem 4, with ideas in [EI22]. An analogous result holds in the context of quantum computing, where role of the function f : ΩnK → C is taken by a quantum observable on K-level qudits. Theorem 9 (Low-degree Qudit Learning). Let A be a degree-d (or d-local) observable on n K-level qudits with kAkop ≤ 1. Then there is a collection S of product states such that with a number O  KkAkop C·d2 2 −d−1 dε   log nδ 11 of samples of the form (ρ, tr[Aρ]), ρ ∼ S, we may with confidence 1 − δ learn e 2 ≤ ε. an observable Ae with kA − Ak 2 See Section 6.4 for the full definition and explanation. I am very grateful to my coauthors Lars Becker, Ohad Klein, Alexander Volberg, and Haonan Zhang for their collaboration in the various papers constituting Part I. 12 Chapter 2 From learning theory to discretization inequalities We chart a path from motivations in quantum and classical learning theory to our discretization inequality, Theorem 1. Boolean functions whose multilinear expansions are low degree (think of deg(f ) as much less than dimension n, or even constant) are ubiquitous in computer science [ODo14] and it is a fundamental task to learn them from random samples. Task (Low-degree learning). Using R-many uniformly-random samples n oR x(r) , f (x(r) ) r=1 of an unknown Boolean function f : {±1}n → [−1, 1] of degree d, produce with probability 2/3 a function g : {±1}n → R such that kf − gk2 ≤ ε. Here the norm k · k2 is with respect to the uniform probability measure on {±1}n , and the figure of merit for the task is sample complexity, i.e., the dependence of R on d, ε, and especially n. e (nd )In 1993 Linial, Mansour, and Nisan [LMN93a] gave a very natural O ε sample algorithm for low degree learning: from samples one may form the empirical Fourier coefficients b g(S) := R1 PR    (r) χS x(r) r=1 f x  for all S ⊆ [n] with |S| ≤ d, and from there define the estimator g := P b S. |S|≤d g(S)χ The analysis is textbook: to obtain, say, kf − gk2 ≤ 0.01 = ε, it suffices to get b |fb(S) − g(S)| ≤ 0.001/nd for each S ⊆ [n], |S| ≤ d, (2.0.1) for then kf − gk2 (Plancherel) = b 2 ≤ kfb − gk b 1 = kfb − gk X |S|≤d b |fb(S) − g(S)| ≤ 0.01. 13 And standard concentration arguments say (2.0.1) can be accomplished with e d ) samples with high probability. O(n For almost 30 years this sample complexity stood without improvement, and for good reason: a natural heuristic calculation suggests no improvement is available. Degree-d polynomials have roughly nd Fourier coefficients, and each is bounded by kfbk∞ ≤ 1, so we might expect an ε-net of such f has cardinality d at least (1/ε)n —which would require Ω(nd log(1/ε)) samples to learn. e (nd−1 )-sample algorithm Yet, 28 years after [LMN93a], [Iye+21] found a O ε based on improved bounds on the Fourier growth of low degree polynomials, i.e., P bounds on |S|=k |fb(S)| for degree-d f as a function of d, k, and n. Although in some sense a modest improvement on the performance of the basic low degree learning algorithm, the work of [Iye+21] already contradicts the heuristic lower bound just described. So what is wrong with our back-of-the-envelope calculation? It turns out we failed to take into account the interaction of the degree constraint and the L∞ bound constraint on f . Soon after the work of [Iye+21], Eskenazis and Ivanisvili [EI22] showed that these two constraints e (log n)-sample algorithm was possible. interact so strongly that actually an O ε,d The breakthrough of Eskenazis–Ivanisvili Eskenazis and Ivanisvili leveraged a theorem about bounded low-degree functions coming from classical harmonic analysis, Bohnenblust–Hille (BH) inequality, that is at this point quite classical [BH31]. The original BH inequality applies to degree-d analytic polynomials on Dn . Theorem 10 (Bohnenblust–Hille inequality [BH31]). Let f : Tn → C be an analytic polynomial of degree at most d. Then kfbk 2d ≤ Cd kf kTn . d+1 Here and throughout the norm kgbkp denotes the `p norm of the Fourier coefficients of g. The two key aspects of the BH inequality are (i) the independence of the constant Cd from dimension n, and (ii) that the `p norm on the left-hand side is for a p strictly less than 2. For p = 2 the comparison of course holds with constant 1 for all functions (Plancherel); actually p = 2d/(d + 1) is the smallest p admitting a dimension-free comparison of the kind above [BH31]. Together the properties (i) and (ii) place powerful constraints on the Fourier spectrum of bounded, low-degree polynomials, as we will detail in the sequel. We also remark that in many applications the dependence of Cd on d is quite 14 p important; the best bound we know currently is Cd ≤ C d log d due to [BPS14]. See [DGMS19, Ch. 6] for more on the classical Bohnenblust–Hille inequalities. The analogous inequality for Boolean functions has a shorter history. The hypercube BH inequality was originally proved with an implicit constant Cd by Blei [Ble01] and then was only very recently improved to the subexponential p d log d Cd = C [DMP18]. Again, p = 2d/(d + 1) is the best possible. Theorem 11 (Hypercube BH inequality). Let f : {±1}n → R be a Boolean function of degree d. Then kfbk 2d ≤ Cd kf k∞ . (2.0.2) d+1 Moreover, Cd ≤ C p d log d for a universal constant C. How can we make use of such a bound? The insight of Eskenazis–Ivanisvili is as follows. Beginning with some unknown vector v ∈ Rn (such as the vector of Fourier coefficients of f ), if one has an `∞ estimate of v, i.e., a w such that kv − wk∞ ≤ ε, as well as the guarantee that kvk`p is bounded for some p < 2, then w can be improved to an `2 estimate we of v, still controlled by ε: p e 2 . ε1− 2 . kv − wk Specifically, we is obtained from w by replacing small entries by 0—a task that crucially does not need any knowledge of v’s entries.1 Lemma 12. Let p ∈ [0, 2) and ε, B > 0. Suppose v, w ∈ Cn with kv −wk∞ ≤ ε and kvkp ≤ B. Then for we defined as   w j if |wj | ≥ ε 1 + 0 otherwise, wej =   q 2 2−p  (2.0.3) we have the bound p kwe − vk2 ≤ 5Bε1− 2 . 1 With the one exception of choosing the threshold for “small,” which is set in terms of the norm bounds just mentioned. 15 With these pieces in place, the algorithm of Eskenazis–Ivanisvili is straightforward: begin by forming the empirical vector of Fourier coefficients gb as b ∞ ≤ ε. before, using enough samples so that with high probability, kfb − gk From Theorem 11 we know kfbk 2d ≤ O(1) always, so we are exactly in the d+1 situation of Lemma 12: replacing the small entries of gb with 0, we may obtain a b of fb with the guarantee that kfb− hk b new estimator h 2 = kf − hk2 is controlled by a power of ε, without any n-dependence in the inequality. As a result, we b ∞ ≤ poly(ε), which is much only need to gather enough data so that kfb − gk d easier to achieve than the O(ε/n ) requirement of the naive algorithm. In fact, the only source of n dependence ends up being the Od (log n) copies required for concentration of measure to overcome the loss when union-bounding over the events n o b |fb(S) − g(S)| ≤ O(1) . S⊂[n],|S|≤d b for fb could be called a We will not delve into it here, but the estimator h hard-thresholding estimator in statistics. The proof of Lemma 4 appears later on in Section 6.4, but for now we just point out that p being strictly less than 2 is absolutely critical: eventually one needs the `2 norm of the small entries of e to be bounded by a function of ε ≥ kv − wk∞ . v (the ones zeroed-out in w) With t = t(ε, p) the threshold in (2.0.3), X j:|wj |<t |vj |2 = X |vj |2−p |vj |p ≤ (t + ε)2−p j:|wj |<t X |vj |p ≤ (t + ε)2−p B 1/p . j:|wj |<t And recalling t is linear is ε, we recognize the right-hand side of this display as CB,p ε2−p . On the other hand, if we only had the trivial bound kvk`2 ≤ 1, there is no reason the `2 norm of the small entries of v should go to 0 as ε → 0. We also remark that while the algorithm of Eskenazis–Ivanisvili achieves a log(n) sample complexity, the time complexity of the algorithm is still O(nd ) b because each g(S) for |S| ≤ d is compared with the threshold in sequence. It is an interesting open problem to determine if the runtime can be improved to o(nd ), as the related problem of k-junta learning has seen a few improvements of this kind, e.g., [Val15]. Quantum generalizations Soon after the Eskenazis–Ivanisvili breakthrough, Rouzé, Wirth, and Zhang conjectured a quantum generalization [RWZ24a]. The idea is to learn local quantum observables. An (n-qubit) quantum observable is a Hermitian matrix 16 A ∈ M2n ×2n (C). Such an observable models the following process: an n-qubit quantum state ρ is subjected to an unknown transformation (quantum channel) N , and then a numerical measurement (with outcomes in, say, [−1, 1]) is performed on the output. The matrix A captures the statistics of this process, in the sense that tr[Aρ] is the expected measurement outcome for N (ρ). Indeed, P if {Nj }j are the Kraus operators for N (so that N (ρ) = j Nj ρNj† ) and M is the measurement operator of the measurement device, then A= † j Nj M Nj . P It is a natural task to learn a description of the matrix A so that the expectation value map ρ 7→ tr[Aρ] can be predicted for new ρ. Learning descriptions of arbitrary 2n × 2n matrices is very difficult, so one simplifying assumption comes by way of a quantum (or noncommutative) generalization of polynomial degree bounds. Any n-qubit observable A admits a unique Fourier-like decomposition A= X σα := b A(α)σ α, α∈{0,1,2,3}n n O σ αj , j=1 where the σj ’s are the Pauli matrices   1 0 σ0 :=  , 0 1   0 1 σ1 :=  , 1 0   0 −i σ2 :=  , i 0   1 0 σ3 :=  . 0 −1 b The observable A is d-local if A(α) = 0 when |α| > d, where |α| is the number of entries j for which αj 6= 0. This is all looking very familiar. As we explain in the Section 6.4, by obtaining samples of the form (ρ, tr[Aρ]) for a certain distribution of states ρ, it is easy to construct an estimator for the “Pauli coefficients” of A, and one might hope for an Oε,d (log n)-sample learning algorithm for d-local A. The missing piece was a quantum (qubit) version of the Bohnenblust–Hille inequality. Theorem 13 (Qubit BH Inequality). Let A be a d-local observable. Then b 2d ≤ C kAk . kAk d op d+1 This theorem was conjectured in [RWZ24b] and a proof appeared very soon after in [HCP22] with constant Cd = O(dd ). A very different proof that improves the constant to Cd = C d was then given by Volberg and Zhang [VZ23]. 17 This second proof is very efficient and directly reduces to the hypercube BH inequality. In brief, for any d-local observable A the authors identify a degree-d Boolean function fA : {±1}3n → R such that b 2d ≤ 3d kfc k 2d kAk A d+1 d+1 and kfbA k{±1}3n ≤ kAkop . (2.0.4) The hypercube BH inequality joins these two bounds to complete the theorem. To larger product spaces It is natural to ask whether this story continues to larger product spaces (quantum and classical), and this was the starting point for the work presented in Part I of this thesis. Classically, such an extension looks like learning lowdegree functions on products of cyclic groups ZnK , which we shall represent by the multiplicative group of K th roots of unity, ΩnK . As mentioned in the previous chapter, the space ΩnK is sometimes called the “hypergrid” in property testing [CS14; BCS23], is a key setting for studying the hardness of approximation (e.g., the Plurality is Stablest Conjecture of [KKMO07], see also [MOO10]), and appears frequently in coding theory and cryptography. The argument of Eskenazis–Ivanisvili generalizes directly to such functions, provided the appropriate BH inequality can be proved: Conjecture 1 (Cyclic-group Bohnenblust–Hille inequality). Let f : ΩnK → C have degree d. Then kfbk 2d ≤ Cd,K kf kΩnK . d+1 As will be explained soon, it is not possible to mimic the proof of the hypercube (or the original polytorus) BH inequality for functions on ΩnK . This conjecture is the main sticking point for extending classical low-degree learning to larger product spaces. The quantum case of generalized low-degree learning is just as important, both for the study of fundamental physics via quantum simulation (e.g., [Kur+21; Gon+22]) and in the operation and validation of quantum computers. In both contexts, gains in efficiency are possible when the underlying hardware system is composed of higher-dimensional subsystems, sometimes carrying an algorithm from theoretical fact to practical reality in the NISQ era [Gon+22]—and this benefit may very well remain as quantum computing advances. Such devices are called multilevel system, or qudit, quantum computers [WHSK20]. While the qubit case gives a conceptual sense of the possibilities 18 for learning on qudit systems, it is of practical value to derive guarantees and algorithms that work directly in the native dimension of the quantum system. To formulate a Bohnenblust–Hille inequality for qudit systems, we must decide on a qudit generalization of Pauli matrices. A natural choice are the Heisenberg–Weyl (HW) matrices, which are their unitary generalizations. We defer a formal definition till Section 6.3. Conjecture 2 (Qudit BH inequality). Let A be a degree-d (or d-local) observable on n-many K-level qudits. Then b 2d ≤ C kAk . kAk d,K op d+1 Following the approach of Zhang and Volberg, there are two steps to proving the qudit BH inequality. The first is, given an n-qudit operator A, to identify a commutative polynomial fA with bounds analogous to Equation (2.0.4). This is technically much more challenging than in the qubit case because the Heisenberg–Weyl matrices have a very intricate commutation structure. Partial results on this part appeared in [KSVZ24] and the full reduction finished in [SVZ24b], the proof of which is included in Section 6.3. The other part is to prove the appropriate commutative BH inequality, which turns out again to be the cyclic-group Bohnenblust–Hille inequality because the eigenvalues of HW matrices are roots of unity. Therefore our goals of generalizing low-degree learning to larger product spaces, both classically and quantumly, dovetail into the task of proving a cyclic-group Bohnenblust–Hille inequality. BH for cyclic groups: the barrier and a first resolution Although a cyclic-group BH inequality is the natural interpolating statement between the now well-understood hypercube BH (polynomials on Ωn2 ) and the classical BH inequality, for polynomials on Tn (or “Ωn∞ ”), there was no proof in the literature. And this is for good reason, as the regime of 2 < K < ∞ faces unique difficulties. At a very coarse level, the hypercube and polytorus BH inequalities are both proved in the following steps [DS14] (using X n to represent either product space): 1. Symmetrization: express f as the restriction of a certain symmetric d-linear form Lf to the diagonal ∆ := {(z, . . . , z) : z ∈ X n }; that is, f (z) = Lf (z, . . . , z). 19 2. BH inequality for multilinear forms: bound the `2d/(d+1) norm of the coefficients of Lf (which are directly related to the coefficients of f ) by the supremum norm of Lf over (X n )d . This step is rather involved and includes several estimates, manipulations, and an application of hypercontractivity and Khintchine’s inequality. 3. Polarization: estimate the supremum norm of Lf on its entire domain (X n )d by the supremum over ∆; that is, kLf k(X n )d . kLf k∆ = kf kX n , where k · kE denotes the supremum norm over some space E. When adapting this proof structure to cyclic groups of order 2 < K < ∞, the main point of failure is in step three, polarization, as is fully worked out in [SVZ24a, Appx. A]. In both the polytorus and hypercube cases, one uses Markov–Bernstein-type inequalities to obtain the intermediate comparison kLf k(X n )d . kf kconv(X)n , where conv(X) denotes the convex hull of X. The passage from conv(X) to X is then immediate for the polytorus by the maximum modulus principle (kf kDn = kf kTn ) and for the hypercube by multilinearity (kf k[−1,1]n = kf k{−1,1}n ). It is at this most trivial step that the BH proof breaks down for K > 3: there is no such easy fact in the setting of the multiplicative cyclic group ΩK := {e2πik/K : k = 0, . . . , K − 1} with 2 < K < ∞ because ΩK is not the entire boundary of conv(ΩK ). Even for n = 1 and K = 3 one can construct example f ’s for which kf kconv(ΩK )n > kf kΩnK . It’s easy to show that kf kconv(ΩK )n and kf kDn = kf kTn are comparable independent of dimension, so essentially one is left wondering whether kf kTn .d,K kf kΩnK . (2.0.5) And in fact, if one could prove such a comparison, then there would be no need to retrace the rest of the BH proof recipe in the first place! One would immediately get the cyclic-group BH inequality via (Classical BH) kfbk 2d d+1 .d (2.0.5) kf kTn .d,K kf kΩnK . The inequality (2.0.5), a certain “discretization of the uniform norm,” and its generalizations are the main contribution of Part I. For now, let us remark that 20 the cyclic-group BH inequality was first proved in [SVZ24a] by a method that actually avoided proving the comparison (2.0.5) and instead made an intricate reduction to the hypercube BH inequality. That argument has two main parts. The first is to prove a cyclic-group BH-type inequality for support-homogeneous polynomials, i.e., those whose monomials all contain the same number of distinct variables. The second is to show the comparison kf(`) kΩnK .d,K kf kΩnK where f(`) is the `-support-homogeneous part of a degree-d polynomial f . The constant in the comparison, while free of dependence on n, was quite large and never estimated explicitly. We have elected to not include the proof of [SVZ24a] in this thesis as the techniques involved were later sharpened and used again in [SVZ25], the first proof of (2.0.5). The discretization inequality: a lingering desire Although the cyclic-group BH inequality was established in [SVZ24a], the question of whether a comparison of the type (2.0.5) is possible still lingered. And from the point of view of the larger project of harmonic analysis on ΩnK , there are other motivations for proving (2.0.5). New difficulties arise when trying to get other standard estimates in this space too. For example, consider the boundedness of k-level Fourier projections. Concretely, given a polynomial f : ΩnK → C of total degree at most d and individual degree at most K − 1, we seek to control its degree-` homogeneous part f` as follows: kf` kΩnK .d,K kf kΩnK . (2.0.6) This is a typical kind of dimension-free estimate in harmonic analysis, and something that is very easy to accomplish both on Ωn2 and on Tn . When the domain is the polytorus Tn , this comparison is a standard Cauchy estimate and bears constant 1: given an analytic function f : Tn → C and z ∗ a maximizer of |f` |, define the univariate function Q(t) = f (tz1∗ , . . . , tzn∗ ) for t ∈ T. Taking the `th derivative and appealing to the Cauchy’s integral formula, we find |Q(`) (0)| 1 Z Q(t) kf` kTn = = dt ≤ kQkT ≤ kf kTn . `! 2π T t`+1 (2.0.7) For the hypercube (K = 2), this estimate is similar and usually attributed to Figiel [MS86, §14.6]. Given f : Ωn2 → R, deg(f ) ≤ d, and with x∗ ∈ Ωn2 a maximizer of |f` |, one considers the polynomial Q(t) := f (tx∗1 , . . . , tx∗n ) for 21 t ∈ [−1, 1] (f is extended to [−1, 1]n as a multilinear function). A Markov– Bernstein-type estimate gives |Q(`) (0)| ≤ C(d, `)kQk[−1,1] ≤ C(d, `)kf k[−1,1]n , `! √ with optimal constant C(d, `) ≤ (1 + 2)d [DMP18, Lemma 1.3 (4)]. The final step is to recognize that kf k[−1,1]n = kf k{−1,1}n because the extension of f to [−1, 1]n is affine in each coordinate. However, as soon as K = 3 it is quite unclear how to proceed. For example, one could analogize the argument from above, constructing a polynomial Q(t) with t now in the disk D, to obtain via the Cauchy estimate kf` kΩn2 = kf` kΩnK = |Q(`) (0)| ≤ kQkD = kQkT . `! Unfortunately, there is no simple way to relate kQkT to kf kΩn3 . Certainly kQkT ≤ kf kTn , but then it seems we would again need the dimension-free comparison kf kTn .d,K kf kΩnK , a reappearance of the discretization inequality. And so we see that many roads lead to a dimension-free discretization of the uniform norm on Tn , which would form a “bridge” from analysis on ΩnK to established theory on Tn . The settling of Theorem 1 brings the state of analysis over ΩnK closer to matching what we understand on the hypercube, and we discuss it in detail next. 22 Chapter 3 The discretization inequality and its accoutrements Aspects of optimality of Theorem 1, extensions, consequences, proof techniques, and related literature. The discretization inequality (2.0.5) was first proved in [SVZ25] with a large (but dimension-free) constant, which was then improved by a different argument in [KSVZ24]. Theorem (Theorem 1 restated [SVZ25; KSVZ24]). Let f be an n-variate analytic polynomial of individual degree at most K − 1 and degree at most d. Then kf kTn ≤ O(log K)d kf kΩnK . One might wonder how important the specific grid of points ΩnK is for getting a dimension-free estimate. In [Bec+25] we extended the proof in [KSVZ24] to show the answer is actually “not so much.” Theorem 14 ([Bec+25]). Let f : Dn → C be an analytic polynomial of degree Q d and individual degree K − 1. Let X = nj=1 Xj ⊂ Dn such that for all j, |Xj | = K. Put η = min min |x − y| . j x,y∈Xj Then kf kDn ≤ (Cη,K )d kf kX , (3.0.1) for a universal constant Cη,K independent of n. As above, when X = ΩnK we may take Cη,K ≤ O(log K). With Theorem 1 in hand, both the Bohnenblust–Hille inequality and the boundedness of level-` Fourier projections on ΩnK become one-line arguments. Corollary 15. Let f be a polynomial of degree d and individual degree K − 1. Then n kfbk 2d ≤ O(log K)d · BH≤d Tn · kf kΩK , d+1 where BH≤d Tn ≤ C p d log d is the best constant in the polytorus BH inequality. 23 f ( , ,⋯, ) Figure 3.1: A visual depiction of Theorem 14. As long as there are K wellspaced red points in each of the coordinate discs, one may control the supremum norm of any individual-degree-(K − 1) polynomial f over the polydisk Dn by its absolute supremum over the finite grid of red points, in a dimension-free way. Proof. Any such polynomial f has the same Fourier expansion with respect to the groups ΩnK and Tn . Therefore, (Classical BH) kfbk 2d d+1 ≤ d ≤d n BH≤d Tn kf kTn ≤ (O(log K)) BHTn kf kΩK . Corollary 16. With f as before let f` be its `-homogeneous part. Then kf` kΩnK ≤ O(log K)d kf kΩnK . Proof. Again because f has the same Fourier expansion over Tn and ΩnK , (2.0.7) kf` kΩnK ≤ kf` kTn ≤ kf kTn ≤ O(log K)d kf kΩnK . Remarks and refinements We now describe in what senses Theorem 1 and its generalization Theorem 14 are optimal, what aspects we do not yet understand, and certain extensions. This discussion is from [Bec+25]. Sharp degree-dependence of the constant The best constant in the comparison Theorem 14 has exponential dependence on d. For simplicity we will argue this with Yn = ΩnK . Consider the univariate inequality kf kT ≤ C(K)kf kΩK (3.0.2) for polynomials f with degree at most K − 1, and where C(K) is the best constant. A Lagrange interpolation argument shows C(K) > 1 for any K ≥ 3 [SVZ24a, Appendix B]. Let g be any extremizer of this inequality and put Qd/(K−1) f (z) = j=1 g(zj ), assuming K − 1 divides d for simplicity. Then  kf kTn = C(K) d/(K−1) kf kYn =: D(K)d kf kYn 24 which is exponential in d. On the other hand, for this specific construction one may calculate that D(K) > 1 does not grow in K. It remains an interesting question to determine the optimal K-dependence of the constant C(η, K) in (3.0.1). Question 1. What is the optimal dependence on K in the constant in (3.0.1) of Theorem 14? On the cardinality of the sampling set Minimal cardinality of product sampling sets The cardinality of Yn in Theorem 14 is optimal in the following sense. If the Q sampling sets are of the product form Yn = 1≤j≤n Zj and one expects (3.0.1) to hold at least for polynomials of individual degree at most K − 1, then each Zj must have cardinality at least K and so Yn contains at least K n points. If |Yn | were any smaller, there would exist a j such that Zj has at most K − 1 Q points, and no such inequality can hold: the polynomial fj (z) := ξ∈Zj (zj − ξ) is of degree at most K − 1 but kfj kYn = 0. Contrapositively, if the sampling set has a product set structure Yn = Qn j=1 Zj with |Zj | = K, then the individual degree constraint on f is of course necessary. Improvements for non-product sets On the other hand, if we remove the product constraint on our sampling set, we can do better. Indeed, in [Bec+25, §4] we show that one may take a Q “small” part of nj=1 Zj and retain a dimension-free constant. Theorem 17. Let K ≥ 2. Consider {Zj ⊂ D}j≥1 a sequence of sets such that for all j ≥ 1 we have |Zj | = K and η := min min |z − z 0 | > 0. 1≤j≤n z6=z 0 ∈Zj Then for any ε > 0 one can find a subset Yn of size at most C1 (d, ε)(1 + ε)n Q contained in nj=1 Zj such that for any analytic polynomial f : Dn → C of degree d and individual degree K − 1, kf kDn ≤ C2 (d, K, η, ε)kf kYn , where C1 (d, ε) ≤ d ε !100d . (3.0.3) 25 Furthermore, if 0 < ε ≤ 1/2, then   C2 (d, K, η, ε) ≤ exp C3 (d, K, η) ε−1 log(ε−1 ) d  , for some constant C3 (d, K, η) depending on d, K, and η. We do not print the argument in this thesis as it is similar in spirit to the proof of Theorem 14 (and the [KSVZ24] proof of Theorem 1): instead of beginning with a univariate interpolation formula (as is done in the proof of Theorem 14), we begin with a multivariate interpolation on a small number of coordinates, and then apply the probabilistic tensorization ideas from Theorem 14 to get to n coordinates. Sharp dependence of sampling set cardinality on dimension n On the third hand, the cardinality of Yn cannot be sub-exponential in n. It suffices to prove this for d = 1; the general d ≥ 1 case follows immediately by definitions and the case d = 1. Theorem 18. Suppose that the uniform norm discretization (3.0.1) holds for sampling set Vn ⊂ Dn with d = 1, K = 2; that is, kf kDn ≤ C0 kf kVn holds for all multi-affine polynomials f of degree 1 with C0 > 1 being the best constant, then |Vn | ≥ C1 C2n , where C1 > 0 is universal and C2 > 1 depends on C0 . See Section 18 for the proof of Theorem 18. Uniform separation In Theorem 14, the constant C(K, η)d grows with η −1 , where η is the minimum pairwise distance between points in the Zj ’s. In fact, this is unavoidable; uniform separation (i.e., independence of η from n) is required to retain the dimension-freeness of the inequality of Theorem 14. This is easy to see in one dimension, nor can it be avoided in higher dimensions, as illustrated by the following example. Suppose Y1 , Y2 , . . . is a sequence of sets with Yn ⊂ Dn and c(n) is a sequence of coordinates; that is, 1 ≤ c(n) ≤ n for all n. Let Pn = {zc(n) : z ∈ Yn } ⊂ D 26 be the projection of Yn onto the c(n)-th coordinate. Suppose |Pn | = K for all n and lim min |z − z 0 | = 0 . 0 n→∞ z6=z ∈Pn For each Pn we may then choose a subset An ⊂ Pn with |An | = K − 1 and an excluded point ζn∗ such that Pn = An t ζn∗ and min |ζn∗ − ζ| ≤ εn , ζ∈An where limn→∞ εn = 0. Now consider the sequence of polynomials fn (z) := Q ζ∈An (zc(n) − ζ) . Certainly kfn kDn is at least as large as any of its coefficients, so we have kfn kDn ≥ 1. On the other hand, kfn kYn is very small: fn (z) = 0 for all z ∈ Yn except those z with zc(n) = ζn∗ , and for these z we have |fn (. . . , ζn∗ , . . .)| = Q ∗ ζ∈An (ζn − ζ) ≤ εn · 2K−2 , which tends to 0 as n → ∞. Therefore no dimension-free uniform norm discretization like Theorem 3 is available for such (Yn )n≥1 . 3.1 Theorem 1, sketch of Proof I: via Fourier multipliers The historically-first proof of Theorem 1 appears in [SVZ25], and is based on Fourier multiplier techniques which split f into certain special polynomials that are more amenable to direct argument. We sketch it now. Let us begin with the following idea, which is a somewhat familiar starting place for these sorts of comparisons: suppose we could find a scaled-down copy of the circle, ηT ⊂ D for η = η(K) > 0, such that for any z ∈ ηT, there is a probability distribution µz on ΩK with zm = E ξm ξ∼µz for all m = 0, 1, . . . , K − 1 . (3.1.1) Then we would essentially be done: for example, for f a d-homogeneous polynomial the argument would be as simple as kf kTn = η −d kf kηTn = η −d maxn z∈ηT E ξj ∼µzj :j∈[n] f (ξ) ≤ η −d kf kΩnK , because the µzj ’s are supported on ΩK . (The non-homogeneous case is not much worse.) 27 Unfortunately, this argument does not work for us because such a µz does not always exist in our setting. Examining the constraints on µz in (3.1.1), it turns out that to satisfy them all we need about twice as many degrees of freedom in µz as is afforded by making supp(µ) = ΩK . We are able to recover the property (3.1.1) only by letting µz be supported on Ω2K instead. So this approach gets us to kf kTn .d,K kf kΩn2K , (3.1.2) and it remains to compare kf kΩn2K with kf kΩnK . Given the apparent saturation of degrees of freedom in the Ωn2K comparison (3.1.2), this second step required a fully novel argument and came as a surprise to the authors. The second comparison is achieved as follows. Fix a z ∗ that maximizes |f | over Ω2K . For some coordinates j, we already have zj∗ ∈ ΩK (the goal domain), so let us forget about those. By a change of variables of the form zj 7→ ξzj for ξ ∈ ΩK , we can assume the remaining coordinates in z ∗ are all equal to √ ω := exp(πi/K). Thus it actually would suffice to prove the two-point comparison ? √ √ |g( ω, . . . , ω)| .d,K |g(1, . . . , 1)| (3.1.3) for all g of degree at most d and individual degree at most K − 1 (noting that the right-hand side is of course at most kgkΩnK ). When g is homogeneous (3.1.3) is an equality: just multiply by the appropriate root of unity. However, it is entirely unclear how to reduce the general case of (3.1.3) to the homogeneous case. With gk the k-homogeneous part of g, a typical approach might be d X √ √ √ √ |g( ω, . . . , ω)| ≤ |gk ( ω, . . . , ω)| = ≤ k=0 d X k=0 d X |gk (1, . . . , 1)| kgk kΩnK k=0 ? .d,K kgkΩnK , which would work provided the level-k Fourier projections are bounded independent of dimension: kgk kΩnK .d,K kgkΩnK . But before this work that question was also open and seemed just as difficult as the discretization inequality. The central technical contribution of [SVZ25] 28 is to introduce a special class of Fourier projections (denoted by D in the proof) P that allow us to write g = j hj for a small number of polynomials hj which... i. Are bounded by g independent of dimension: khj kΩnK .k,D kgkΩnK , and ii. Are not homogeneous but, by a very serendipitous identity for “half-roots” of unity (4.2.13), nevertheless satisfy √ √ |hj ( ω, . . . , ω)| = |hj (1, . . . , 1)|. The proof is completed by replacing the gk ’s in the previous display with our hj ’s. A byproduct of this technique is a rich class of Fourier multipliers (actually, projections onto certain fine-grained classes of monomials) that are bounded independent of dimension. The class has only an implicit description for general K, but when K is prime we may leverage some results from transcendental number theory to get the following characterization. For an S ⊂ {0, 1, . . . , K − 1}n we denote by fS the S-part of f : fS := X fb(α)z α . α∈S Theorem (Theorem 2 restated: Bounded Fourier projections, prime K ≥ 3). Suppose K is an odd prime and let S be a maximal subset of {0, 1, . . . , K − 1}n such that for all α, β ∈ S: • Degrees are equal: Pn j=1 αj = j=1 βj . Pn • Individual degree symmetry: there is a bijection π : [n] → [n] such that for all j ∈ [n], αj = βπ(j) or αj = K − βπ(j) . Then for any n-variate analytic polynomial f of degree at most d and individual P degree at most K − 1, the S-part of f , fS (x) := α∈S fb(α)z α , satisfies kfS kΩn .d,K kf kΩnK . K Theorem 2 and related techniques do not seem to follow from the later argument in [KSVZ24; Bec+25] and can be considered as one of the main contributions of the work [SVZ25]. 29 3.2 Theorem 1, sketch of Proof II: via interpolation The second proof, presented in [KSVZ24] for the domain ΩnK and in [Bec+25] for more general domains, takes a probabilistic view of interpolation. In one coordinate, polynomial interpolation (Lagrange interpolation) admits a probabilistic interpretation of the form f (z) = D · E[R · f (W )], (3.2.1) where D = D(K) > 1 is a constant and R and W are correlated random variables taking values in Ω4 and ΩK respectively. Repeating (3.2.1) coordinatewise gives the identity hQ f (z) = Dn E n j=1 Rj   i (3.2.2) f W1 , . . . , Wn , which immediately implies a discretization inequality of the desired form, except with exponential dependence on n. The idea is to notice that (3.2.2) is an expectation over n-many independent pairs of variables (Rj , Wj ), while f is of   bounded total degree d and thus is not very “aware” that (R1 , W1 ), . . . , (Rn , Wn ) is a product distribution. It turns out that by introducing certain correlations among the Wj ’s, we can reduce the power on D at the expense of picking up an error term: hQ f (z) = Dd E d j=1 Sj   f ,...,W f f W 1 n i + errorf,z . (3.2.3) f ’s are still over Ω , but now the joint Here the Sj ’s are i.i.d. over Ω4 and the W j K f f distribution (W1 , . . . , Wn ) has an intricate dependence structure. If we only had the first term we would be done of course, and with the right d-dependence in the constant. To remove the error term, we will use special features of the error’s relationship to the introduced correlations. Specifically, the correlation construction actually defines a family of identities similar to (3.2.3) of the form f (z) = Dm E hQ m (m) j=1 Sj   (m) f f W 1 f (m) ,...,W n i   + errorf,z m1 , for any integer m > 1, and where errorf,z is a fixed polynomial in 1/m of degree at most d − 1 and with no constant term. These properties imply there is an affine combination of these identities for m = 1, . . . , d that eliminates the error term: f (z) = 2d−1 X m=d am f (z) = 2d−1 X hQ am D m E (m) m j=1 Sj   (m) f f W 1 i f (m) , ,...,W n m=d (3.2.4) and where the absolute sum of the am ’s is suitably small. Placing | · |’s in the right spots finishes the theorem. 30 A new interpolation formula All the expectations in (3.2.4) are over finite probability spaces, so we actually have proved a new interpolation formula: Theorem 19. Let f, X, η be as in Theorem 14. Then for any z ∈ Dn , there P (z) (z) exist explicit coefficients {ax }x∈ΩnK such that x |ax | ≤ (Cη,K )d and f (z) = X a(z) x f (x) . (3.2.5) x∈X Comparing (3.2.5) to classical multivariate polynomial interpolation formulas, we obtain coefficients with dimension-free absolute sum at the expense of sampling more points than strictly necessary. As a result the linear combination (3.2.5) is not unique, and it is interesting to understand whether this flexibility can lead to sharpenings of Theorem 1. In the full proof, the identity (3.2.5) appears in detail as Equation (5.1.9). We hope this interpolation formula can have future applications and offer as a first example usage a short proof of a dimension-free discretization inequality for Lp norms, 1 ≤ p < ∞, as we describe next. Lp discretization Let Lp (Tn ) and Lp (ΩnK ) denote the Lp -space with respect to the uniform probability measures on Tn and ΩnK , respectively. When Yn = ΩnK , one way to prove a dimension-free Lp discretization inequality for p < ∞ would be to use hypercontractivity over Tn [Jan97] and over ΩnK [Wei80; JPPP17]. Hypercontractivity is a workhorse of high dimensional analysis [BGL14; Hu17] and implies dimension-free L2 -Lp comparisons for bounded-degree polynomials when 1 ≤ p < ∞ (see [ODo14, Chapter 9.5] and [Def+11; DGMS19, Chapter 8.4] for discussion). For example, with 2 ≤ p < ∞, and f a degree-d function on ΩnK , the argument is kf kLp (Tn ) .d,p kf kL2 (Tn ) = kf kL2 (ΩnK ) ≤ kf kLp (ΩnK ) , where hypercontractivity on the polytorus is applied in the first inequality. (N.b. that this hypercontractivity argument does not work for p = ∞.) In Section 6.1 we show a proof that avoids hypercontractivity altogether by making use of the interpolation formula (3.2.5) (or more concretely, Equation (5.1.9)). The main result of Section 6.1 is the following. 31 Theorem 20. Let d, n ≥ 1, K ≥ 2. Let 1 ≤ p ≤ ∞. Then for each polynomial f : Tn → C of degree at most d and individual degree at most K − 1, the following holds: kf kLp (Tn ) ≤ C(d, K)kf kLp (ΩnK ) with C(d, K) ≤ d(C1 log(K) + C2 )d with universal C1 , C2 > 0. We remark that the constant in the inequality of Theorem 20 is independent from p but dependent on d, so has a different character from Marcinkiewicz– Zygmund inequalities, where the constant depends on p but is typically required to be independent from the total degree d for 1 < p < ∞. The proof combines the interpolation formula with group-invariance of the uniform measure on ΩnK . 3.3 Relationships to other literatures Theorem 14 in the context of approximation theory The context and history for discretization inequalities in approximation theory begins in dimension n = 1. For 1 < p < ∞, the so-called Marcinkiewicz– Zygmund inequality [Zyg02, Chapter X, Theorem (7.5)] states that for all analytic polynomials f of degree at most K − 1, one has Cp−1 · K1 X z∈ΩK |f (z)|p ≤ Z T |f (z)|p dz ≤ Cp · K1 X |f (z)|p . (3.3.1) z∈ΩK Here Cp is a constant depending only on p (independent of K), and T = {z ∈ C : |z| = 1} denotes the unit circle. The inequality (3.3.1) is an example of a discretization of the Lp -norm, and integral norm inequalities of this type are usually called Marcinkiewicz-type theorems. At the endpoint p = ∞ this type of inequality is often called a Bernstein-type theorem or a discretization of the uniform norm (see [Ber31; Ber32] and [Zyg02, Chapter X, Theorem (7.28)]). In our notation, the p = ∞ endpoint of (3.3.1) reads kf kΩK ≤ kf kT ≤ C(K)kf kΩK . (3.3.2) In this p = ∞ case (and unlike 1 < p < ∞) we emphasize the right-hand side inequality cannot have constant independent of K. See for example [OS07, Theorem 5]. We refer to surveys [DPTT19; KKLT22] and references therein for more historical background about norm discretizations. Bernstein-type discretization 32 theorems also have some overlap with discrete Remez-type inequalities as we discuss below. Now let us return to the high-dimensional case, where Theorem 1 can be understood as a Bernstein-type discretization inequality for bounded-degree multivariate polynomials in many dimensions n. In this setting there are intricate tradeoffs between the cardinality (and structure) of the sampling set, the constant in the discretization inequality, and the function space to which the estimate applies. Recently there has been very important progress on understanding the minimum cardinality of sampling sets when one demands a universal constant (independent from any notion of degree or dimension) in the inequality. In [KKT23], Kashin, Konyagin, and Temlyakov give a discretization of the uniform norm that applies to any N -dimensional subspace of continuous functions on a compact subset of Rn , achieving a universal constant 2 with a sampling set of cardinality 9N . Moreover, as the authors show, this is essentially the best possible sampling set cardinality for a Bernstein-type discretization inequality at this level of generality. On the other hand, much smaller sampling sets—again for L∞ norm discretizations with universal constants—can be had when one fixes the function space to be polynomials of degree at most d. A significant recent work along these lines is [DP24]. Here Dai and Prymak resolved an important problem of Kroó [Kro11] in real approximation theory by showing there are discretizations of the uniform norm for n-variate polynomials of (total) degree at most d over any convex domain in Rn , with universal constant 2 and a sampling set of cardinality Cn dn in our notation.1 When degree d is large in comparison to dimension n, this cardinality Cn dn matches the dimension of the set of such polynomials, and is therefore the best possible. (N.b. our primary interest is in the opposite of their regime, d  n.) Our motivating application to functions on ΩnK —that is, to obtain a comparison kf kTn .d,K kf kΩnK for analytic polynomials f of individual degree at most K −1 and total degree at most d—is in some ways more demanding, and in other ways more relaxed, than the parameter regime traditionally considered in approximation theory. On the N.b., in the notation of [DP24] it will be Cd nd where they used d for the dimension and n for the degree. 1 33 one hand, we are restricted by the sampling set ΩnK , which is a fixed product set of small cardinality. Existing Bernstein-type estimates do not seem to apply in the parameter regime K < d, which is the setting dictated by applications to harmonic analysis in the high-dimensional realm of combinatorics, computer science, and learning theory. On the other hand, we do not require an absolute constant; indeed, as we discuss in below, dependence of the constant on degree d is unavoidable under these constraints. Remez-type inequalities in many dimensions Consider J a finite interval in R and a subset E ⊂ J with positive Lebesgue measure µ(E) > 0. Let f : R → R be a real polynomial of degree at most d. The classical Remez inequality [Rem36] states that max |f (x)| ≤ x∈J 4µ(J) µ(E) !d max |f (x)|. x∈E (3.3.3) Despite a large literature extending (3.3.3), we are not aware of any direct multi-dimensional generalizations that are dimension-free. Multi-dimensional versions of the Remez inequality are considered in the papers of Brudnyi and Ganzburg [BG73], Ganzburg [Gan17], Kroó and Schmidt [KS97] but they are not at all dimension-free: it is instructive to take a look at inequality (23) in [KS97] and see how the estimates blow up with dimension (called m in [KS97]). If one abandons the L∞ norm on the left-hand side of (3.3.3) then something can be said; there are distribution function inequalities for volumes of level sets of polynomials that are dimension-free, see [Fra09; NSV02; NSV03]. But those are distribution function estimates, not L∞ estimates. Some other related results include Nazarov’s extension [Naz93] of Turán’s inequality [Tur53], as well as more generalizations [Fon06; FY13]. The lack of a dimension-free multi-dimensional Remez inequality of the form (3.3.3) is not surprising: there is no hope for such an inequality phrased in terms of µ(E) for any positive-measure E ⊆ J. This can already be seen P when J is a unit ball in Rn and fn (x) = 1 − nj=1 x2j . For large n, most of the volume of the ball is concentrated in a neighborhood of the unit sphere where fn is very small. However, this observation does not preclude the existence of certain sets E giving multi-dimensional analogues of (3.3.3) that are dimension-free. Indeed, Lundin [Lun85], and later Aron–Beauzamy–Enflo [ABE93] and Klimek [Kli95], show this is possible in certain cases of (J, E) with convex E, such as for 34 bounded-degree polynomials over the polydisk J = Dn and the real cube E = [−1, 1]n . As an explicit example, with the prevailing notation, Klimek [Kli95] showed that for n-variate analytic polynomials of degree d, we have the √ comparison kf kDn ≤ (1 + 2)d kf k[−1,1]n . On the other hand, it was not at all clear when dimension-free Remez inequalities should exist in non-convex settings like J = Tn and E ⊂ Tn . The arguments in [Lun85; ABE93; Kli95] make essential use of the convexity of the testing set E and do not seem to suitably generalize. In comparison, for our application to functions on products of cyclic groups f : ΩnK → C, we have no choice but to use the non-convex grid ΩnK as our E. That our E is discrete and indeed finite is another interesting feature. Remez-type estimates with discrete E were known before; notably, Yomdin [Yom11] (see also [BY16]) identifies a geometric invariant which directly replaces the Lebesgue measure in (3.3.3) and is positive for certain finite sets E—though the comparison is not dimension-free. 35 Chapter 4 Proof I: Discretization by Fourier Multipliers Recall our goal is to prove the following. Theorem 21 (Theorem 1, implicit constant). Let f : Tn → C be an analytic polynomial of degree at most d and individual degree at most K − 1. Then kf kTn .d,K kf kΩnK . Our approach has two steps: Step 1. kf kTn .d,K kf kΩn2K , and Step 2. kf kΩn2K .d,K kf kΩnK . 4.1 Step 1 Proposition 1 (Torus bounded by Ω2K ). Let d, n ≥ 1, K ≥ 3. Let f : Tn → C be an analytic polynomial of degree at most d and individual degree at most K − 1. Then d kf kTn ≤ CK kf kΩn2K , where CK ≥ 1 is a universal constant depending on K only. To prove this proposition, we need the following lemma. Lemma 22. Fix K ≥ 3. There exists ε = ε(K) ∈ (0, 1) such that, for all z ∈ C with |z| ≤ ε, one can find a probability measure µz on Ω2K such that zm = E ξm, ξ∼µz ∀ 0 ≤ m ≤ K − 1. (4.1.1) Proof. Put θ = 2π/2K = π/K and ω = ω2K = eiθ . Fix a z ∈ C. Finding a probability measure µz on Ω2K satisfying (4.1.1) is equivalent to solving P 2K−1   pk = 1    k=0 P2K−1 m k=0 pk cos(kmθ) = <z     P2K−1 p sin(kmθ) = =z m k=0 k 1≤m≤K −1 (4.1.2) 1≤m≤K −1 with non-negative pk = µz ({ω k }) for k = 0, 1, . . . , 2K − 1. Note that the pk ’s are non-negative and thus real. 36 For this, it is sufficient to find a solution p~ = p~z to DK p~ = ~vz with each entry of p~ = (p0 , . . . , p2K−1 )> being non-negative. Here DK is a 2K × 2K real matrix given by  1   1   .. .   DK =  1   1   .. .  1 cos(θ) .. . 1 cos(2θ) .. . ··· ··· .. . 1 cos((2K − 1)θ) cos(Kθ) cos(2Kθ) ··· cos((2K − 1)Kθ) sin(θ) .. . sin(2θ) .. . ··· .. . sin((2K − 1)θ) .. .          ,         1 sin((K − 1)θ) sin(2(K − 1)θ) · · · sin((2K − 1)(K − 1)θ) and ~vz = (1, <z, . . . , <z K−1 , <z K , =z, . . . , =z K−1 )> ∈ R2K . Note that (4.1.2) does not require the (K + 1)-th row (1, cos(Kθ), cos(2Kθ), . . . , cos((2K − 1)Kθ)) (4.1.3) of DK . The matrix DK is non-singular. To see this, take any ~x = (x0 , x1 , . . . , x2K−1 )> ∈ R2K such that DK ~x = ~0. Then 2K−1 X (ω k )m xk = 0, 0 ≤ m ≤ K. (4.1.4) k=0 This is immediate for 0 ≤ m ≤ K − 1 by definition, and m = K case follows from the “additional” row (4.1.3) together with the fact that sin(kKθ) = 0, 0 ≤ k ≤ 2K − 1. Conjugating (4.1.4), we get 2K−1 X (ω k )m xk = 0, K ≤ m ≤ 2K. (ω k )m xk = 0, 0 ≤ m ≤ 2K − 1, k=0 Altogether, we have 2K−1 X k=0 that is, V ~x = ~0, where V = VK = [ω jk ]0≤j,k≤2K−1 is a 2K × 2K Vandermonde matrix given by (1, ω, . . . , ω 2K−1 ). Since V has determinant det(V ) = Y (ω j − ω k ) 6= 0 , 0≤j<k≤2K−1 37 we get ~x = ~0. So DK is non-singular. Therefore, for any z ∈ C, the solution to (4.1.2), thus to (4.1.1), is given by   −1 p~z = p0 (z), p1 (z), . . . , p2K−1 (z) = DK ~vz ∈ R2K . Notice one more thing about the rows of DK . As 2K−1 X (ω k )m = 0, m = 1, 2, . . . , 2K − 1 , k=0 we have automatically that vector p~∗ :=  1 1 , . . . , 2K 2K  ∈ R2K gives DK p~∗ = (1, 0, 0, . . . , 0)T =: ~v∗ . For any k-by-k matrix A denote kA~v k∞ . v k∞ ~06=~v ∈Rk k~ kAk∞→∞ := sup −1 So with p~∗ := DK v~∗ we have −1 k~pz − p~∗ k∞ ≤ kDK k∞→∞ k~vz − ~v∗ k∞   −1 = kDK k∞→∞ max max |<z k |, max |=z k | 1≤k≤K 1≤k≤K−1 −1 ≤ kDK k∞→∞ max{|z|, |z|K }. That is, max 0≤j≤2K−1 pj (z) − 1 −1 ≤ kDK k∞→∞ max{|z|, |z|K }. 2K −1 −1 Since DK ~v∗ = p~∗ , we have kDK k∞→∞ ≥ 2K. Put 1 # 1 ε∗ := ∈ 0, . −1 (2K)2 2KkDK k∞→∞ Thus whenever |z| < ε∗ < 1, we have max 0≤j≤2K−1 pj (z) − 1 1 −1 −1 ≤ |z|kDK k∞→∞ ≤ ε∗ kDK k∞→∞ ≤ , 2K 2K so in particular pj (z) ≥ 0 for all 0 ≤ j ≤ 2K − 1. Now we are ready to prove Proposition 1. 38 Proof of Proposition 1. Let ε∗ be as in Lemma 22. With a view towards applying the lemma we begin by relating sup |f | over the polytorus to sup |f | over a scaled copy. Recalling that the homogeneous parts fk of f are trivially bounded by f over the torus: kfk kTn ≤ kf kTn (a standard Cauchy estimate). Thus we have kf k Tn ≤ = ≤ d X k=0 d X k=0 d X kfk kTn ε−k ∗ sup |fk (ε∗ z)| z∈Tn ε−k ∗ sup |f (ε∗ z)| k=0 z∈Tn ≤ (d + 1)ε−d ∗ sup |f (ε∗ z)| z∈Tn = (d + 1)ε−d ∗ kf k(ε∗ T)n . (4.1.5) Let z = (z1 , . . . , zn ) ∈ (ε∗ T)n . Then for each coordinate j = 1, 2, . . . , n there exists by Lemma 22 a probability distribution µj = µj (z) on Ω2K for which Eξj ∼µj [ξjk ] = zjk for all 0 ≤ k ≤ K −1. With µ = µ(z) := µ1 ×· · ·×µn , this implies for a monomial ξ α with multi-index α ∈ {0, 1, . . . , K −1}n , Eξ∼µ(z) [ξ α ] = z α , or more generally by linearity Eξ∼µ(z) [f (ξ)] = f (z) for z ∈ (ε∗ T)n and f under consideration. So sup |f (z)| = z∈(ε∗ T)n sup E f (ξ) ≤ z∈(ε∗ T)n ξ∼µ(z) sup E |f (ξ)| ≤ kf kΩn2K . (4.1.6) z∈(ε∗ T)n ξ∼µ(z) Combining observations (4.1.5) and (4.1.6) we conclude −d d kf kTn ≤ (d + 1)ε−d ≤ CK kf kΩn2K . ∗ kf k(ε∗ T)n ≤ (d + 1)ε∗ kf kΩn 2K The last inequality follows from the fact that ε∗ depends only on K. 4.2 Step 2 Now we turn to Step 2’s estimate, kf kΩn2K .d,K kf kΩnK . (4.2.1) We will find it useful to rephrase this question as one about the boundedness at the single point √ √ √ f ( ω, . . . , ω) =: f ( ω) . 39 √ Here and in what follows, ω := ωK = e2πi/K , and ω will be used as shorthand to denote the root eπi/K . It turns out the following proposition is enough to give (4.2.1). Proposition 2. Let d, n ≥ 1, K ≥ 3. Let f : Tn → C be an analytic polynomial of degree at most d and individual degree at most K − 1. Then √ |f ( ω)| .d,K kf kΩnK . To explain why Proposition 2 suffices for Step 2, let us finish the proof of Theorem 21 given Proposition 1 and assuming Proposition 2. Proof of Theorem 21. Fix a z ∗ ∈ arg maxz∈Ωn |f (z)|. Then there exist w = 2K √ (w1 , . . . , wn ) ∈ ΩnK and y ∗ ∈ {1, ω}n such that wj yj∗ = zj∗ , j ∈ [n] , where [n] := {1, 2, . . . , n}. Define fe : Tn → C by fe(z) = f (w1 z1 , w2 z2 , . . . , wn zn ) . We therefore have |fe(y ∗ )| = kf kΩn2K and kfekΩnK = kf kΩnK . (4.2.2) (4.2.3) Equation (4.2.2) holds by the definition of y ∗ , and (4.2.3) holds by the group property of ΩK (recall w ∈ ΩnK ). √ Now let S = {j : yj∗ = ω} and m = |S|. Let π : S → [m] be any bijection. Define the “selector” function sy∗ : Tm → Tn coordinate-wise by  sy∗ (z)  j =   y ∗ j if j 6∈ S  zπ(j) if j ∈ S . Finally, define g : Tm → C by g(z) = fe(sy∗ (z)) . Then we observe that g is analytic with degree at most d and individual degree at most K − 1, and √ √ √ (4.2.2) |g( ω, ω, . . . , ω)| = |fe(y ∗ )| = kf kΩn2K (4.2.4) kgkΩm ≤ kfekΩnK K (4.2.3) = kf kΩnK , (4.2.5) 40 with the inequality holding because we are optimizing over a subset of points. From (4.2.4) and (4.2.5) we see Theorem 21 would follow if we could prove √ √ √ |g( ω, ω, . . . , ω)| .d,K kgkΩm , K independent of m ≥ 1. This is precisely Proposition 2. The proof of Proposition 2 is the subject of the rest of this subsection. Our P approach is to split f into parts f = j gj such that each part gj has the properties A and B: Property A kf kΩnK &d,K Property B kgj kΩnK &d,K √ |gj ( ω)| . (4.2.6) Such splitting gives X X X √ √ |f ( ω)| ≤ |gj ( ω)| .d,K kgj kΩnK .d,K kf kΩnK . j j j So as long as the number of gj ’s is independent of n such a splitting with Properties A and B entails the result. We will split f via an operator that was first employed to prove the Bohnenblust–Hille inequality for cyclic groups [SVZ24a]. We will only need the basic version of the operator here; a generalized version is considered in [SVZ24a]. Recall that any polynomial f : ΩnK → C has the Fourier expansion aα z α . X f (z) = α∈{0,1,...,K−1}n Recall the support of a monomial z α is supp(α) := {j : αj 6= 0}, and the support size |supp(α)| refers to the cardinality of supp(α). Definition 1 (Maximum support pseudoprojection). For any multi-index α ∈ {0, 1, . . . , K − 1}n define the factor Y τα = (1 − ω αj ) . j:αj 6=0 For any polynomial on ΩnK with the largest support size ` ≥ 0 f (z) = aα z α , X |supp(α)|≤` we define Df : ΩnK → C via Df (z) = X |supp(α)|=` τα aα z α . 41 The operator D can be considered a Fourier multiplier, and this somewhat technical definition is motivated by the following key property, the L∞ → L∞ boundedness when restricted to certain polynomials. Lemma 23 (Boundedness of maximum support pseudoprojection). Let f : ΩnK → C be a polynomial and ` be the maximum support size of monomials in f . Then √ kDf kΩnK ≤ (2 + 2 2)` kf kΩnK . (4.2.7) The proof of Lemma 23 is given in [SVZ24a]. We repeat it here in a slightly simplified form for convenience. 2πi Proof. Let ω = e K . Consider the operator G:  G(f )(x) = f 1+ω 1−ω 1+ω 1−ω + x1 , . . . , + xn , 2 2 2 2  x ∈ Ωn2 that maps any function f : {1, ω}n ⊂ ΩnK → C to a function G(f ) : Ωn2 → C. Then by definition kf kΩnK ≥ kf k{1,ω}n = kG(f )kΩn2 . (4.2.8) Fix m ≤ `. For any α we denote mk (α) := |{j : αj = k}|, 0 ≤ k ≤ K − 1. Then for α with |supp(α)| = m, we have m1 (α) + · · · + mK−1 (α) = |supp(α)| = m. For z ∈ {1, ω}n with zj = 1+ω + 1−ω xj , xj = ±1, note that 2 2 α zj j =  αj 1+ω 1−ω 1 + ω αj 1 − ω αj + xj = + xj . 2 2 2 2  So for any A ⊂ [n] with |A| = m, and for each α with supp(α) = A, we have for z ∈ {1, ω}n : zα = Y α zj j j:αj 6=0 = Y  1 + ω αj j:αj 6=0 = 2 + Y  1 − ω αj  j:αj 6=0 2 =2−m τα xA + · · · 1 − ω αj xj 2 · xA + · · ·  42 where xA := j∈A xj is of degree |A| = m while · · · is of degree < m. Then for P f (z) = |supp(α)|≤` aα z α we have Q   1 X  X G(f )(x) = τα aα xA + · · · , m m≤` 2 |A|=m supp(α)=A X x ∈ Ωn2 . Again, for each m ≤ `, · · · is some polynomial of degree < m. So G(f ) is of degree ≤ ` and the `-homogeneous part is nothing but   1 X X τα aα xA . 2` |A|=` supp(α)=A Consider the projection operator Q that maps any polynomial on Ωn2 onto its highest level homogeneous part; i.e., for any polynomial g : Ωn2 → C with deg(g) = m we denote Q(g) its m-homogeneous part. Then we just showed that   1 X X Q(G(f ))(x) = ` τ α aα  x A . 2 |A|=` supp(α)=A (4.2.9) It is known that [DMP18, Lemma 1 (iv)] for any polynomial g : Ωn2 → C of degree at most d > 0 and gm its m-homogeneous part, m ≤ d, we have the estimate √ kgm kΩn2 ≤ (1 + 2)d kgkΩn2 . Applying this estimate to G(f ) and combining the result with (4.2.8), we have √ √ kQ(G(f ))kΩn2 ≤ ( 2 + 1)` kG(f )kΩn2 ≤ (1 + 2)` kf kΩnK and thus by (4.2.9)   X  |A|=` √ ≤ (2 + 2 2)` kf kΩnK . τα aα xA X supp(α)=A Ωn 2 The function on the left-hand side is almost Df . Observe that ΩnK is a group, so we have sup z,ξ∈Ωn K X aα z α ξ α = sup α z∈Ωn K X aα z α . α Thus we have actually shown  sup  X X  z∈Ωn ,x∈Ωn 2 |A|=` K supp(α)=A Setting x = ~1 gives (4.2.7). √ τα aα z α xA ≤ (2 + 2 2)` kf kΩnK . 43 Note that Df is exactly the part of f composed of monomials of maximum support size, except where the coefficients aα have picked up the factor τα . The relationships among the τα ’s can be intricate: while in general they are different for distinct α’s, this is not always true. Consider the case of K = 3 and the two monomials z β := z12 z2 z3 z4 z5 z6 z7 z8 , 0 z β := z12 z22 z32 z42 z52 z62 z72 z8 . Then τβ = (1 − ω)7 (1 − ω 2 ) = (1 − ω)(1 − ω 2 )7 = τβ 0 , which follows from the identity (1 − ω)6 = (1 − ω 2 )6 for ω = e2πi/3 . Understanding precisely when τα = τβ seems to be a formidable task in transcendental number theory. When K is prime there is a relatively simple characterization (see Section 4.3) but for composite K the situation is much less clear. Nevertheless, it turns out that for the purposes of Theorem 21 we do not need a full understanding. Indeed, our gj ’s shall be defined according to the τ ’s. Definition. Two monomials z α , z β with associated multi-indexes α, β ∈ {0, 1 . . . , K − 1}n are called inseparable if |supp(α)| = |supp(β)| and τα = τβ . When m and m0 are inseparable, we write m ∼ m0 . Inseparability is an equivalence relation among monomials. We may split P any polynomial f into parts f = j gj according to this relation. That is, any two monomials in f are inseparable if and only if they belong to the same gj . Call these gj ’s the inseparable parts of f . It is these inseparable parts that are our gj ’s in (4.2.6). We shall formally check it later, but it is easy to see the number of inseparable parts is independent of n. We formulate and prove Properties A & B next. Property A: Boundedness of inseparable parts Repeated applications of the operator D enable splitting into inseparable parts. Proposition 3 (Property A). Fix K ≥ 3 and d ≥ 1. Suppose that f : ΩnK → C is a polynomial of degree at most d with maximum support size L. For 0 ≤ ` ≤ L 44 let f` denote the part of f composed of monomials of support size `, and let g(`,1) , . . . , g(`,J` ) be the inseparable parts of f` . Then there exists a universal constant Cd,K independent of n and f such that for all 0 ≤ ` ≤ L and 1 ≤ j ≤ J` , kg(`,j) kΩnK ≤ Cd,K kf kΩnK . Proof. We first show the proposition for g(L,j) , 1 ≤ j ≤ JL . Suppose that aα z α . X f (z) = α:|supp(α)|≤L Inductively, one obtains from Lemma 23 that for 1 ≤ k ≤ JL , Dk f = ταk aα z α X |supp(α)|=L √ k with D f Ωn K (4.2.10) kL ≤ (2 + 2 2) kf k Ωn K . By definition there are JL distinct values of τα among the monomials of fL ; label them c1 , . . . , cJL . Then aα z α = X fL (z) = ταk aα z α = X g(L,j) (z), and 1≤j≤JL |supp(α)|=L Dk f (z) = X X ckj g(L,j) (z), k ≥ 1. 1≤j≤JL |supp(α)|=L Let us confirm JL is independent of n. Consider α with |supp(α)| = L. We may count the support size of α by binning coordinates according to their degree: |supp(α)| = L, X |{s ∈ [n] : αs = t}| = L ≤ d, 1≤t≤K−1 so JL ≤ |{(m1 , . . . , mK−1 ) ∈ {0, . . . , L}K−1 : m1 + · · · + mK−1 = L}| (4.2.11) ! K −1+L−1 ≤ ≤ (K + d)d . L−1 According to (4.2.10), we have    c2 c22 .. .   cJ1 L cJ2 L Df c    1  2   2 Df   c  = 1  ..   ..  .   .  DJLf |  · · · cJ L · · · c2JL . . . .. . JL · · · cJ L {z =: VL  g  (L,1)     g(L,2)     .. .  .   } g(L,JL )  45 The JL × JL modified Vandermonde matrix VL has determinant  det(VL ) =  JL Y   Y cj   j=1 (cs − ct ). 1≤s<t≤JL Since the cj ’s are distinct and nonzero we have det(VL ) 6= 0. So VL is invertible and in particular g(L,j) is the j th entry of VL−1 (D1f, . . . , DJLf )> . Letting η (L,j) = (L,j) (ηk )1≤k≤JL be the j th row of VL−1 , this means g(L,j) = X (L,j) ηk Dkf. 1≤k≤JL As η (L,j) depends on d and K only, so for all 1 ≤ j ≤ JL , kg(L,j) kΩnK ≤ (L,j) X ηk · Dkf 1≤k≤JL Ωn K  √  JL d ≤ kη (L,j) k1 2 + 2 2 kf kΩnK , (4.2.12) where we used (4.2.10) in the last inequality. The constant √ kη (L,j) k1 (2 + 2 2)JL d ≤ C(d, K) < ∞ for appropriate C(d, K) that is dimension-free and depends only on d and K only. This finishes the proof for the inseparable parts in fL . We now repeat the argument on f −fL to obtain (4.2.12) for the inseparable parts of support size L−1. In particular, there are vectors η (L−1,j) , 1 ≤ j ≤ JL−1 of dimension-free 1-norm with kg(L−1,j) kΩnK ≤ C(d, K)kη (L−1,j) k1 kf − fL kΩnK .d,K kf − fL kΩnK . This can be further repeated to obtain for 0 ≤ ` ≤ L and 1 ≤ j ≤ J` , the vectors η (`,j) with dimension-free 1-norm such that kg(`,j) kΩnK .d,K f − X `+1≤k≤L fk . Ωn K It remains to relate kf − `+1≤k≤L fk kΩnK to kf kΩnK . Note that with VL as originally defined, by considering (1 1 . . . 1)VL−1 (D1 f, . . . , DJL f )> we obtain a constant DL = DL (d, K) independent of n for which P kfL kΩnK ≤ DL kf kΩnK . This means kf − fL kΩnK ≤ (1 + DL )kf kΩnK . 46 Notice the top-support part of f −fL is exactly fL−1 , so repeating the argument above on f − fL yields a constant DL−1 = DL−1 (d, K) such that kfL−1 kΩnK ≤ DL−1 kf −fL kΩnK ≤ DL−1 (1+DL )kf kΩnK = (DL−1 +DL−1 DL )kf kΩnK . Continuing, for 1 ≤ ` ≤ L we find X kfL−` kΩnK ≤ DL−` kf − fk kΩnK L−`+1≤k≤L X ≤ DL−` (1 + DL−`+1 )kf − fk kΩnK L−`+2≤k≤L ≤ DL−` Y (1 + DL−k )kf kΩnK . 0≤k≤`−1 We have found for each `-support-homogeneous part of f , kf` kΩnK .d,K kf kΩnK , so we have kf − .d,K kf kΩnK as well. `+1≤k≤L fk kΩn K P √ Property B: Boundedness at ω for inseparable parts √ 2πi Here we argue g( ω) is bounded for inseparable g. Recall that ω = e K √ πi and ω = e K . √ Proposition 4 (Property B). If g is inseparable then |g( ω)| ≤ kgkΩnK . Proof. We will need an identity for half-roots of unity. For k = 1, . . . , K − 1 we have √ 1 − ωk ( ω)k = i , (4.2.13) |1 − ω k | √ following from the orthogonality of ( ω)k and 1 − ω k in the complex plane. We claim that for two monomials m and m0 √ √ m ∼ m0 =⇒ m( ω) = m0 ( ω) . By definition m ∼ m0 means m and m0 have the same support size (call it `) and Q Q αj βj j:αj 6=0 (1 − ω ) = j:βj 6=0 (1 − ω ) . Dividing both sides by the modulus and multiplying by i` allows us to apply (4.2.13) to find √ αj Q √ Q = j:βj 6=0 ( ω)βj , j:αj 6=0 ( ω) 47 as desired. √ Now let ζ = m( ω) ∈ T for some monomial m in g. Then because ζ is √ P P independent of m, with g = α∈S aα z α , we have g( ω) = ζ α∈S aα and √ P |g( ω)| = | α∈S aα | = |g(~1)| ≤ kgkΩnK . We may now prove Proposition 2. Proof of Proposition 2. Write f = 0≤`≤L 1≤j≤J` g(`,j) in terms of inseparable parts, where g(`,j) , 1 ≤ j ≤ J` , 0 ≤ ` ≤ L are as in Proposition 3. Then by Propositions 3 (Property A) and 4 (Property B) P √ |f ( ω)| ≤ X P X √ |g(`,j) ( ω)| 0≤`≤L 1≤j≤J` ≤ X X kg(`,j) kΩnK (Property B) J` . (Property A) 0≤`≤L 1≤j≤J` .d,K kf kΩnK X 0≤`≤L √ In view of (4.2.11) and L ≤ d, we obtain |f ( ω)| .d,K kf kΩnK . 4.3 Aside: characterizing inseparable parts for prime K Although it is not required for the proof of Theorem 21, it is interesting to understand what are the parts g of f for which kgkΩnK .d,K kf kΩnK (4.3.1) via our Property A (Proposition 3)? Recall that (4.3.1) holds when g is a part of f containing all monomials in f from an equivalence class of the inseparability equivalence relation ∼. Thus we are led to ask for a characterization of inseparability. It turns out that for prime K this can be done completely via connections to transcendental number theory including Baker’s theorem [Bak22]. Proposition 5. Suppose K ≥ 3 is prime and α, β ∈ {0, 1, . . . , K − 1}n . Then two monomials z α , z β are inseparable if and only if • Support sizes are equal: |supp(α)| = |supp(β)|, • Degrees are equal mod 2K: |α| = |β| mod 2K, • Individual degree symmetry: there is a bijection π : supp(α) → supp(β) such that for all j ∈ supp(α), αj = βπ(j) or αj = K − βπ(j) . 48 Proof. Recall that by definition, two monomials z α and z β are inseparable if and only if they have the same support size and τα = τβ ; that is, Y (1 − ω αj ) = j:αj 6=0 Y (1 − ω βj ) , j:βj 6=0 where ω = e2πi/K . For these quantities to be equal, their respective moduli and arguments must coincide. To compare arguments, observe that for any multi-index σ ∈ {0, 1, . . . , K − n 1} , by the identity (4.2.13) we may normalize τσ like so: n √ Y √ τσ = i−|supp(σ)| ( ω)σj = i−|supp(σ)| ( ω)|σ| , |τσ | j=1 √ where as before ω = eπi/K . It is given that |supp(α)| = |supp(β)|, so √ √ the arguments of τα and τβ are equal exactly when ( ω)|α| = ( ω)|β| , or equivalently, |α| = |β| mod 2K. As for the moduli, using the identity |1 − ω k | = 2 sin(kπ/K) we find for any multi-index σ that |τσ | = Y Y 2 sin(σj π/K) = j:σj 6=0 2 sin(min{σj , K − σj } · π/K) , j:σj 6=0 where the last step follows from the symmetry of sine about π/2. So when are |τα | and |τβ | equal? By the last display, certainly they are the same if there is a bijection π : supp(α) → supp(β) such that for all j ∈ supp(α), αj = βπ(j) or αj = K − βπ(j) . Is this the only time |τα | = |τβ |? Returning to σ, define for 1 ≤ k ≤ (K − 1)/2 the quantity σb (k) = |{j : σj = k or σj = K − k}| . Then (K−1)/2 log(|τσ |) = X σb (k) · log(2 sin(kπ/K)). k=1 Therefore if the numbers {bk := log(2 sin(kπ/K)), k = 1, . . . , (K − 1)/2} were linearly independent over Z, the only way |τα | = |τβ | is the existence of a bijection π as above. Conveniently, the question of the linear independence of the bk ’s has already appeared in a different context, concerning an approach of Livingston to resolve 49 a folklore conjecture of Erdös on the vanishing of certain Dirichlet L-series. It was answered in [Pat17] in the positive for K ≥ 3 prime and in the negative for all composite K ≥ 4 using several tools including Baker’s celebrated theorem on linear forms in logarithms of algebraic numbers [Bak22]. Finally, recalling (e.g., Corollary 16) that Theorem 21 implies kfk kΩnK .d,K kf kΩnK for all k-homogeneous parts fk of f , 0 ≤ k ≤ d, we may conclude by Proposition 3: Corollary 24. Suppose K is an odd prime and let S be a maximal subset of {0, 1, . . . , K − 1}n such that for all α, β ∈ S: • Support sizes are equal: |supp(α)| = |supp(β)|. • Degrees are equal: |α| = |β|. • Individual degree symmetry: there is a bijection π : supp(α) → supp(β) such that for all j ∈ supp(α), αj = βπ(j) or αj = K − βπ(j) . Then for any n-variate analytic polynomial f of degree at most d and individual P degree at most K − 1, the S-part of f , i.e., fS = α∈S fb(α)z α , satisfies: kfS kΩnK .d,K kf kΩnK . 50 Chapter 5 Proof II: Discretization by interpolation Here we give a proof of Theorem 1 (or Theorem 14 for X = ΩnK ) via interpolation, follow [KSVZ24]. The more general case of Theorem 14, as proved in [Bec+25], is more complicated in notation only. 5.1 The proof A natural approach to proving Theorem 1 is to consider a specific maximizer z ∈ Tn of |f | and write f (z) as a linear combination of evaluations of f at points in ΩnK . We might begin with this lemma for a single coordinate: Lemma 25. Suppose z ∈ T. Then there exists c := (c0 , . . . , cK−1 ) such that for all k = 0, 1, . . . , K − 1, zk = K−1 X cj (ω j )k . j=0 Moreover, kck1 ≤ B log(K) for a universal constant B. Proof. Let ω = exp(2πi/K). The discrete Fourier transform (DFT) of the array A = (1, z, . . . , z K−1 ) yields K complex numbers ce0 , . . . , c] K−1 so that z k = Ak = X 1 K−1 cej ω jk K j=0 for all k = 0, . . . , K − 1. Using cj := K1 cej we get k z = K−1 X cj ω jk . j=0 Recall the DFT coefficients are given by cej = K−1 X Ak ω −kj . k=0 Since Ak = z k we have cej = K−1 X k=0 z k ω −kj = 1 − (z/ω j )K . 1 − (z/ω j ) (5.1.1) 51 By the triangle inequality, ! 2 |cej | ≤ min K, j . |ω − z| Using that the harmonic number HK = it is elementary to see that we have K−1 X PK k=1 1/k satisfies HK ≤ log(K) + 1, |cej | ≤ BK log K j=0 for B a sufficiently large constant. That is, kck1 = K−1 X |cj | = j=0 X 1 K−1 |cej | ≤ B log(K) . K j=0 In a single coordinate, Lemma 25 provides the desired inequality as follows. With z ∈ T a maximizer of |f (z)| we have kf kT = |f (z)| = | d X k ak z | = | d K−1 X X j k ak cj (ω ) | = | k=0 j=0 k=0 ≤ kck1 kf kΩK ≤ C log(K)kf kΩK . K−1 X cj f (ω j )| j=0 (Hölder) However, in higher dimensions, repeating this argument coordinatewise introduces an exponential dependence on n. We circumvent this by taking a probabilistic view of the foregoing display: the sum over j can be interpreted as an expectation over a (complex-valued) measure on ΩK . When it is repeated in several dimensions, this is like taking an expectation over n independent random variables. The key insight is that this independence is more than we need: by correlating the random variables, we “save on randomness” (which reduces the multiplicative constant) while retaining control of the error. Lemma 26. Let f be a degree-d n-variate polynomial and z ∈ Tn . Then there is a univariate polynomial p = pf,z such that for any positive integer m there are (dependent) random variables R, W taking values in Ω4 and ΩnK respectively such that f (z) = Dm E [Rf (W )] + p(1/m) . (5.1.2) R,W Moreover, p has deg(p) < d and zero constant term, and D = D(K) is a universal constant depending on K only. 52 Lemma 26 is the crux of our argument and we are not aware of a similar statement in the literature. Theorem 1 follows quickly, though it is interesting to note that instead of clearing the error term by taking m → ∞ (which would indeed make p(1/m) → 0 but also send Dm → ∞), we will end up using algebraic features of p (namely, low-degree-ness) to remove it. But first, the lemma: Proof of Lemma 26. We will argue Lemma 26 for f (z) = z α , a monomial of degree at most d. The claim extends to general degree-d f by linearity. We begin by examining a single coordinate with the aim of rewriting Lemma 25 in a probabilistic form. To that end, we first decouple the angle and magnitude information of the cj ’s. Fix z ∈ T and let cj be as in Lemma 25. We may write a decomposition 3 X (0) (1) (2) (3) (s) cj = 1 · cj + i · cj + (91) · cj + (9i) · cj = is · cj , s=0 (s) (0) (2) (1) (3) with all cj ∈ R≥0 and cj cj = cj cj = 0. This can be done for all j so that, with C := B log K from Lemma 25, kc(s) k1 ≤ C (5.1.3) (s) (s) is satisfied for each s ∈ {0, 1, 2, 3}, where c(s) = (c1 , . . . , cn ). So we have for all k = 0, . . . , K − 1, k z = K−1 3 X X (s) is · cj · (ω j )k . j=0 s=0 We now rewrite the sum in Lemma 25 in probabilistic form. Put D = 4C + 1 and define r : [0, D] → C by r(t) =           1 0 ≤ t ≤ C + 1, i C + 1 < t ≤ 2C + 1,    −1      −i 2C + 1 < t ≤ 3C + 1, 3C + 1 < t ≤ 4C + 1 = D . Also define a piecewise-constant function w : [0, D] → ΩK as follows. Consider (s) any collection of disjoint intervals Ij , 0 ≤ j ≤ K − 1, 0 ≤ s ≤ 3 such that (s) Ij ⊂ [0, D], s ∈ {0, 1, 2, 3}, j ∈ {0, 1, . . . , K − 1} 53 (s) (s) (s) and for each s and j, Ij ⊆ [sC + 1, (s + 1)C + 1] and |Ij | = cj . Disjointness is possible because for each s, |[sC + 1, (s + 1)C + 1]| = C ≥ PK−1 j=0 (s) cj (s) by (5.1.3). Now assign w(Ij ) = ω j and in the remaining region of [0, D] (that (s) is, [0, D]\ ts,j Ij ) let w take on each element of ΩK with in equal amount (w.r.t. the uniform measure). Claim 1. Let T be sampled uniformly from [0, D]. Then for all k = 0, 1, . . . , K− 1, z k = D E[r(T )w(T )k ] . (5.1.4) T Proof of Claim 1. Let us begin with k = 0, which simplifies to D E[r(T )] = 1 . T (5.1.5) This can be seen by direct computation: E[r(T )] = T 1 1 (1 + 1 · C + i · C + (91) · C + (9i) · C) = . D D For k ≥ 1, consider the joint distribution of (r(T ), w(T )k ), whose product appears in (5.1.4). Fix s ∈ {0, 1, 2, 3}, and condition on r(T ) = is . The conditional distribution of w(T ) has two parts. One part, corresponding to (s) (s) tj Ij , has w(T ) = ω j over Ij with the probability Pr[r(T ) = is ∧ w(T ) = ω j ] (s) equal to cj /D, while the other has w(T ) uniformly distributed in ΩK . The P j k latter part contributes 0 to the expectation E[r(T )w(T )k ], since K−1 j=0 (ω ) = 0 for k = 1, 2, . . . , K − 1. The former part contributes s i · (s) K−1 X cj j=0 D ω jk . Summing this display over s ∈ {0, 1, 2, 3} and rearranging, we get that E[r(T )w(T )k ] = completing proof of (5.1.4). K−1 X cj j k 1 (ω ) = z k , D j=0 D ♦ We return to the multivariate setting. Fix z := (z1 , . . . , zn ) ∈ Tn and define the functions w1 , . . . , wn corresponding to the above construction applied to 54 each coordinate z1 , . . . , zn . If each coordinate were to receive a fresh copy of T this would lead to an identity with exponential constant: hQ z α = Dn E iid n α` `=1 r(T` )w` (T` ) i . T` ∼ T, 1≤`≤n iid Instead, we consider only m independent copies of T : T1 , . . . , Tm ∼ U[0, D]. The decision of which coordinates are integrated with respect to which T` is also made randomly, via a uniformly random function P : [n] → [m]. We finally arrive at the definitions of R and W : R := m Y with R` := r(T` ), 1 ≤ ` ≤ m R` `=1       W := w1 TP (1) , w2 TP (2) , . . . , wn TP (n)  =: (W1 , . . . , Wn ) . When P is injective on supp(α), we easily achieve the smaller constant. Claim 2. Consider m ≥ |supp(α)|. Then E [R · W α | P is injective on supp(α)] = D−m z α . R,W Proof of Claim 2. It suffices to prove this for an arbitrary projection Pe that is injective on supp(α). Consider the partition of [n] given by Pe −1 ([m]) and write S` = Pe −1 (`) for ` ∈ [m]. By independence, the expectation splits over these S` ’s: h Q i Q αk E [R · W α | P = Pe ] = m . (5.1.6) `=1 E R` k∈S` Wk R,W Because Pe is injective on supp(α), every S` contains one or zero elements of supp(α). By Claim 1, in the latter case we have h E R` Q i αk = E[R` ] = k∈S` Wk 1 , D and in the former case we have h E R` Q i α αk = E[R` Wj j ] = k∈S` Wk 1 αj z , D j for the specific j for which {j} = S` ∩ supp(α). Substituting these observations into (5.1.6) completes the argument. ♦ When P is not injective, we still have some control. Let S = {Sj } be a partition of supp(α). We say P induces S if {P −1 (j) ∩ supp(α) : j ∈ [m]} = S . 55 Claim 3. For each partition S of supp(α) there is a number E(S) such that for all m ≥ |S|, E [R · W α | P induces S] = D−m E(S). R,W Proof of Claim 3. Condition again on a specific Pe that induces S. There are two types of ` ∈ [m]: those that W α depends on (that is, Pe (supp(α))), and those that only R depends on. Call these sets L = Pe (supp(α)) and Lc respectively. Then by independence of the T` ’s, E [R · W α | P = Pe ] = E [( R,W R,W Q Q `∈Lc R` )( = D−m+|S| E [( R,W Q `∈L R` ) · W `∈L R` ) · W | α α | P = Pe ] | P = Pe ] . {z ∗ } We observe that the expectation (∗) does not depend on the specific Pe inducing S, nor on m. Thus we may define E(S) by setting D−|S| E(S) equal to (∗). ♦ To summarize claims 2 and 3, we have that for all partitions S of supp(α) there is a number E(S) such that for all m ≥ |S|, E[R · W α |P induces S] = D−m E(S). n o And using S ∗ to denote the singleton partition {j} , we additionally j∈supp(α) ∗ α have E(S ) = z . Now we consider the unconditional expectation E[R · W α ] with P ∼ U([m][n] ). Elementary combinatorics give that for all partitions S and all m ≥ 1, with s = |S|, Pr[P induces S] = =: m(m − 1) · · · (m − s + 1) m|supp(α)|    1 + qs       qs 1 m 1 m if s = |supp(α)| if s < |supp(α)| , for polynomials qs with zero constant term and deg(qs ) < d. Of course P can only induce S for |S| ≤ m, so by the law of total probability, E [R · W α ] = R,W X S,|S|≤min(m,|supp(α)|) E[R · W α | P induces S ] Pr[P induces S] . 56 Consider first the case m ≥ |supp(α)|. We obtain E [R · W α ] = X R,W E[R · W α | P induces S ] Pr[P induces S] S =D −m ∗    E(S ) 1 + q|supp(α)| m1   D−m E(S) · q|S| m1 X + S,|S|<|supp(α)| =D −m  α z + X   E(S) · q|S| m1 (5.1.7) . S Now when m < |supp(α)|, we combine the fact that Pr[P induces S] = 0 for |S| > m with the definition of qs to see E [R · W α ] = 0 + R,W E[R · W α | P induces S ] Pr[P induces S] X S,|S|≤m = D−m E(S) Pr[P induces S] + X S,|S|>m X D−m E(S) Pr[P induces S] S,|S|≤m    = D−m E(S ∗ ) 1 + q|supp(α)| m1 X +   D−m E(S) · q|S| m1 |supp(α)|>|S|>m +   D−m E(S) · q|S| m1 X m≥|S|  = D−m z α + X   E(S) · q|S| m1 (5.1.8) . S Noting that (5.1.8) and (5.1.7) are identical, we rearrange to find z α = Dm E[R · W α ] − X   E(S) · q|S| m1 , S and the second part is in total a polynomial in m1 with no constant term and degree < d.   Finally, the error term p m1 is removed by considering several values of m. Proof of Theorem 1. Suppose there were some coefficients am ∈ C with dm=1 am = 1, so that for any polynomial p of degree < d and p(0) = 0 we would have P Pd 1 m=1 am p( m ) = 0. 57 We could then sum (5.1.2) for m = 1, . . . , d, weighted by am , and get d X f (z) = m=1 d X = m=1 d X = am f (z) am Dm E[Rm f (Wm )] + d X   am p m1 m=1 am Dm E[Rm f (Wm )], (5.1.9) m=1 where Rm , Wm are those R, W from (5.1.2) marked with explicit dependence on m. Well, these coefficients am can be arranged, since the monomial vectors (1/mt )m=1,...,d for t = 0, . . . , d − 1 are linearly independent (Vandermonde). Since always |Rm | ≤ 1, we deduce |f (z)| ≤ d X |am Dm | · kf kΩnK ≤ m=1 maxdm=1 |am | · Dd kf kΩnK . 1 − 1/D An explicit formula for the am ’s is given by am = (−1)d−m md , m!(d − m)! and it is evident that maxdm=1 |am | ≤ exp(O(d)), and specifically maxdm=1 |am | ≤ exp(1.28d). Without loss of generality, we may assume D ≥ 11 and so 1/(1−1/D) ≤ 1.1. We conclude |f (z)| ≤ (4D)d kf kΩnK = (4B log(K) + 4)d kf kΩnK . 5.2 Nonexistence of subexponential-cardinality meshes Proof of Theorem 18. For any ε = (ε1 , . . . , εn ) ∈ {−1, 1}n , consider the polyP nomials fε (x) = nj=1 εj xj on {−1, 1}n of degree at most 1. Then by definition, n = kfε k{±1}n ≤ Ckfε kVn . In other words, we have for all ε ∈ {−1, 1}n that max v=(v1 ,...,vn )∈Vn n X vj εj ≥ 2δn Λv :=  δ= j=1 So we have the inclusion {−1, 1}n ⊂   with S 1 ∈ (0, ∞). 2C v∈Vn Λv , where ε ∈ {−1, 1}n : |fv (ε)| = n X j=1   vj εj ≥ 2δn ,  v ∈ Vn . 58 For each v ∈ Vn and i.i.d. Bernoulli random variables ε1 , . . . , εn , we have by Hoeffding’s inequality that  |Λv | = 2n Pr n X  vj εj ≥ 2δn j=1  ≤ 2n Pr n X   <(vj )εj ≥ δn + 2n Pr j=1 ≤2  =(vj )εj ≥ δn j=1 2 2 n+1 n X δ n exp − 2kak22 ! ! +2 n+1 δ 2 n2 exp − , 2kbk22 where a = <v and b = =v are real vectors. Recalling that v ∈ Vn ⊂ Dn , we have kak22 ≤ n and kbk22 ≤ n. Therefore,    2 |Λv | ≤ 2n+2 exp −δ 2 n/2 = 4 2e−δ /2 n . All combined, we just proved 2n = |{−1, 1}n | ≤ X v∈Vn This gives the bound 1 δ2 n |Vn | ≥ e 2 , 4 as desired.  2 |Λv | ≤ 4|Vn | 2e−δ /2 n . 59 Chapter 6 Applications 6.1 Lp discretization inequalities Theorem 27. Let 1 ≤ p ≤ ∞ and f : ΩnK → C of degree at most d. Then kf kLp (Tn ) ≤ dO(log K)d kf kLp (ΩnK ) . Proof. Beginning with the interpolation formula (5.1.9), it is evident that for any ξ ∈ ΩnN , f (ξ z) = d X am Dm E[Rm f (ξ Wm )], m=1 where denotes coordinatewise multiplication. Thus by Jensen we have z)|p ≤ dp · |f (ξ d 1 X |am Dm |p E[|f (ξ d m=1 ≤ dp O(log K)dp · d1 d X Wm )|p ] E[|f (ξ Wm )|p ] . m=1 Now consider ξ sampled uniformly from ΩnN . For any fixed value of Wm , the distribution ξ Wm is also uniform on ΩnN , so we have E [|f (ξ ξ∼Ωn N z)|p ] ≤ dp O(log K)dp E n [|f (ξ)|p ] . ξ∼ΩN Let A = e[0,1)·2πi/N be the arc on T from the first to the second N th root of unity. Observe that the random variable ξz with z ∼ A, ξ ∼ ΩN is distributed uniformly on T, and accordingly the random variable ξ z with z ∼ An , ξ ∼ ΩnN is distributed uniformly on Tn . Therefore, E [|f (z)|p ] = z∼Tn E z∼An ,ξ∼Ωn N [|f (ξ z)|p ] ≤ dp O(log K)dp E n [|f (ξ)|p ], ξ∼ΩN which finishes the argument. 60 6.2 Junta theorem for functions on the hypergrid We say a function f : ΩnK → D is a k-junta if it depends on only k coordinates. The concept of juntas is a central tool in the analysis of Boolean functions [ODo14]. So-called “junta theorems” show that functions which are simple in some way are close to juntas. For functions from the hypercube to the range {−1, 1} these include Friedgut’s theorem (which constrains the functions’ total influence) and the FKN Theorem and Bourgain’s junta theorem, both of which constrain P the weight of the Fourier tail |S|≥k fb(S)2 . When the image of f is allowed to lie in [−1, 1], we have the following result [DFKO07]. Theorem 28 ([DFKO07]). Let f : Ωn2 → [−1, 1], ε and k be such that X   fb(S)2 ≤ exp − O(k 2 log k)/ε . |S|>k Then f is ε-close to a (2O(k) /ε2 )-junta. In the case of functions on ΩnK , there is a statement along the lines of Friedgut’s theorem for functions whose image is {0, 1} [Ben+16]. Here we obtain a junta theorem for general f : ΩnK → D (in loose analogy to [DFKO07]) using the cyclic-group Bohnenblust–Hille inequality. Theorem 29. If f : ΩnK → D has degree at most d, then there exists another function h : ΩnK → D such that kf − hk2 ≤ ε and h is a k-junta for  2d BH2d ΩK   k≤d ε O(log K)d ≤d ε !2d . This argument is similar to the junta theorem for qubits in [VZ23], which is credited to Eskenazis. It is also a good warmup for the learning results in the sequel. Proof. Denote the heavy Fourier coefficients of f by S = {α : |fb(α)| ≥ t} and define the function ft (z) = X α∈{0,1,...,K−1}n 1α∈S · fb(α)z α . 61 Using the definition of S and the cyclic group BH inequality we have ! |S| = X α∈S   2d 2d 2d X 2d 2d |α| d+1 d+1 = t− d+1 |α| d+1 ≤ t− d+1 BH≤d , ΩK |α| all α so ft is a k-junta with  k ≤ d 1t · BH≤d ΩK  2d d+1 . Let cl(x) = x/ max{1, |x|} be a clamp function. Then cl(ft ) has image in D and is a k-junta for the same k. Because |cl(ft )(x) − f (x)| ≤ |ft (x) − f (x)| pointwise, we also have kf − cl(ft )k22 ≤ kf − ft k22 = X 2 |fb(α)|2 ≤ t d+1 α6∈S  X 2d 2  |α| d+1 ≤ t d+1 BH≤d ΩK  2d d+1 . all α Now with the choice t = εd+1 BH≤d ΩK a k-junta for −d , we have kf − cl(ft )k2 ≤ ε with cl(ft ) 2d  BH≤d ΩK   k=d . ε 6.3 Qudit Bohnenblust–Hille in the Heisenberg–Weyl basis Definition 2 (Heisenberg–Weyl Basis). Fix K ≥ 2 and let ω = ωK = exp(2πi/K). Define the K-dimensional clock and shift matrices respectively via Z |ji = ω j |ji , X |ji = |j + 1i for all j ∈ ZK . Here ZK := {0, 1, . . . , K − 1} denotes the additive cyclic group of order K. Note that X K = Z K = I. See more in [AEHK16]. Then the Heisenberg–Weyl basis for MK (C) is HW(K) := {X ` Z m }`,m∈ZK . Any observable A ∈ MK (C)⊗n has a unique Fourier expansion with respect to HW(K) as well: A= X `1 m1 b ~ A( `, m)X ~ Z ⊗ · · · ⊗ X `n Z mn , (6.3.1) n ~ `,m∈Z ~ K b ~ where A( `, m) ~ ∈ C is the Fourier coefficient at (~`, m). ~ We say that A is of b ~ degree at most d if A( `, m) ~ = 0 whenever |(~`, m)| ~ := n X (`j + mj ) > d. j=1 62 Here, 0 ≤ `j , mj ≤ K − 1. Noting that the eigenvalues of Heisenberg–Weyl matrices are the roots of unity, it is natural to pursue a reduction to a scalar BH inequality over ΩnK , the multiplicative cyclic group of order K—precisely the inequality needed for classical learning on functions on ΩnK . This reduction works well when K is prime. Theorem 30 (Qudit Bohnenblust–Hille, Heisenberg–Weyl Basis: prime case). Fix a prime number K ≥ 2 and suppose d ≥ 1. Consider an observable A ∈ MK (C)⊗n of degree at most d. Then we have b 2d ≤ C(d, K)kAk , kAk op d+1 (6.3.2) with C(d, K) ≤ (K + 1)d BH≤d ΩK . When K is non-prime, the reduction still works under modifications, namely the degree may jump from d up to (K − 1)d. Theorem 31 (Qudit Bohnenblust–Hille, Heisenberg–Weyl Basis: non-prime case). Fix a non-prime number K ≥ 4 and suppose d ≥ 1. Consider an observable A ∈ MK (C)⊗n of degree at most d. Then we have b 2(K−1)d ≤ C(d, K)kAk , kAk op (6.3.3) (K−1)d+1 ≤(K−1)d with C(d, K) ≤ K 2d BHΩK . In fact, the constant K 2d can be replaced by |ΣK |d with |ΣK | being the cardinality of ΣK = {(`, m) ∈ ZK × ZK : ` and m are coprime}. The proofs of Theorems 30 and 31 are contained in Section 6.3. The full strength of Theorems 30 and 31 relies on the BH inequality for the cyclic groups ΩnK , i.e., the finiteness of the Bohnenblust–Hille constant BH≤d ΩK for cyclic groups which we quote here for reference. Fix K ≥ 3 and denote ω := e2πi/K . Let ΩK := {1, ω, ω 2 , . . . , ω K−1 }. Then any function f : ΩnK → C admits the unique Fourier expansion f (z) = X fb(α)z α , (6.3.4) α where α = (α1 , . . . , αn ) are vectors of non-negative integers and each αj ≤ K −1. We say f is of degree at most d if fb(α) = 0 whenever |α| > d. The following result was proved in [SVZ24a; SVZ25; KSVZ24; Bec+25]. 63 Theorem 32 (Cyclic Bohnenblust–Hille). Fix K ≥ 3 and d ≥ 1. There exists C(d, K) > 0 such that for all n ≥ 1 and for all f : ΩnK → C of degree at most d, we have kfbk 2d ≤ C(d, K) sup |f (z)|. (6.3.5) d+1 z∈Ωn K Denote by BH≤d ΩK the best constant C(d, K) in (6.3.5). An upper bound ≤d d BHΩK ≤ O(log K) was obtained in [KSVZ24; Bec+25]. ∗ ∗ ∗ In this section we prove Theorems 30 and 31 via reduction to the BH inequality for cyclic groups, Theorem 32. We collect first a few facts about the Heisenberg–Weyl basis {X ` Z m }. Fix K ≥ 3. Recall that gcd(a, b) denotes the greatest common divisor of a and b. For (`, m) ∈ ZK × ZK , gcd(`, m) is understood as when `, m ∈ {1, 2, . . . , K}, i.e. we do not mod K freely here. For example if K = 6, then gcd(0, 2) is understood as gcd(6, 2) = 2. For a group G, we use the convention that hgi is the abelian subgroup generated by g ∈ G. So for any (`, m) ∈ ZK × ZK , we have h(`, m)i = {(k`, km) : k ∈ ZK }. (6.3.6) In the sequel, we denote ω = ωK = e2πi/K and use the notation ω 1/2 := ω2K = eπi/K . Lemma 33. We have the following: 1. {X ` Z m : `, m ∈ ZK } form a basis of MK (C). 2. For all k, `, m ∈ ZK : 1 (X ` Z m )k = ω 2 k(k−1)`m X k` Z km and for all `1 , `2 , m1 , m2 ∈ ZK : X `1 Z m1 · X `2 Z m2 = ω `2 m1 −`1 m2 X `2 Z m2 · X `1 Z m1 . 3. If gcd(`1 , m1 ) = 1 and (`, m) ∈ / h(`1 , m1 )i, then X `1 Z m1 · X ` Z m = ω `m1 −`1 m X ` Z m · X `1 Z m1 with ω `m1 −`1 m 6= 1. (6.3.7) 64 4. If gcd(`, m) = 1, then the set of eigenvalues of X ` Z m is either ΩK or Ω2K \ ΩK . Proof. 1. Suppose that X `,m ` m = 0. For any j, k ∈ ZK , we have `,m a`,m X Z P a`,m hX ` Z m ej , ej+k i = X ak,m ω jm = 0. m Since the Vandermonde matrix associated to (1, ω, . . . , ω K−1 ) is invertible, we have ak,m = 0 for all k, m ∈ ZK . 2. It follows immediately from the identity ZX = ωXZ which can be verified directly: for all j ∈ ZK ZXej = Zej+1 = ω j+1 ej+1 = ω j+1 Xej = ωXZej . 3. It is a direct consequence of (2) and the following fact: for (`1 , m1 ) ∈ ZK × ZK such that gcd(`1 , m1 ) = 1 and (`2 , m2 ) ∈ ZK × ZK , we have `1 m2 − `2 m1 ≡ 0 mod K if and only if (`2 , m2 ) ≡ (k`1 , km1 ) mod K for some k ∈ ZK . The “if” direction is obvious. To show the “only if” part, recall that by Bézout’s lemma, there exist integers α and β such that α`1 + βm1 = gcd(`1 , m1 ) = 1. Take k ≡ α`2 + βm2 mod K. Then `2 = `2 (α`1 + βm1 ) ≡ α`1 `2 + β`1 m2 ≡ k`1 mod K, (6.3.8) where we used `1 m2 ≡ `2 m1 mod K. Similarly, m2 = m2 (α`1 + βm1 ) ≡ α`2 m1 + βm1 m2 ≡ km1 mod K, (6.3.9) as desired. This finishes the proof of the fact. 4. By (2), we have (X ` Z m )2K = ω K(2K−1)`m X 2`K Z 2mK = I. So the eigenvalues of X ` Z m must be roots of unit of order 2K. Then the proof is finished as soon as we prove the following claim: for gcd(`, m) = 1, if λ is an eigenvalue of X ` Z m , then so is ωλ. To prove the claim, recall that by Bézout’s lemma, there exist integers α and β such that α` + βm = gcd(`, m) = 1. By (2), we get X ` Z m X β Z −α = ω α`+βm X β Z −α X ` Z m = ωX β Z −α X ` Z m . (6.3.10) 65 Suppose ~0 6= ξ is an eigenvector of X ` Z m with eigenvalue λ. Then X ` Z m X β Z −α ξ = ω α`+βm X β Z −α X ` Z m ξ = ωλX β Z −α ξ, (6.3.11) implying that X β Z −α ξ (non-zero since X β Z −α is invertible) is an eigenvector of X ` Z m with eigenvalue ωλ. This finishes the proof of the claim. Let us record the following observation as a lemma. Lemma 34. Suppose that k ≥ 1, A, B are two unitary matrices such that B k = I, AB = λBA with λ ∈ C and λ 6= 1. If ξ 6= ~0 is an eigenvector of B with eigenvalue µ (µ 6= 0 since µk = 1), then hξ, Aξi = 0. Proof. By assumption µhξ, Aξi = hξ, ABξi = λhξ, BAξi. Since B † = B k−1 , B † ξ = B k−1 ξ = µk−1 ξ = µξ. Thus µhξ, Aξi = λhξ, BAξi = λhB † ξ, Aξi = λµhξ, Aξi. Hence, µ(λ − 1)hξ, Aξi = 0. This gives hξ, Aξi = 0 as µ(λ − 1) 6= 0. The prime K case In this subsection we prove Theorem 30. When K is prime, the basis {X Z m } has nicer properties. ` Lemma 35. Fix K ≥ 3 a prime number. Consider the set of generators ΣK := {(1, 0), (1, 1), . . . , (1, K − 1), (0, 1)}. (6.3.12) Then the group ZK × ZK is the union of subgroups ZK × ZK = [ h(`, m)i, (6.3.13) (`,m)∈ΣK where each two subgroups intersects with the unit (0, 0) only. Moreover, for any (`, m) ∈ ΣK , the set of eigenvalues of each X ` Z m is exactly ΩK . 66 Proof. To prove the first statement, take any (`, m) ∈ ZK × ZK . If ` = 0, then (`, m) = (0, m) ∈ h(0, 1)i. If ` 6= 0, then gcd(`, K) = 1. So by Bézout’s lemma, there exists `0 such that ``0 ≡ 1 mod K. Thus (`, m) = (`, ``0 m) ∈ h(1, `0 m)i. The statement about the intersection is clear since otherwise, the cardinality of the union of these subgroups is strictly smaller than 1 + (K + 1)(K − 1) = K 2 which leads to a contradiction. The second statement follows from the proof of Lemma 33. In fact, when K is odd, (K − 1)/2 is an integer and we have by Lemma 33 (2) that 1 (X ` Z m )K = ω 2 K(K−1)`m X `K Z mK = I. So the eigenvalues of X ` Z m must be roots of unity of order K. This, together with the claim in the proof of Lemma 33 (4) and the fact that gcd(`, m) = 1, (`, m) ∈ ΣK , completes the proof of the lemma. Now we are ready to prove Theorem 30: 2πi Proof of Theorem 30. Fix a prime number K ≥ 2. Recall that ω = e K . Consider ΣK defined in (6.3.12). For any (`, m) ∈ ΣK , by Lemma 35 any z ∈ ΩK is an eigenvalue of X ` Z m and we denote by e`,m the corresponding unit z (K+1)n eigenvector. For any vector ω ~ ∈ ΩK of the form (noting that |ΣK | = K + 1) ω ~ `,m = (ω1`,m , . . . , ωn`,m ) ∈ ΩnK , ω ~ = (~ω `,m )(`,m)∈ΣK , (6.3.14) we consider the matrix ρ(~ω ) := ρ1 (~ω ) ⊗ · · · ⊗ ρn (~ω ), where ρk (~ω ) := X 1 `,m |e`,m `,m ihe `,m |. ω ωk K + 1 (`,m)∈ΣK k Then each ρk (~ω ) is a density matrix and so is ρ(~ω ). Fix (`, m) ∈ ΣK and 1 ≤ k ≤ K − 1. We have by Lemma 33 1 `,m − 2 k(k−1)`m `,m tr[X k` Z km |e`,m hez , (X ` Z m )k e`,m z ihez |] = ω z i 1 = ω − 2 k(k−1)`m z k , z ∈ ΩK . On the other hand, for any (`, m) 6= (`0 , m0 ) ∈ ΣK , we have (k`, km) ∈ / h(`0 , m0 )i by Lemma 35. From our choice gcd(`0 , m0 ) = 1. So Lemma 33 gives 0 0 0 0 0 0 X k` Z km X ` Z m = ω k` m−k`m X ` Z m X k` Z km 67 0 0 with ω k` m−k`m 6= 1. This, together with Lemma 34, implies 0 0 0 0 0 0 0 0 tr[X k` Z km |e`z ,m ihe`z ,m |] = he`z ,m , X k` Z km e`z ,m i = 0, z ∈ ΩK . for any 1 ≤ k ≤ K − 1. All combined, for all 1 ≤ k ≤ K − 1, (`, m) ∈ ΣK and 1 ≤ j ≤ n we get * k` tr[X Z km X 0 0 1 k` km `0 ,m0 ρj (~ω )] = e` `,m Z e `0 ,m0 0 ,m0 , X ωj K + 1 (`0 ,m0 )∈ΣK ωj 1 k` km `,m = e`,m Z eω`,m `,m , X ω K +1 j j 1 1 = ω − 2 k(k−1)`m (ωj`,m )k . K +1  +  Note that by Lemma 35 any polynomial in MK (C)⊗n of degree at most d is a linear combination of monomials A(~k, ~`, m; ~ ~i) := · · · ⊗ X k1 `1 Z k1 m1 ⊗ · · · ⊗ X kκ `κ Z kκ mκ ⊗ · · · , where • ~k = (k1 , . . . , kκ ) ∈ {1, . . . , K − 1}κ with 0 ≤ Pκ j=1 kj ≤ d; • ~` = (`1 , . . . , `κ ), m ~ = (m1 , . . . , mκ ) with each (`j , mj ) ∈ ΣK ; • ~i = (i1 , . . . , iκ ) with 1 ≤ i1 < · · · < iκ ≤ n; • X kj `j Z kj mj appears in the ij -th place, 1 ≤ j ≤ κ, and all the other n − κ elements in the tensor product are the identity matrices I. (K+1)n So for any ω ~ ∈ ΩK that of the form (6.3.14) we have from the above discussion tr[A(~k, ~`, m; ~ ~i)ρ(~ω )] = κ Y tr[X kj `j Z kj mj ρij (~ω )] j=1 = ω − 12 Pκ j=1 kj (kj −1)`j mj (K + 1)κ (ωi`11 ,m1 )k1 · · · (ωi`κκ ,mκ )kκ . Thus ω ~ 7→ tr[A(~k, ~`, m; ~ ~i)ρ(~ω )] is a monomial on (ΩK )(K+1)n of degree at most Pκ j=1 kj ≤ d. Now for general polynomial A ∈ MK (C)⊗n of degree at most d: A= X ~k,~ `,m, ~ ~i c(~k, ~`, m; ~ ~i)A(~k, ~`, m; ~ ~i), 68 where the sum runs over the above (~k, ~`, m; ~ ~i). This is the Fourier expansion of A and each c(~k, ~`, m; ~ ~i) ∈ C is the Fourier coefficient. So 1/p  |c(~k, ~`, m; ~ ~i)|p  X b = kAk p . ~k,~ `,m, ~ ~i (K+1)n To each A we assign the function fA on ΩK given by fA (~ω ) = tr[Aρ(~ω )] 1 = ω− 2 X Pκ kj (kj −1)`j mj j=1 c(~k, ~`, m; ~ ~i) (K + 1)κ ~k,~ `,m, ~ ~i (ωi`11 ,m1 )k1 · · · (ωi`κκ ,mκ )kκ . Note that this is the Fourier expansion of fA since the monomials (ωi`11 ,m1 )k1 · · · (ωi`κκ ,mκ )kκ differ for distinct (~k, ~`, m; ~ ~i)’s. Therefore, for p > 0 p 1/p  kfcA kp =  X c(~k, ~`, m; ~ ~i)  (K + 1)κ ~ ~k,~ `,m, ~ i 1/p  X 1  ≥ |c(~k, ~`, m; ~ ~i)|p  d (K + 1) ~ ~ ~ k,`,m, ~ i = 1 b . kAk p (K + 1)d According to Theorem 32, one has kfcA k 2d ≤ BH≤d ΩK kfA kΩ(K+1)n d+1 K for some BH≤d ω ) is a density matrix, we have by duality ΩK < ∞. Since each ρ(~ that kfA kΩ(K+1)n = sup | tr[Aρ(~ω )]| ≤ kAkop . K ω ~ ∈(ΩK )(K+1)n All combined, we obtain b 2d ≤ (K+1)d kfc k 2d ≤ (K+1)d BH≤d kf k (K+1)n ≤ (K+1)d BH≤d kAk . kAk A A Ω op ΩK ΩK d+1 d+1 K The non-prime K case This subsection is devoted to the proof of Theorem 31. Throughout this part, K ≥ 4 is a non-prime integer. We start with a substitute of ΣK in (6.3.12) for non-prime K. 69 Lemma 36. Fix non-prime K ≥ 4. Consider ΣK := {(`, m) ∈ ZK × ZK : gcd(`, m) = 1}. (6.3.15) Then ZK × ZK is the union of subgroups generated by elements in ΣK : ZK × ZK = [ h(`, m)i. (6.3.16) (`,m)∈ΣK Proof. The proof is direct: any (`, m) ∈ ZK × ZK belongs to h(`1 , m1 )i for (`1 , m1 ) = (`/ gcd(`, m), m/ gcd(`, m)) ∈ ΣK . Recall that when K is prime, for two different subgroups h(`1 , m1 )i 6= h(`2 , m2 )i one has the singleton set {(0, 0)} as their intersection. However, this is no longer the case when K is not prime. For example, for K = 6, we have h(1, 0)i 6= h(2, 3)i while h(1, 0)i ∩ h(2, 3)i = {(0, 0), (2, 0), (4, 0)}. This difference will make the proof of Theorem 31 more involved. Proof of Theorem 31. Fix non-prime K ≥ 4. Consider ΣK defined in (6.3.15). Then we know by Lemma 36 that the set of eigenvalues of X ` Z m is either ΩK or Ω2K \ ΩK . In either case, suppose that z is an eigenvalue of X ` Z m . We denote by e`,m the unit eigenvector of X ` Z m corresponding to z. z − Write ΣK = Σ+ K ∪ ΣK with ` m Σ+ is ΩK } K := {(`, m) ∈ ΣK : the set of eigenvalues of X Z (6.3.17) and ` m Σ− is Ω2K \ ΩK }. (6.3.18) K := {(`, m) ∈ ΣK : the set of eigenvalues of X Z |Σ |n As before, for any ω ~ ∈ ΩK K ω ~ = (~ω `,m )(`,m)∈ΣK , of the form ω ~ `,m = (ω1`,m , . . . , ωn`,m ) ∈ ΩnK , (6.3.19) we shall consider ρ(~ω ) := ρ1 (~ω ) ⊗ · · · ⊗ ρn (~ω ) (6.3.20) where each ρj (~ω ) is the average of some eigen-projections of X ` Z m , (`, m) ∈ ΣK . ` m If (`, m) ∈ Σ+ has ωj`,m ∈ ΩK as an eigenvalue with e`,m being K , then X Z ω `,m j `,m the unit eigenvector. If (`, m) ∈ Σ− ∈ ΩK is not an eigenvalue of K , then ωj X ` Z m . In this case, X ` Z m has ω 1/2 ωj`,m ∈ Ω2K \ ΩK as an eigenvalue with e`,m being the unit eigenvector. ω 1/2 ω `,m j 70 For each 1 ≤ j ≤ n, consider X X 1 1 `,m |e`,m |e`,m `,m i he`,m |. `,m i he `,m | + ωj ω 1/2 ωj`,m |ΣK | (`,m)∈Σ+ ωj |ΣK | (`,m)∈Σ− ω1/2 ωj ρj (~ω ) := K K (6.3.21) By definition, each ρj (~ω ) is a density matrix and so is ρ(~ω ). For any (0, 0) 6= (`0 , m0 ) ∈ ZK × ZK and any (`, m) ∈ ΣK , either (`0 , m0 ) ∈ / h(`, m)i or (`0 , m0 ) = (k`, km) for some k ∈ ZK . If (`0 , m0 ) ∈ / h(`, m)i, then by Lemma 33 0 0 0 0 0 0 X ` Z m · X ` Z m = ω `m −` m X ` Z m · X ` Z m (6.3.22) 0 0 with ω `m −` m 6= 1. So Lemma 34 gives 0 0 `,m tr[X ` Z m |e`,m z i hez |] = 0, (6.3.23) for any eigenvalue z of X ` Z m . If (`0 , m0 ) = (k`, km) for some k ∈ ZK , then by Lemma 33 0 1 0 X ` Z m = X k` Z km = ω − 2 k(k−1)`m (X ` Z m )k . (6.3.24) So for any eigenvalue z of X ` Z m . 0 1 0 `,m − 2 k(k−1)`m k tr[X ` Z m |e`,m z . z i hez |] = ω (6.3.25) All combined, we have for any ω ~ ∈ (ΩK )|ΣK |n that h 0 0 i tr X ` Z m ρj (~ω ) = X 1 |ΣK | (`,m)∈Σ+ :(`0 ,m0 )=(k `,m `,k`,m m) K + 1 |ΣK | (`,m)∈Σ− :(`0 ,m0 )=(k X 1 |ΣK | (`,m)∈Σ+ :(`0 ,m0 )=(k `,m `,k`,m m) 1 ω − 2 k`,m (k`,m −1)`m (ωj`,m )k`,m `,m `,k`,m m) K + 1 ω − 2 k`,m (k`,m −1)`m (ω 1/2 ωj`,m )k`,m X K = 1 ω − 2 k`,m (k`,m −1)`m (ωj`,m )k`,m 1 |ΣK | (`,m)∈Σ− :(`0 ,m0 )=(k 1 K 1 ω − 2 k`,m (k`,m −1)`m+ 2 k`,m (ωj`,m )k`,m . X `,m `,k`,m m) Here in the summation, when (`0 , m0 ) ∈ h(`, m)i we write (`0 , m0 ) = (k`,m `, k`,m m) h 0 i 0 |Σ |n with 1 ≤ k`,m ≤ K − 1. So ω ~ 7→ tr X ` Z m ρj (~ω ) is a polynomial on ΩK K of degree at most K − 1, and all the (non-zero) coefficients are of modulus |ΣK |−1 . To compare, in the prime K case, we have only one non-zero term in the above summation. In the non-prime K case, there might be more than one term. 71 That is, we may have different (`1 , m1 ) and (`2 , m2 ) in ΣK such that (`0 , m0 ) = (k1 `1 , k1 m1 ) = (k2 `2 , k2 m2 ) with 1 ≤ k1 , k2 ≤ K − 1. For example for K = 6, we have (0, 3) = (3 · 0, 3 · 1) = (3 · 2, 3 · 3) = (3 · 4, 3 · 5) = (3 · 2, 3 · 1) = (3 · 4, 3 · 3). 0 0 0 0 Though tr[X ` Z m ρj (~ω )] is no longer a monomial of degree deg(X ` Z m ), it is still a non-zero polynomial of degree at most K − 1. Now for any monomial in MK (C)⊗n of degree at most d admitting the form A(~`, m; ~ ~i) := · · · ⊗ X `1 Z m1 ⊗ · · · ⊗ X `κ Z mκ ⊗ · · · , (6.3.26) where κ ≤ d and • ~` = (`1 , . . . , `κ ), m ~ = (m1 , . . . , mκ ) with each (0, 0) 6= (`j , mj ) ∈ ZK × ZK ; • ~i = (i1 , . . . , iκ ) with 1 ≤ i1 < · · · < iκ ≤ n; • and each X `j Z mj appears in the ij -th place, and all the other n − κ elements in the tensor product are the identity matrices I. According to our previous discussion, tr[A(~`, m; ~ ~i)ρ(~ω )] = Y tr[X `j Z mj ρij (~ω )] (6.3.27) 1≤j≤κ is a linear combination of monomials (ωia11 ,b1 )c1 · · · (ωiaκκ ,bκ )cκ of degree at most (K − 1)κ ≤ (K − 1)d, with (aj , bj ) ∈ ΣK and 1 ≤ cj ≤ K − 1 such that (cj aj , cj bj ) ≡ (`j , mj ) mod K. This implies that these monomials ~ 0 ; ~i0 ), the remember the profile (~`, m; ~ ~i) well; i.e., for distinct (~`, m; ~ ~i) 6= (`~0 , m ~ 0 ; ~i0 )ρ(~ω )] do not corresponding polynomials tr[A(~`, m; ~ ~i)ρ(~ω )] and tr[A(`~0 , m admit common monomials. Moreover, the coefficients of the those monomials are all of the modulus |ΣK |−κ ≥ |ΣK |−d . For general A ∈ MK (C)⊗n of degree at most d admitting X A= c(~`, m; ~ ~i)A(~`, m; ~ ~i) (6.3.28) ~ `,m, ~ ~i as the Fourier expansion, consider the polynomial fA (~ω ) = tr[Aρ(~ω )] = X ~ `,m, ~ ~i c(~`, m; ~ ~i) tr[A(~`, m; ~ ~i)ρ(~ω )] (6.3.29) 72 |Σ |n on ΩK K . From the above discussion, the `p -norm kfcA kp of Fourier coefficients of fA satisfies 1/p  kfcA kp ≥ |ΣK |−d  X |c(~`, m; ~ ~i)|p  b , = |ΣK |−d kAk p p > 0. ~ `,m, ~ ~i Moreover, fA is of degree at most (K − 1)d. So Theorem 32 implies ≤(K+1)d kfcA k 2(K−1)d ≤ BHΩK kfA k∞ . (K−1)d+1 Recall that each ρ(~ω ) is a density matrix, so by duality kfA k∞ = | tr[Aρ(~ω )]| ≤ kAkop . sup ω ~ ∈(ΩK )(K+1)n All combined, we prove that ≤(K+1)d b 2(K−1)d ≤ |Σ |d kfc k 2(K−1)d ≤ |Σ |d BH kAk K A K ΩK (K−1)d+1 6.4 (K−1)d+1 ≤(K+1)d kfA k∞ ≤ |ΣK |d BHΩK kAkop . Learning Here we give learning algorithms for low-degree functions on ΩnK and local qudit observables. This work is published in [KSVZ24]. We begin by extracting the estimation lemma implicit in [EI22] that will allow us to use our new Bohnenblust–Hille-type inequalities. Theorem 37 (Generic Eskenazis–Ivanisvili). Let d ∈ N and η, B > 0. Suppose v, w ∈ Cn with kv − wk∞ ≤ η and kvk 2d ≤ B. Then for we defined as d+1 wej = wj 1[|wj |≥η(1+√d+1)] we have the bound 1 kwe − vk22 ≤ (e5 η 2 dB 2d ) d+1 . Proof. Let t > 0 be a threshold parameter to be chosen later. Define St = {j : |wj | ≥ t} and note from the triangle inequality in C that |vj | ≥ |wj | − |vj − wj | = t − η for j ∈ St (6.4.1) |vj | ≤ |wj | + |vj − wj | = t + η for j 6∈ St . (6.4.2) We may also estimate |St | as |St | = X |vj | (6.4.1) j∈S |vj | 2d ≤ (t−η)− d+1 X j∈[n] 2d 2d 2d 2d 2d − d+1 |vj | d+1 ≤ (t−η)− d+1 kvk d+1 B d+1 . 2d ≤ (t−η) d+1 (6.4.3) 73 With we (t) := (wj 1[wj ≥t] )nj=1 , we find X kwe (t) − vk22 = |wj − vj |2 + (6.4.2) 2 |vj |2 ≤ |St |η 2 + (t + η) d+1 j6∈St j∈St (6.4.3) X  2d X 2d |vj | d+1 j∈[n]  2 2d ≤ B d+1 η 2 (t − η)− d+1 + (t + η) d+1 . √ Choosing t = η(1 + d + 1) then yields we with error, after some careful scalar estimates, 1 kwe − vk22 ≤ (e5 η 2 dB 2d ) d+1 . See [EI22, Eqs. 18 & 19] for details on the scalar estimates. In the context of low-degree learning, v is the true vector of Fourier coefficients, and w is the vector of empirical coefficients obtained through Fourier sampling. Cyclic group learning Theorem 38. Let f : ZnK → D be a degree-d function. Then with 2 (log K)O(d ) log(n/δ)ε−d−1 independent random samples (x, f (x)), x ∼ U(ZnK ), we may with confidence 1 − δ learn a function fe : ZnK → C with kf − fek22 ≤ ε. Proof. Let f = α fb(α)z α be the Fourier expansion. For a number of samples iid s to be specified later, sample x(1) , . . . , x(s) ∼ U({0, 1, . . . , K − 1}n ) and for each α ∈ ZnK with |α| ≤ d, form the empirical Fourier coefficient P s 1X − wα := f (x(j) )ωK s j=1 2πi (j) Pn `=1 (j) α ` x` , (j) where ωK = e K and x(j) = (x1 , . . . , xn ). Then wα is a sum of bounded i.i.d. random variables with expected value fb(α), so Chernoff gives h  2 2  i + = fb(α) − wα ≥ η 2 h   h   √ i √ i ≤ Pr |< fb(α) − wα | ≥ η/ 2 + Pr |= fb(α) − wα | ≥ η/ 2 Pr[|fb(α) − wα | ≥ η] = Pr < fb(α) − wα   ≤ 4 exp −sη 2 /4 . So the probability we simultaneously estimate all nonzero Fourier coefficients of f to within η is h i Pr |fb(α) − wα | < η for all α with |α| ≤ d ≥ 1 − 4 d X k=0 ! ! n −sη 2 exp , k 4 74 which in turn we will require to be ≥ 1 − δ. Now applying Theorem 37 to obtain we and recalling kfbk 2d ≤ BH≤d ZK kf k∞ = d+1 e BH≤d ZK we have that with probability 1−δ, the function f (x) := has L2 error (Parseval) kfe − f k2 = X e α |2 ≤ |fb(α) − ω 2  α 2d e η d(BH≤d ZK ) 5 2 P eα αw  1 d+1 . α j xj j=1 ωK Qn (6.4.4) −2d So in order to achieve kfe−f k22 ≤ ε it is enough to pick η 2 = εd+1 e−5 d−1 (BH≤d , ZK ) which entails by standard estimates that taking a number of samples s with   2d 4e5 d2 (BH≤d 4en ZK ) log εd+1 δ s≥ suffices. Qudit learning We will find it more convenient to use a different orthonormal basis for qudit learning, the so-called Gell–Mann matrices. Definition 3 (Gell-Mann Basis). Let K ≥ 2. Put Ejk = |jihk| , 1 ≤ j, k ≤ K. The generalized Gell-Mann Matrices are a basis of MK (C) and are comprised of the identity matrix I along with the following generalizations of the Pauli matrices: symmetric: Ajk = q  Ejk + Ekj antisymmetric: Bjk = q  − iEjk + iEkj diagonal: where Γm := K 2 K 2  for 1 ≤ j < k ≤ K  for 1 ≤ j < k ≤ K Pm Cm = Γm ( k=1 Ekk − mEm+1,m+1 ) for 1 ≤ m ≤ K − 1, q K . We denote m2 +m GM(K) := {I, Ajk , Bjk , Cm }1≤j<k≤K,1≤m≤K−1 . An observable A ∈ MK (C)⊗n has expansion in the GM basis as A= X α∈Λn K b A(α)M α = X b A(α) Nn j=1 Mαj α∈Λn K for some index set ΛK (so {Mα }α∈ΛK = GM(K)). Letting |α| = |{j : Mαj 6= I}|, b we say A is of degree at most d if A(α) = 0 for all α with |α| > d. In [SVZ24b] we find the Gell-Mann BH inequality enjoys a reduction to the 2 Boolean cube BH inequality on {−1, 1}n(K −1) and obtain the following. 75 Theorem 39 (Qudit Bohnenblust–Hille, Gell-Mann Basis). Fix any K ≥ 2 and d ≥ 1. There exists C(d, K) > 0 such that for all n ≥ 1 and GM observable A ∈ MK (C)⊗n of degree at most d, we have b 2d ≤ C(d, K)kAk . kAk op d+1 Moreover, we have C(d, K) ≤  d 3 (K 2 − K) 2 (6.4.5) BH≤d {±1} . In particular, for K = 2 we recover the main result of [VZ23] exactly. Theorem (Low-degree Qudit Learning, restatement of Theorem 9). Let A be a degree-d observable on n qudits with kAkop ≤ 1. Then there is a collection S of product states such that with a number O  KkAkop C·d2 2 −d−1 dε   log nδ of samples of the form (ρ, tr[Aρ]), ρ ∼ U(S), we may with confidence 1 − δ e 2 ≤ ε. learn an observable Ae with kA − Ak 2 Here kAk2 denotes the normalized L2 norm induced by the inner product hA, Bi := K −n tr[A† B]. Also, we choose to include explicit mention of kAkop here as it will be useful later. For applications it is natural to assume kAkop is bounded independent of n. Proof. We will first pursue an L∞ estimate of the Fourier coefficients in the 2 iid ~ 1, . . . , x ~ s ∼ {−1, 1}n(K −1) . As in the Gell-Mann basis. To that end, sample x ~ we partition indices as x ~ = (x1 , . . . , xn ) ∈ proof of Theorem 39, for any such x K 2 −1 n ({−1, 1} ) with each x` , 1 ≤ ` ≤ n, corresponding to a qudit. Each x` is further partitioned as K K x` = (x(`) , y (`) , z (`) ) ∈ {−1, 1}( 2 ) × {−1, 1}( 2 ) × {−1, 1}K−1 , with each sub-coordinate associated with a specific Gell-Mann basis element for that qudit. ~ , for each qudit ` ∈ [n] form the mixed state Again for each x   K−1 (`) (`) X X X (x ) (y ) 1 (`) √ 1 r(x(`) , y (`) , z (`) ) = K   Ajkjk + Bjkjk + zm C + K−1 · I. 2 2K m 3 2 1≤j<k≤K m=1 1≤j<k≤K ~ the n qudit mixed state Then we may define for x r(~x) = n O `=1 r(x(`) , y (`) , z (`) ) 76 and consider the function fA (~x) := tr[A · r(~x)]. Let S(α) denote the index map from the GM basis to subsets of [n(K 2 − 1)]. With these states in hand and in view of the identity q   b fcA S(α) = c|α| A(α) with c := K/2   3 K2 < 1, we may now define the empirical Fourier coefficients W(α) = c−|α| · s s Y Y 1X 1X fA (~xt ) xj = c−|α| tr[A · r(~x)] xj . s t=1 s t=1 j∈S(α) j∈S(α) The coefficient W(α) is a sum of bounded i.i.d. random variables each with b expectation A(α), so by Chernoff we have h i b Pr |W(α) − A(α)| ≥ η ≤ 2 exp(−sη 2 c|α| ) . Taking the union bound, we find as before the chance of achieving `∞ error η is h i b Pr |W(α) − A(α)| < η for all α with |α| ≤ d ≥ 1 − 2 d X k=0 ! n exp(−sη 2 cd ) , k which again we shall require to be ≥ 1 − δ. f and recalling kAk b 2d ≤ BH≤d Applying Theorem 37 to obtain W GM(K) kAkop d+1 we find the estimated operator Ae := X f W(α)M α α has L2 -squared error kAe − Ak22 (Parseval) X = 2  1  2d d+1 f b W(α) − A(α) ≤ e5 η 2 d(BH≤d . GM(K) kAkop ) α −2d Thus to obtain error ≤ ε it suffices to pick η 2 = εd+1 e−5 d−1 (BH≤d , GM(K) kAkop ) which entails by standard estimates that the algorithm will meet the requirements with a sample count of   2d 2en s ≥ e6 K 3/2 d2 (BH≤d ε−d−1 . GM(K) kAkop ) log δ Part II Other applications 77 78 Published as [Slo24] Joseph Slote. “Parity vs. AC0 with Simple Quantum Preprocessing”. In: 15th Innovations in Theoretical Computer Science Conference, ITCS 2024. Ed. by Venkatesan Guruswami. Vol. 287. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, 92:1–92:21. doi: 10.4230/LIPICS.ITCS.2024.92. [CNS25] Matthias Caro, Preksha Naik, and Joseph Slote. “Testing classical properties from quantum data”. In: TQC 2025 (to appear) (2025). arXiv: 2411.12730 [quant-ph]. 79 Chapter 7 Introduction This part concerns two other applications of discrete harmonic analysis in (quantum) complexity theory. The first, “Parity vs. AC0 with simple quantum preprocessing” concerns concrete complexity theory of quantum circuits. Compared to the community’s understanding of shallow classical circuits, small quantum circuits are still very mysterious. This paper introduces a circuit model, termed AC0 ◦ QNC0 , which is arguably the simplest possible model of constant-depth quantum computation that is still capable of solving nontrivial decision problems and makes initial progress in proving bounds on its capabilities. From a technical perspective, the work identifies certain basic mysteries related to nonlocal games that appear in quantum circuits under restriction. From a conceptual standpoint, the work suggests that while constant-depth quantum circuits have dramatic advantage over their classical counterparts for total search (or multi-output) problems, they may not have much advantage for decision problems—matching a dichotomy that was recently identified in the context of query complexity [YZ22]. This project also provided inspiration for later influential works [NPVY23; ADOY25] on a related model of constant-depth quantum computation known as QAC0 (Warning: it is not known whether classical AC0 ⊂ QAC0 and is widely expected to be false). The second application, “Testing classical properties with quantum data” introduces a novel mode of property testing and opens the door to a new category of quantum advantage. In classical complexity theory, property testing is traditionally the home of super-fast algorithms which use queries to determine whether a black-box Boolean function f has a certain property √ (for example, whether f is monotone, which has an O( n)-query algorithm). Unfortunately, for applications to data analysis and machine-learning, when   a testing algorithm only has access to random samples x, f (x) , it becomes much harder to test properties of f , often requiring just as much data as learning the whole function. (For the example of monotonicity, we have the √ lower bound of 2Ω( n) samples [Bla24]). This work shows that for a wide range of properties, if a tester is provided 80 with data in certain quantum encodings, ultrafast testers are again available. And this quantum data shares “physical” properties with classical data, in that it can be collected far in advance and is independent of the property to be tested. There is much that remains to be understood about the power of quantum data for testing, and the work lays out several directions for further research. We now discuss the specific contributions of these works in more detail. Application I: Parity vs. AC0 with simple quantum preprocessing A recent line of work [BGK18; WKST19; GS20; BGKT20; WP23] has shown the unconditional advantage of constant-depth quantum computation, or QNC0 , over NC0 , AC0 , and related models of classical computation. Problems exhibiting this advantage include search and sampling tasks related to the parity function, and it is natural to ask whether QNC0 can be used to help compute parity itself. Namely, we study AC0 ◦ QNC0 —a hybrid circuit model where AC0 operates on measurement outcomes of a QNC0 circuit—and we ask whether Par ∈ AC0 ◦ QNC0 . We believe the answer is negative. In fact, we conjecture AC0 ◦ QNC0 cannot even achieve Ω(1) correlation with parity. As evidence for this conjecture, we prove: • When the QNC0 circuit is ancilla-free, this model can achieve only negligible correlation with parity, even when AC0 is replaced with any function having LMN-like decay in its Fourier spectrum. • For the general (non-ancilla-free) case, we show via a connection to nonlocal games that the conjecture holds for any class of postprocessing functions that has approximate degree o(n) and is closed under restrictions. Moreover, this is true even when the QNC0 circuit is given arbitrary quantum advice. By known results [BKT19], this confirms the conjecture for linear-size AC0 circuits. • Another approach to proving the conjecture is to show a switching lemma for AC0 ◦ QNC0 . Towards this goal, we study the effect of quantum preprocessing on the decision tree complexity of Boolean functions. We find that from the point of view of decision tree complexity, nonlocal channels are no better than randomness: a Boolean function f precomposed with 81 an n-party nonlocal channel is together equal to a randomized decision tree with worst-case depth at most DTdepth [f ]. Taken together, our results suggest that while QNC0 is surprisingly powerful for search and sampling tasks, that power is “locked away” in the global correlations of its output, inaccessible to simple classical computation for solving decision problems. Application II: Testing classical properties from quantum data Many classes of Boolean functions can be tested much faster than they can be learned. However, this speedup tends to rely on query access to the function f . When access is limited to random samples (x, f (x))—the passive testing model and a natural setting for data science—testing can become much harder. Here we introduce quantum passive testing as a quantum version of this “data science scenario”: quantum algorithms that test properties of a function f solely from quantum data in the form of copies of the function state P |f i ∝ x |x, f (x)i. Just like classical samples, function states are independent of the property of interest and can be collected well in advance. Quantum advantage in testing from data: an emerging theme. For three well-established properties—monotonicity, symmetry, and trianglefreeness—we show passive quantum testers are unboundedly- or super-polynomially better than their classical passive testing counterparts, and in fact are competitive with classic query-based testers in each case. Existing quantum testers for k-juntas and linearity can be interpreted as passive quantum testers too and exhibit the same phenomena. Inadequacy of Fourier sampling. Our new testers use techniques beyond quantum Fourier sampling, and it turns out this is necessary: we show a certain class of bent functions can be tested from O(1) function states but has a sample complexity lower bound of √ 2Ω( n) for any tester relying exclusively on Fourier and classical samples. Classical queries vs. quantum data. Our passive quantum testers are competitive with classical query-based testers, but this isn’t universal: we exhibit a testing problem that can be solved from O(1) classical queries but requires Ω(2n/2 ) function state copies. The 82 Forrelation problem provides a separation of the same magnitude in the opposite direction, so we conclude that quantum data and classical queries are “maximally incomparable” resources for testing. Towards lower bounds. We also begin the study of lower bounds for testing from quantum data. For quantum monotonicity testing, we prove that the ensembles of [Gol+00; Bla24], which give exponential lower bounds for classical sample-based testing, do not yield any nontrivial lower bounds for testing from quantum data. New insights specific to quantum data will be required for proving copy complexity lower bounds for testing in this model. 83 Chapter 8 Parity vs. AC0 with simple quantum preprocessing In 2017, Bravyi, Gosset, and König [BGK18] proved a breakthrough unconditional separation between constant-depth quantum circuits, or QNC0 , and constant-depth bounded fan-in classical circuits, or NC0 . The authors showed that for a certain search problem solvable by QNC0 circuits, any randomized NC0 circuit solving the same problem with high probability must have logarithmic depth. The realization that unconditional proofs of quantum advantage were possible—albeit over weak models of classical computation— inspired an exciting series of results strengthening and generalizing the work of Bravyi, Gosset, and König. There are now separations against stronger classical circuit models such as constant depth circuits with unbounded fan-in, or AC0 [WKST19], average-case separations [Gal20], separations between more intricate interactive models [GS20], separations that remain even for quantum circuits subject to noise (e.g., [BGKT20]), and separations for sampling problems with no input [WP23], among others. Although these separations are for comparatively weak models of computation, they are concrete non-oracle, non-query separations, and are free from complexity-theoretic assumptions, making them important companions to the query complexity and conditional separations studied since the founding of quantum computer science. One notable feature of these QNC0 separations, however, is that they are all for search or sampling problems; decision separations appear to be absent from this list. On the surface, there is a somewhat trivial reason for this: QNC0 cannot solve interesting decision problems alone. Indeed, any single output qubit in a constant-depth quantum circuit can only depend on constantly-many input qubits, so any QNC0 circuit with one output bit may be simulated by randomized NC0 . However, this “lightcone barrier” may be removed by instead measuring all qubits in the quantum circuit and then applying a classical Boolean function f to the result. As long as f depends on all of its inputs, it might be possible for f to leverage QNC0 ’s search and sampling prowess for decision-making ends. Given Bene Watts et al.’s search separation between QNC0 and AC0 [WKST19], a natural class of Boolean functions to choose for 84 this postprocessing is AC0 itself. This gives rise to the following definition, which does not appear to have been studied before. Definition. Let AC0 ◦ QNC0 denote the model of computation composed of a QNC0 circuit C, followed by a computational basis measurement, and then an AC0 function f applied to the result. This process defines the randomized Boolean function f ◦ C : {0, 1}n → M({−1, 1}) from the hypercube to the set M({−1, 1}) of probability measures on {−1, 1}. In this chapter we take a QNC0 circuit to be a polynomial-size constantdepth quantum circuit composed of arbitrary 2-qubit unitary gates. Ancilla qubits are allowed and are initialized in the state |0m i for m ∈ poly(n). No geometric locality or clean computation constraints are assumed. A formal definition appears later as Definition 8.2. Certainly QNC0 ⊆ AC0 ◦ QNC0 , so the search separation between QNC0 and AC0 in Bene Watts et al. is also a search separation between AC0 ◦ QNC0 and AC0 . Moreover, this modification obviates the lightcone barrier mentioned above and allows us to ask meaningful questions about decision separations between concrete models of quantum and classical computation. Specifically, Bene Watts et al. [WKST19] show exponential advantage of QNC0 over AC0 for (a variant of) the “parity halving problem”: Parity halving. Given x ∈ {0, 1}n with the promise |x| ≡ 0 mod 2, output any even string if |x| ≡ 0 mod 4 and any odd string otherwise. Given the form of this problem, it is natural to ask whether parity is itself computable by a hybrid model such as AC0 ◦ QNC0 . Before summarizing our progress on this question, we pause to note another reason to study AC0 ◦ QNC0 coming from the rich subject of quantum-classical interactive proofs. A central project in this area is the classical verification of quantum computations [GKK18]. In a landmark 2018 work, Mahadev gave a cryptographic protocol for this task [Mah18]; however, whether or not this task may be accomplished without cryptographic hardness assumptions remains open despite many efforts [GKK18]. It therefore makes sense to consider the question in simpler contexts, such as where the prover and verifier are replaced with QNC0 and AC0 respectively and interact for constantly-many rounds to 85 establish the correctness of a QNC0 computation. With this perspective we see that AC0 ◦ QNC0 models the first round of interaction in such a proof system. Parity vs. AC0 ◦ QNC0 : Overview and organization We conjecture that AC0 ◦ QNC0 cannot approximate parity (Parn ) on average, over both choice of uniformly random input x ∼ U ({0, 1}n ) and the randomness in f ◦ C. It is convenient to take Par and f ◦ C to be (±1)-valued and phrase this in terms of the correlation E [(f ◦ C)(x) · Par(x)], x proportional to the advantage of f ◦ C over random guessing for computing parity. Conjecture 3. AC0 ◦ QNC0 cannot achieve correlation Ω(1) with the parity function. That is, fix a polynomial size bound p(n) and constant depth d. Then for all sequences {(fn , Cn )}n of circuits such that size(fn ), size(Cn ) ≤ p(n) and depth(fn ), depth(Cn ) ≤ d, we have E[(fn ◦ Cn )(x) · Parn (x)] → 0 as x n → ∞. Although proving correlation bounds against AC0 is a well-understood topic with many techniques (among them Håstad’s switching lemma [Hås86] and Razborov-Smolensky [Raz87; Smo87a]), when QNC0 precomputation is added these approaches cannot be used directly. The pursuit of new techniques leads us to connections with many-player nonlocal games, approximate degree bounds, and new directions for generalizing Håstad’s switching lemma. Evidence for Conjecture 3 is laid out as follows. The ancilla-free case In Section 8.1 we prove Conjecture 3 when QNC0 is restricted to be ancillafree. A key feature of such QNC0 circuits is that they correspond to unitary transformations, and we find in this case the correlation of f ◦ C with Par is controlled by the Fourier tail of f . Recall the k th Fourier tail of a Boolean function f is given by W≥k [f ] := 2 b |S|≥k f (S) . P Appealing to the Linial-Mansour-Nisan-type (LMN-type) estimates of the Fourier tail of AC0 [LMN93b], we obtain the following strong correlation bound. 86 Theorem 40 (Ancilla-free QNC0 , general AC0 case). If C is an ancilla-free QNC0 circuit and f is an AC0 function then E [(f ◦ C)(x) · Parn (x)] ≤ 2−n/polylog(n) . x This is proved as Corollary 44 in Section 8.1. The full statement holds for any Boolean function f with sufficient decay in the tail of the Fourier spectrum, including those outside of AC0 . However, as we explain in the end of Section 8.1, the proof technique of Theorem 40 cannot extend to the case of general QNC0 and we must find a different approach. Reducing to nonlocal games To move beyond ancilla-free QNC0 , in Section 8.2 we reduce Conjecture 3 to a question about the value of a certain class of nonlocal games, which we call n-player parity games and which are parameterized by a postprocessing Boolean function f . Through a connection to the notion of k-wise indistinguishability introduced in [BIVW16], we show the quantum value of a parity game is controlled by the approximate degree of the associated f . Recall for ε > 0 the ε-approximate degree of a (0, 1)-valued1 Boolean function f is given by g [f ] = min{deg(g) | g : {0, 1}n → R a polynomial with kf − gk ≤ ε} . deg ∞ ε g [f ] ≤ n for any n-variate f and ε > 0. By convention deg[f g ] := Of course, deg ε g [f ]. A function class F = (F ) deg n n≥1 is a sequence of sets Fn of n-variate 1/3 Boolean functions, and we extend approximate degree to function classes g g ]. With this notation, we have the following via deg[F](n) := maxf ∈Fn deg[f theorem. Theorem 41 (Corollary 51, Section 8.2). Suppose function class F is closed g under inverse-polynomial-sized restrictions. Then if deg[F] ∈ o(n), F ◦ QNC0 cannot achieve Ω(1) correlation with Parn , even if QNC0 is given arbitrary quantum advice. It follows from Theorem 41 that Conjecture 1 would be confirmed in full 0 g generality if deg[AC ] ∈ o(n), a notorious open problem [BT22]. Such a bound 1 For (±1)-valued f , we use the same definition after making the standard identification +1 7→ 0, −1 7→ 1. 87 is already known for large subclasses of AC0 , however: for example, for AC0 circuits of size O(n) (termed LC0 ), we may appeal to the recent bounds of [BKT19] to conclude: Theorem 42 (General QNC0 , linear-size AC0 case). Suppose f ∈ AC0 has size O(n). Then f ◦ QNC0 achieves correlation at most 1/ poly(n) with Parn . This holds even if QNC0 is given arbitrary quantum advice. That is, E[(LC0 ◦ QNC0 /qpoly) · Parn ] ∈ negl(n) . (This is proved as Corollary 52 in Section 8.2). Is the difficulty of proving approximate degree bounds for AC0 a barrier for resolving Conjecture 3? It seems unlikely: the reduction to approximate degree bounds is via a series of substantial relaxations and it would be surprising if all the required converses held. In fact, we conclude Section 8.2 with a self-contained approximation theory question (Question 2) concerning a notion 0 g of blockwise approximate degree which may be easier to solve than deg[AC ] but would still imply Conjecture 3. Towards an AC0 ◦ QNC0 switching lemma In Section 8.3 we chart a different route to resolving Conjecture 3, aiming to prove a switching lemma for our hybrid AC0 ◦ QNC0 circuits. Recall that Håstad’s original switching lemma is used to argue that (very roughly) randomly fixing a large fraction of inputs to an AC0 circuit with high probability yields a function that can be computed by a shallow decision tree. At the same time, Par retains maximum decision tree complexity under the same restrictions, so this leads to AC0 correlation bounds. In comparison to Håstad’s switching lemma and its descendants, a challenge with AC0 ◦ QNC0 circuits is that QNC0 can correlate, spread out, and bias random restrictions before they reach the bottom layer of DNFs or CNFs in the AC0 circuit. If QNC0 were replaced with randomized NC0 this problem could be readily addressed by considering each deterministic circuit in the distribution, applying standard arguments there, and computing the expected correlation with parity across circuits in the distribution. But unlike randomized computation, and as discussed e.g., in [AIK22], a recurring theme in quantum complexity theory is the impossibility of “pulling out the quantumness” from a quantum circuit. 88 Contrary to this theme, however, we show that when QNC0 is replaced by an n-party nonlocal channel N , it is possible to pull out the quantumness in a particular sense: Theorem (Theorem 53, restated). Let f : {0, 1}m → {0, 1} be any Boolean function and consider an n-party nonlocal channel N , where the ith party P receives one bit and responds with mi ≥ 0 bits, such that i mi = m. Then the random function f ◦ N is equal to a randomized decision tree Γ such that depth(T ) ≤ DTdepth [f ] for all T ∈ Supp(Γ). (This theorem is proved in Section 8.3 as Theorem 53.) By an n-party nonlocal channel we mean the channel corresponding to a quantum strategy in an n-player nonlocal game: parties receive one bit of input each and may measure disjoint systems of a shared quantum state as part of their responses, but they are not allowed to communicate. A formal definition appears as Definition 8.2. In fact, Theorem 53 is true not only for nonlocal channels, but for any channel where parties obey the no-signaling property; that is, the output of any subset S ⊂ [n] of the parties is a function only of the inputs to those parties in S. A formal definition of no-signaling channels appears as Definition 8.3. The regime where Theorem 53 is truly interesting is when DTdepth [f ] ≥ log(n). Then f may depend on all the input coordinates and (potentially) make great use of the processing power afforded by no-signaling channels. Theorem 53 says that to the contrary, precomposition of f by any no-signaling channel has no effect on the (randomized) decision tree complexity of f . How does Theorem 53 connect to Conjecture 3? As we detail in Section 8.2, the replacement of QNC0 by the channel N is essentially without loss of generality from the point of view of Conjecture 3. Unfortunately, however, AC0 circuits can easily have maximum decision tree complexity, so Theorem 53 cannot be immediately applied. Instead, we believe Theorem 53 stands as a striking example of the inability of classical postprocessing to make use of the search and sampling power of quantum and super-quantum models of computation. Additionally, we hope that this theorem’s proof technique, which involves tracking the interplay between a decision tree for f and the no-signaling channel N , represents the style of argument that could eventually lead to a switching lemma for AC0 ◦ QNC0 . 89 Outlook Taken together, these results suggest QNC0 cannot render its power in a way AC0 or other simple models of classical computation can access for the purpose of making decisions. Several questions for further research are posed in Section 8.4. Related work Unlike the quantum-classical separations surveyed in the introduction, which show quantum upper bounds and classical lower bounds, this chapter aims to prove a lower bound against a concrete model of quantum computation. The pursuit of lower bounds against quantum circuits for computational problems is a nascent area and very little is known. One quantum circuit model where lower bounds have received some concerted study is QAC0 [Moo99; H03; PFGT20; Ros21; NPVY23]. A superset of QNC0 circuits, QAC0 additionally allows for arbitrarily-large Toffoli gates, E |x1 , . . . , xk , xk+1 i 7→ x1 , . . . , xk , xk+1 ⊕ (∧ki=1 xi ) , which are quantum analogues of classical AND gates with unbounded fan-in. In this setting correlation with parity is also a central open question, and there is growing evidence that QAC0 cannot achieve Ω(1) correlation with parity either. Recent work has shown negligible correlation bounds between QAC0 and parity when a) the QAC0 circuit is restricted to depth 2 [Ros21], and b) when the QAC0 circuit is of any depth d and is restricted to O(n1/d )-many ancillas [NPVY23]. In fact, the second result is a corollary to a Pauli-basis analogue of the LMN theorem for the same subclass of QAC0 [NPVY23]. The relationship between QAC0 and AC0 ◦ QNC0 is rather unclear, and they are likely incomparable as decision classes. In fact, as far as we know, it is even open whether AC0 ⊆ QAC0 , let alone whether AC0 ◦ QNC0 ⊆ QAC0 (noting the trivial containment AC0 ⊆ AC0 ◦ QNC0 ). The difficulty in comparing these models stems from a subtlety concerning the difference between unbounded fan-in and unbounded fan-out when implemented coherently. AC0 circuits have no restriction on the fan-out of their gates, while the definition of QAC0 appears to strongly limit outward propagation of information. If one augments QAC0 with the so-called fan-out gate—which is a CNOT gate with any number of target qubits, |x1 , . . . , xk i 7→ |x1 , x1 ⊕ x2 , . . . , x1 ⊕ xk i , 90 one obtains the circuit model QAC0f , and it is known QAC0f can compute parity exactly in depth 3 [Moo99]. In view of existing lower bounds against QAC0 , it is expected that QAC0 is strictly contained in QAC0f , and assuming this holds we immediately have that the function version of AC0 ◦ QNC0 is not in the function version of QAC0 . This follows, for example, from the fact that multi-output AC0 circuits easily implement the classical reversible fan-out gate, (x1 , . . . , xk ) 7→ (x1 , x1 ⊕ x2 , . . . , x1 ⊕ xk ). It is safe to say the interaction of nonlocal gates with QNC0 —whether that interaction is coherent as in QAC0 and QAC0f , or preceded by measurement as in AC0 ◦ QNC0 —is only beginning to be understood. A separate area where concrete quantum circuit lower bounds have been very successfully developed is for state preparation problems. We do not attempt a survey here, but just mention they were crucial to the resolution of the NLTS conjecture [ABN23] and make use of ideas from error correction, which partially originate in sampling lower bounds from classical complexity [LV11]. However, it is not clear how to transfer these methods to quantum circuit lower bounds for computational problems in the AC0 ◦ QNC0 model. 8.1 Lower bounds when QNC0 is ancilla-free Here we show any Boolean function f with small Fourier tail retains a small top-degree coefficient when composed with ancilla-free QNC0 . By the celebrated work of Håstad [Hås86] and Linial, Mansour, and Nisan [LMN93b], any f ∈ AC0 is an example—but this theorem addresses a broader set of functions. On the other hand, as we discuss at the end of the section, once ancillas are allowed, the theorem no longer holds for such a general class of functions. Recall a function f : {−1, 1}n → R admits a unique Fourier decomposition f= fb(S)χS , X S⊆[n] where χS (x) := i∈S xi is the S th Fourier character (see e.g., [ODo14] for more). We will later make use of the familiar Plancherel theorem, which states for any f, g : {−1, 1}n → R that Q E[f (x)g(x)] = x X S⊆[n] fb(S)gb(S) . 91 Let us briefly connect this perspective to quantum observables. Given a Boolean function f : {±1}n → R we define its Von Neumann observable as Mf := X f (x) |xihx| . x An identity we will use is MχS = Z S , 0 where the operator Z here is the Pauli operator ( 10 −1 ), and generally for any 1-qubit operator A we use the notation AS :=   OA i  1 if i ∈ S otherwise. Any Von Neumann observable M (that is, any Hermitian operator) has expectation value on state ρ given by hM iρ := tr[M ρ] , and when M = Mf and x ∈ {0, 1}n we note the identity hMf ix := hMf i|xihx| = f (x) . With this notation, we prove the following. Theorem 43 (Correlation bound for ancilla-free QNC0 ). Let f : {±1}n → R and U an ancilla-free QNC0 circuit of depth t. Then the correlation of f ◦ U and Par is bounded as E[hU † Mf U ix · Parn (x)] ≤ x  −t n W≥2 [f ] 1/2 . For example, when f is an AC0 circuit, we may use an LMN-type Fourier concentration bound, such as from [Tal17], to a obtain: Corollary 44. If U is an ancilla-free QNC0 n-qubit circuit of depth t, and f : {±1}n → {±1} is implemented by an AC0 circuit of depth d and size s, we have ! √ −n † E[hU Mf U ix · Parn (x)] ≤ 2 · exp t+1 . x 2 O(log s)d−1 The proof of Theorem 43 relies on two brief lemmas. The first says that when measuring correlations, we could just as well have compared the correlation of f alone to the random function Parn ◦ U † , defined by applying Parn to the output of U † |xi. 92 Lemma 45 (Symmetry of correlation). Let f, g : {±1}n → {±1} and U any n-qubit unitary. Then E[hU † Mf U ix · g(x)] = E[f (x) · hU Mg U † ix ] x x = 2−n tr[Mf U Mg U † ] . Proof. Expanding the trace we have tr[Mf U Mg U † ] = X = X hz| P  P y f (y)|yihy| U  † x g(x)|xihx| U |zi z f (y)g(x) hz|yi hy| U † |xi hx| U |zi x,y,z = X f (y)g(x) hy| U |xi hx| U † |yi , (8.1.1) x,y while expanding the expectations we see E[hU † Mf U ix · g(x)] = x 1 X f (y)g(x) hx| U † |yi hy| U |xi = E[f (y) · hU Mg U † iy ] . y 2n x,y Identifying the center expression with (a multiple of) (8.1.1) and changing variables completes the lemma. The second lemma roughly says when Fourier characters ZS and ZT correspond to sets S, T of very different cardinality, they remain orthogonal (with respect to the inner product hA, Bi = tr[A† B]) after an application of U . Lemma 46 (Lightcone lemma). Suppose U is a depth-t ancilla-free quantum circuit and |S|2t < n. Then tr[Z[n] U ZS U † ] = 0. Proof. The number of qubits on which ZS acts nontrivially at most doubles upon conjugation by each layer in U . Therefore the number of non-identity coordinates in U ZS U † is at most |S|2t . Now if |S|2t < n, then there is at least one coordinate j such that U ZS U † = V[n]\j ⊗ 1j for some (n − 1)-qubit unitary V[n]\j , so tr[Z[n] U ZS U † ] = tr[Z[n] (V[n]\j ⊗ 1j )] = tr[Z[n]\j V[n]\j ] · tr[Z] = 0 because Z is traceless. With these lemmas in hand, we can give the proof of Theorem 1 in a single display: 93 Proof of Theorem 43. E [hU † Mf U ix · χ[n] (x)] = E [f (x) · hU Z[n] U † ix ] x x = X (Lemma 45) fb(S) · hU\ Z[n] U † i(S) (Plancherel) S⊆[n] = fb(S) E[hU Z[n] U † ix · χS (x)] X x S⊆[n] | X = {z } = 2−n tr[Z[n] U † ZS U ] (Lemma 45) = 0 if |S|2t < n (Lemma 46) † Z U i(S) fb(S) · hU\ [n] S⊆[n] |S|≥2−t n 1/2   ≤ X fb(S)2  |S|≥2−t n 1/2 X  † Z U i(S)2  hU\ [n] |S|≥2−t n (Cauchy-Schwarz) −t n  ≤ W≥2 [f ] 1/2 . One may ask whether this proof approach extends to QNC0 circuits with ancillas. Although it might be possible to prove slight generalizations, we present an example demonstrating that any proof approach using an LMN-type theorem as a black box will fail for general QNC0 circuits. This is essentially because functions with Fourier decay are not closed under composition. Example 47. Consider the following “Trojan horse” function on an even number of bits n = 2m: h : {±1}2m → {±1} x 7→   χ[m] (x) if x[m+1,2m] = 11 · · · 1  1 otherwise . By direct computation one finds the Fourier coefficients of h are given by b h(S) =    1 − 2−m       −2−m    2−m      0 S = ∅, S ⊆ [m], S 6= ∅ [m + 1, 2m] ⊆ S otherwise . 94 This means for any t ≥ 1, the tth Fourier tail of h is W≥t [h] ∈ O(2−n/2 ). Thus by Theorem 43, for any ancilla-free QNC0 circuit C, h ◦ C has negligible correlation with parity. On the other hand, consider the (deterministic) function C : {±1}m → {±1}2m given by x 7→ x11 · · · 1. Certainly C can be implemented in QNC0 , and we have h ◦ C = χ[m] = Parm . This example shows that exponential Fourier decay of f is not sufficient to entail Conjecture 3 for general AC0 ◦ QNC0 circuits. We must take a different approach that exploits finer structural properties of AC0 and QNC0 . 8.2 Lower bounds against AC0 ◦ QNC0 via nonlocal games Here we pass from QNC0 to nonlocal games to make an argument that works for general QNC0 . First let us fix ideas about QNC0 . Definition (QNC0 ). An n-input, depth-d QNC0 circuit C is a quantum circuit composed of d layers of arbitrary 2-qubit gates, acting on an input register of n qubits and an ancilla register of m ∈ poly(n) qubits initialized to |0m i. Via measurement of the entire output of C in the computational basis, the circuit C effects a randomized mapping from n bits of input to n + m ∈ poly(n) bits of output. A QNC0 circuit with v qubits of quantum advice, has v out of m ancilla qubits initialized to a v-qubit state, not necessarily a product state. For general v ∈ poly(n), this is denoted by the class QNC0 /qpoly. We will show a reduction from QNC0 circuits to nonlocal channels. Definition. (Nonlocal channel) Let n, k ≥ 1 and m ≥ 0. An (n, k, m) nonlocal channel is the randomized mapping defined by a quantum strategy in a nonlocal game where n parties receive one bit of input each and respond with k bits each, along with a referee response of m bits. Concretely, each party i ∈ [n] is assigned a local Hilbert space Hi and for each b ∈ {0, 1}n , a POVM n y M(i,b) = M(i,b) : y ∈ {0, 1}k o on Hi . There is also a referee Hilbert space Href with a fixed POVM n o y Mref = Mref : y ∈ {0, 1}m . The definition of the nonlocal channel is completed by a choice of shared state N  n |ψi ∈ i=1 Hi ⊗ Href and works as follows. Upon receipt of an input string 95 x ∈ {0, 1}n , the n players and one referee perform the joint measurement   M(1,x1 ) , . . . , M(n,xn ) , Mref on |ψi, resulting in the outcomes y1 , . . . , yn , and yref . The output of the channel is the (nk + m)-bit string y = y1 || · · · ||yn ||yref . Definition (No-signaling channel). An (n, k, m) no-signaling channel is defined analogously, except the correlations among parties may be general no-signaling correlations. (A very detailed definition of such channels is given in Definition 8.3.) Definition (Parity games). Let n, k, m be fixed and consider f : {0, 1}kn+m → {0, 1}. The (n, k, f ) parity game is played by n entangled and non-communicating players, with the ith player receiving input bit xi from x drawn uniformly from {0, 1}n . A (quantum) parity game strategy is an (n, k, m) nonlocal channel with output string y. Players win when f (y) = Par(x). We say a parity game strategy has advantage ε if its winning probability is at least 1/2 + ε. As a final piece of notation, for Boolean f let ¬f denote its negation. We are prepared to give our reduction to parity games. Lemma 48. Fix n ≥ 1, m ∈ poly(n), let C be a n-qubit, depth-d QNC0 circuit with m ancilla and arbitrary quantum advice, and let f : {0, 1}n+m → {0, 1} be any Boolean function. Suppose f ◦ C has correlation ε with Parn . Then for some n0 ≥ n/(2d + 1) there is a quantum strategy for the (n0 , 2d , f ) or (n0 , 2d , ¬f ) parity game with advantage ε/2. Proof. Suppose f ◦ C has correlation ε with Par. For each input qubit j denote by Lj the set of output qubits in the forward lightcone of j. Consider the graph with vertices the input qubits [n] and edges drawn between qubits j and k when Lj and Lk have nonempty intersection. Then G has degree at most 2d , so there exists an independent set S ⊆ [n] of size at least n/(2d + 1). c For each y ∈ {0, 1}S , define the circuit Cy to be C but where for j ∈ S c , the j th input is hardcoded to yj . Then Cy is a circuit on at least n/(2d + 1) variables such that the forward lightcones of input qubits are pairwise disjoint. Such a circuit defines an (n0 , 2d , m0 ) nonlocal channel for some n0 ≥ 2−d + 1 and m0 = n + m − n0 2d . (Note this m0 is without loss of generality because we may freely assign a player some output bits of the referee if their lightcone is smaller than 2d .) 96 As a result, this restriction represents a strategy for the (n0 , 2d , f ) parity game. Moreover, we have E[(f ◦ C)(x) · Par(x)] = x = [f ◦ Cy (z) · Par(y||z)] E E E Par(y) y∼{0,1}S c z∼{0,1}S y∼{0,1}S c E z∼{0,1}S [f ◦ Cy (z) · Par(z)] . Therefore since f ◦ C has ε correlation with parity on n bits, for at least one y, f ◦ Cy or ¬f ◦ Cy must have at least ε correlation (in magnitude) with parity on n/d bits. This is exactly half the advantage of the strategy defined by Cy . Lemma 48 shows that bounds on the value of parity games translate into correlation bounds for AC0 ◦ QNC0 with Par. How might we analyze parity games? They are in some sense “flipped” versions of XOR games, where parity is computed on the inputs to the players, rather than the outputs. However, it is not clear whether the rich collection of techniques developed to analyze XOR games is applicable here. Instead, we bound the no-signaling value of the game by taking the perspective of distinguishability. For any (n, k, 0) no-signaling channel N , begin by rewriting the correlation as E[(f ◦ N )(x) · Par(x)] = E[(f ◦ N )(x) | x even] − E[(f ◦ N )(x) | x odd] . 2 Let Ueven and Uodd denote the uniform distribution on even and odd bitstrings of length n respectively, and consider the pushforwards of Ueven and Uodd through N:     µ := N Ueven and ν := N Uodd . So µ and ν are distributions on strings of length N := nk, and E[(f ◦ N )(x) · Par(x)] = E[f (µ)] − E[f (ν)] = Pr[f (µ) = 1] − Pr[f (ν) = 1] . 2 Therefore the correlation of f ◦ N with parity can be phrased in terms of f ’s ability to distinguish the distributions µ and ν. What can be said about µ and ν? We claim that on every set S ⊂ [N ] of size at most N/k − 1 = n − 1, we must have µ S = νS . (8.2.1) Here the notation µS denotes the marginal distribution of µ on the coordinates in S. To see (8.2.1), let T ⊂ [n] be the set of players whose outputs overlap S. 97 Then by the no-signaling property of N , the marginal µS (resp. νS ) is entirely determined by the marginal input distribution on T ; that is, (Ueven )T (resp. (Uodd )T ). And for any T a strict subset of [n], (Ueven )T = (Uodd )T = U({0, 1}|T | ), so we must have µS = νS . So all small marginals of µ and ν are information-theoretically indistinguishable. This is exactly k-wise indistinguishability, a generalization of k-wise independence introduced by Bogdanov et al. [BIVW16] and first used in the context of secret sharing. Definition (k-wise indistinguishability [BIVW16]). Two distributions µ and ν on {±1}N are k-wise indistinguishable if for all S ⊂ [N ] with |S| ≤ k, µS = νS . Additionally, for f : {0, 1}n → {0, 1}, we say f is ε-fooled by k-wise indistinguishability if for any pair µ, ν of k-wise indistinguishable distributions, | Pr[f (µ) = 1] − Pr[f (ν) = 1]| ≤ ε . It turns out k-wise indistinguishability over the hypercube is intimately connected to approximate degree. By a linear programming duality argument, Bogdanov et al. proved the following. Theorem 49 ([BIVW16, Theorem 1.2]). Let f : {0, 1}n → {0, 1} and ε > 0. g [f ] ≤ k . Then f is ε-fooled by k-wise indistinguishability if and only if deg ε/2 With this fact, Lemma 48, and the above discussion, we are ready prove the main theorem in this section. We say a class of Boolean functions F = (Fn )n≥1 is closed under inversepolynomial restrictions if for all f ∈ Fn and all S ⊆ [n] with n ∈ poly(|S|), fixing the bits in S c yields a function still in F: f S c ←x ∈ F|S| c ∀x ∈ {0, 1}|S | . Note that AC0 is closed under inverse-polynomial restrictions. Theorem 50. Suppose F is a class of Boolean functions closed under negations and inverse-polynomial restrictions. Let m be fixed and suppose there is an f ∈ F on N = poly(m) variables and an m-input QNC0 circuit C of depth d, with N − m ancilla qubits, and receiving arbitrary quantum advice, such that f ◦ C achieves correlation ε with Parm . Then there is a g ∈ F on n ≥ m/2 g [g] ≥ n/2d − 1. variables with deg ε/2 98 Proof. By Lemma 48, there is an m0 ≥ m/(2d + 1) and an (m0 , 2d , N − 2d m0 ) nonlocal channel N such that f ◦ N or ¬f ◦ N achieves correlation ε with Parm0 . Suppose the referee measures their system and obtains outcome string r. This event leads to an updated state shared among the parties in N and thereby defines an (m0 , 2d , 0) nonlocal channel N R←r . By a similar averaging argument to the one used in Lemma 48, there is at least one outcome r of the referee register such that N R←r still yields correlation ε with Par. Define g := f R←r or g := ¬f R←r as appropriate and put E := N R←r . Then g ∈ F is a function on n := 2d m0 bits and E [(g ◦ E)(x) · Par(x)] ≥ ε . x Therefore, by the discussion above, we see g can ε-distinguish (n/2d − 1)-wise indistinguishable distributions. Applying Theorem 49 we conclude that g [g] ≥ deg ε/2 n − 1. 2d Corollary 51. Suppose function class F is closed under inverse-polynomialg sized restrictions. Then if deg[F] ∈ o(n), F ◦ QNC0 cannot achieve Ω(1) correlation with Par. 0 g The burning question, then, is whether deg[AC ] ∈ o(n). In fact, the 0 approximate degree of AC is a longstanding open problem and its resolution would lead to several consequences in complexity theory [BT22]. To get a sense of the difficulty of this question, consider that on one hand, a sublinear upper bound is known for a large subclass of AC0 . Theorem ([BKT19, Theorem 5]). Let p(n) ∈ poly(n). Then the class of AC0 circuits of linear size, denoted by LC0 , has 0 g deg 1/p(n) [LC ] ∈ o(n). Yet on the other hand, a series of works, most recently [She22], show the following: g ] ∈ Ω(n1−δ ). Theorem. For any δ > 0, there is a function f ∈ AC0 with deg[f The lower bound of Ω(n1−δ )-for-any-δ is tantalizingly close to the trivial upper bound of n for the approximate degree of any Boolean function, but 99 0 g as it stands it is not unreasonable to guess that deg[AC ] ∈ Θ(n/ log n) either. Several questions—including now Conjecture 3—could be settled if the gap 0 g between Ω(n1−δ )-for-any-δ and n for deg[AC ] were closed. We may combine the sublinear lower bound on LC0 from [BKT19] with Theorem 50 to obtain: Corollary 52. Let C be an n-input, m-ancilla QNC0 circuit with arbitrary advice. Suppose f : {0, 1}n+m → {−1, 1} is defined by an AC0 circuit of size O(n). Then f ◦ C achieves negligible correlation with Parn . Blockwise approximate degree We conclude this section by laying out a self-contained question concerning the approximate degree of AC0 with respect to a modified, “blockwise” notion of approximate degree. This question is sufficient to imply Conjecture 3 in full 0 g generality and may be easier to resolve than deg[AC ]. Fix k ≥ 1 (assuming k divides n for simplicity) and let P be the partition of [n] into “blocks” of size k: n o P := {1, . . . , k}, {k + 1, . . . , 2k}, . . . , {n − k + 1, n} . For a monomial χS = i∈S xi define the (k-)block degree bdegk [χS ] to be the number of distinct blocks B ∈ P having nonempty intersection with S. This definition extends naturally to the k-block degree bdegk [f ] of a Boolean ] k [f ] function f : {0, 1}n → {−1, 1} and to the approximate k-block degree bdeg of f : Q ] k [f ] = min{bdegk [g] | g : {0, 1}n → R a polynomial with kf −gk∞ ≤ 1/3} . bdeg ] k [f ] ≤ n/k for any function. Of course bdeg Question 2. For all constants k, does the following hold? ? ] k [AC0 ] ≤ n/k − 1 . bdeg As we explain below, this would be enough to prove Conjecture 3. Note the following, which are immediate and hold for all f : g ]< deg[f n g ] < n−k. ] k [f ] < n =⇒ deg[f =⇒ bdeg k k Moreover, these implications are sharp in that each one cannot generically imply anything stronger, as witnessed by a parity function on an appropriate subset 100 of [n]. Regarding f ∈ AC0 , the left-hand side holding for arbitrary constant k 0 g is equivalent to deg[AC ] ∈ o(n), while the far right-hand side follows directly from LMN-type Fourier tail bounds for AC0 . Proposition 6. If the resolution to Question 2 is “yes”, then Conjecture 3 is true. Proof sketch. Consider the referee-free nonlocal channel E from the proof of Theorem 50, with n/k players responding with k bits each. Defining µ and ν as the pushforwards of uniform distributions over even and odd bitstrings as before, it is true that µ and ν are (n/k − 1)-wise indistinguishable when viewed as distributions on {0, 1}n . However, they may also be viewed as distributions on the hypergrid [2k ]m for m = n/k. With this perspective, µ and ν are m − 1 indistinguishable. Repeating the proof of [BIVW16, Theorem 1.2] over this larger alphabet, we recover exactly the notion of blockwise degree. The rest of the argument is as before. ? 0 g It is unclear to us whether Question 2 is easier than deg[AC ] ∈ o(n). Because AC0 is closed under permutations of input coordinates [n], we can compare the two questions head-to-head as follows. Let Pk be all the relabelings of P : Pk := n {π(1), . . . , π(k)}, {π(k+1), . . . , π(2k)}, . . . , {π(n−k+1), . . . , π(n)} o . π∈Sn For any P ∈ Pk , let bdegP [f ] be the maximum number of blocks in P overlapped by some monomial in f . Then we have the following characterization, where g ranges over real-valued multilinear polynomials on the hypercube as usual: 0 g deg[AC ] < n/k ⇐⇒ ∀f ∈ AC0 , ∃g, ∀P ∈ Pk , bdegP [g] ≤ n/k and kf − gk∞ ≤ 1/3 ] k [AC0 ] < n/k ⇐⇒ ∀f ∈ AC0 , ∀P ∈ Pk , ∃g, bdegP [g] ≤ n/k and kf − gk∞ ≤ 1/3 . bdeg 8.3 Towards a switching lemma for AC0 ◦ QNC0 Recall that our approach in Section 8.1 fails because circuits with LMN-style Fourier decay are not suitably closed under precomposition by QNC0 . In fact this is true even under precomposition by NC0 , and the proof of the LMN theorem elegantly avoids an induction assumption phrased in terms of Fourier decay. Instead, the proof relies on a structural theorem about the effect of random restrictions on DNFs and CNFs—Håstad’s celebrated switching lemma: 101 Theorem (Håstad [Hås86]). Suppose f is a width-w DNF. Then for any 0 ≤ δ ≤ 1, Pr [DTdepth (f ρ ) > t] ≤ (Cδw)t , ρ∼Rδ where C is a universal constant. Here Rδ is the distribution of random restrictions with star probability δ (see e.g., [ODo14, §4.3] for more). This theorem has received several proofs over time, but each rely on the well-controlled structure of random restrictions. To naively repeat the switching lemma argument directly on AC0 ◦ QNC0 would mean to track the passage of random restrictions through QNC0 —a tall order given that QNC0 can destroy the independence and unbiasedness of random restrictions that switching arguments tend to rely on. The situation may be slightly improved by instead considering a switching lemma for the model studied in Section 8.2. Recalling that f ◦N is a randomized function, we may hope for a switching lemma of the following form: An imagined switching lemma for nonlocal channels. Let m ≥ 0 and k, w, n ≥ 1 and suppose f : {0, 1}kn+m → {0, 1} is a DNF of width w and N is an (n, k, m) nonlocal channel. Then for each restriction ρ there exists a distribution Γρ over decision trees such that (f ◦ N )ρ = {T }T ∼Γρ and Pr Pr [depth(T ) > t] ≤ (Cδw)t . ρ∼Rδ T ∼Γρ By Lemma 48 such a switching lemma would be sufficient to show correlations bounds between f ◦ QNC0 and parity for any DNF (or CNF) f , which in turn are direct prerequisites to proving Conjecture 3. While this imagined switching lemma is currently out of reach, we contend it presents a useful challenge to existing switching lemma proof techniques. As a first step in this direction, we devote this section to a proof of a simpler but related structural result. Theorem. (Informal) Any no-signaling channel N composed with a decision tree τ is equal to a probability distribution Γ of decision trees with depth(τ 0 ) ≤ depth(τ ) for all τ 0 ∈ Supp(Γ). Let us fix some notation. For a finite set X let M(X) denote the set of probability measures on X. The set M(X) is convex, so for ν a probability measure on M(X) we may define the expected distribution   E [µ] := x w.p. Pr Pr [z = x] µ∼ν µ∼ν z∼µ . x∈X (8.3.1) 102 Here we study Boolean channels, or functions of the form N : {±1}n → M({±1}N ). For a probability measure µ on the set of channels from n to N bits, we use EN ∼µ N to denote the channel defined pointwise as   E N (x) := E [N (x)]. N ∼µ (8.3.2) N ∼µ To be clear, N (x) is a probability measure on {±1}N , so in the right-hand side of (8.3.2) we are computing the expected distribution according to (8.3.1). Also, for T ⊆ [N ] define the reduced channel  T N (x) := y w.p.  X Pr[N (x) = z] . y∈{±1}|T | z∈{±1}N zT =y Definition (No-signaling channel). Consider a map N : {±1}n → M({±1}N ) and a ‘backwards lightcone’ function B : [N ] → [n] ∪ {⊥}. The pair (N , B) is a no-signaling channel (NSC) if for all S ⊆ [n], for all x, x0 ∈ {±1}n with −1 −1 xS = x0S , we have N B (S∪⊥) (x) = N B (S∪⊥) (x0 ). That is, a channel is an NSC if for any collection of output indices T , N T (x) −1 is a function of xB(T )\{⊥} only. Note also N B (⊥) is oblivious to the value of x entirely—the outputs B −1 (⊥) could be called the referee outputs. Recall that for a Boolean function f : {±1}N → {±1}, f ◦ N denotes the channel n o f ◦ N (x) = b w.p. Pr [f (y) = b] . b∈{±1} y∼N (x) The restriction structure on NSCs interacts nicely with decision trees: Theorem 53. Given f : {±1}N → {±1} and N : {±1}n → M({±1}N ) an NSC, there exists a distribution Γ over decision trees such that i. For all x the composition f ◦ N (x) = {τ (x)}τ ∼Γ , so E[f ◦ N (x)] = Eτ ∼Γ [τ (x)]; and ii. For all τ ∈ Supp(Γ), DTdepth (τ ) ≤ DTdepth (f ). Recall that f ◦ N is an M({±1})-valued function on the hypercube, so x→ 7 E[f ◦ N (x)] is a [−1, 1]-valued function on the hypercube, and accordingly has a multilinear Fourier expansion E[f ◦ N ] = X S⊂[n] aS χS with h aS := E E[(f ◦ N )(x)] · χS (x)] . x 103 We pause to note the related fact that in terms of the expected output E[f ◦ N ], the degree of any function f does not increase under composition with an NSC: deg(f ) ≥ deg(E[f ◦ N ]). This claim has a very simple direct proof2 and we emphasize that it is not equivalent to Theorem 53. For example, there are Boolean functions g with deg(g) = n2/3 but DTdepth (g) = n (see Example 3 in [BW02]). One could imagine a Boolean function h with deg(h) ≈ DTdepth (h) ∈ o(n) but where E[h ◦ N ] is “g-like”: any decision tree decomposition of E[h ◦ N ] contains a tree of depth n despite having deg(E[h ◦ N ]) ∈ o(n). Theorem 53 says such an h, N pair does not exist; precomposition by an NSC cannot increase the decision tree complexity of a function. The proof of Theorem 53 requiries some bookkeeping. The idea is to begin with τ ’s root vertex variable yi and locally decompose the univariate channel N {i} (x) 7 yi into a distribution of deterministic functions {yi,ω (xi )}ω∼µ . This decomposition of the root vertex induces a probabilistic decomposition {τω0 ◦ Nω0 }ω∼µ of the entire hybrid computation where the root variable yi in τ 0 has been replaced with an xB(i) and the left and right subtrees of τ become compositions not with N , but with conditional versions of N where xB(i) and yi have been fixed to certain values. This conditioning preserves the NSC-ness of the new N 0 s, and the decomposition recurses down the tree. We now introduce a notion of conditioning. For any n-to-N bit Boolean channel N , x ∈ {±1}n , J ⊆ [N ] and Y ∈ Supp(N J (x)) define the conditional channel as n o N (x | yJ = Y ) := y w.p. Pr[N (x) = y | yJ = Y ] y∈{±1}N , and for T ⊆ [N ] the reduced conditional channel  N T (x | yJ = Y ) := y w.p. X Pr[N (x) = z | zJ = Y ] z∈{±1}N  . y∈{±1}|T | zT =y Note that T -reduced conditional no-signaling channels can depend on inputs outside B(T ). Consider for example the n-to-n-bit NSC G(x) =    U{even strings} x even   U{odd strings} x odd. P 2 b χ Consider the Fourier expansion f = Then E[(f ◦ N )(x)] = S⊆[N ] f (S) S . P  P P S b χ b b χ χ E S⊆[N ] f (S) S ◦ N (x) = S f (S) E[ S ◦ N (x)] = S f (S) E[ S ◦ N (x)], a linear combination of functions of at most |S| variables each for |S| ≤ deg(f ). 104 Now G {i} (x) is identically a Rademacher random variable (oblivious to x entirely), but G {i} (x | y[n]\i = 00 · · · 0) = Y  xj w.p. 1 , j the parity of all n bits of x. All the same, some structure remains after conditioning: Proposition 7. For T, J ⊆ [N ], let x, x0 ∈ {±1}n be such that xB(J∪T ) = x0B(J∪T ) . Then for all Y ∈ Supp(N J (x)), N T (x | yJ = Y ) = N T (x0 | yJ = Y ). Proof. Let x, x0 be as in the proposition statement. We have from the definition of NSCs that N J∪T (x) = N J∪T (x0 ). Certainly then N J∪T (x | yJ = Y ) = N J∪T (x0 | yJ = Y ) (we have taken the marginal of two equal distributions). The conclusion then follows from noticing that for any U ⊆ V , N U = (N V )U . This proposition says N T (x | yJ = Y ) is a function of xB(J∪T ) only. Thus if we fix variables xB(J) we recover a smaller NSC: Corollary 54. Consider an n-to-N NSC (N , B), an i ∈ [N ], and X, Y ∈ {±1}. If B(i) =⊥ let N 0 be the n-to-(N − 1) NSC N 0 = N [N ]\{i} (x | yi = Y ) and otherwise let N 0 be the (n − 1)-to-(N − 1) NSC N 0 = N [N ]\{i} (x{B(i)}c | xB(i) = X, yi = Y ). Define a new lightcone function B 0 from B as follows. Put B(j) =⊥ for all j ∈ B −1 (B(i)) and then remove i from the domain of B. Then (N 0 , B 0 ) is an NSC. Finally we introduce an object used internally in the proof of Theorem 53. Definition (Hybrid Decision Tree). A hybrid decision tree T on n variables with ` leaves consists of the data (τ, G1 , . . . , G` ), where i. The first argument τ is a rooted binary tree with ` leaves labeled as follows. Each internal node is assigned xi for some i ∈ [n], the edge to its left child is labeled 1, and the edge to its right child is labeled −1. 105 ii. Each leaf ι of τ is associated with an n-to-1 channel Gι : {±1}n → M{±1}. A hybrid tree defines a channel Tτ (G1 , . . . , G` ) : {±1}n → M{±1} as follows. Computation on input x ∈ {±1}n proceeds just as with standard decision trees until a leaf ι is reached, at which point the distribution Gι (x) is returned. Theorem 53 follows from these three claims. Proofs of the first two are immediate from the definitions. Claim 4. For any hybrid decision tree T ,   h i T G1 , . . . , Gι−1 , E [Gω ], Gι+1 , . . . , G` = E T (G1 , . . . , Gι−1 , Gω , Gι+1 , . . . , G` ) ω∼µ ω∼µ Claim 5. For any hybrid decision trees Tτ (G1 , . . . , G` ) and Tτ 0 (Gι1 , . . . , Gι`0 ),  Tτ G1 , . . . , Gι−1 , Tτ 0 (Gι1 , . . . , Gι`0 ), Gι+1 , . . . , G`  = Tτ ◦ι τ 0 (G1 , . . . , Gι−1 , Gι1 , . . . , Gι`0 , Gι+1 , . . . , G` ), where τ ◦ι τ 0 is τ with the ιth leaf replaced with τ 0 . Claim 6. Suppose τ is a decision tree and (N , B) is an NSC. Then either: i. τ ◦ N = Eω∼µ [τω ◦ Nω ] where depth(τω ) ≤ depth(τ ) − 1, |Supp(µ)| ≤ 2, and each Nω is an NSC, or   ii. τ ◦ N = Eω∼µ Tτ ∗ τωL ◦ NωL , τωR ◦ NωR  , where |Supp(µ)| ≤ 3, τ ∗ has one internal node, depth(τωL ), depth(τωR ) ≤ depth(τ ) − 1, and each NωL , NωR is an NSC; or iii. (Base case) τ ◦ N (x) = {b w.p. 1} for all x, for some fixed b ∈ {±1}. Proof. If τ is the trivial decision tree with no internal nodes, clearly we satisfy case iii. Otherwise, let yi be the variable at the root of τ . There are two cases depending on the value of B(i). Case i), B(i) = ⊥. Observe that N {i} (x) is the same distribution µ over {±1}, independent of x. For ω ∈ {±1} let τω be the subtree of τ attached to the ω-valued edge of yi . Put Nω = N T \{i} (x | yi = ω). Then we have for 106 z ∈ {±1}, i i ω∈{±1} Pr[τ ◦ N (x) = z | D (x) = ω] Pr[N (x) = ω] Pr[τ ◦ N (x) = z] = P = P = P i ω∈{±1} Pr[τω ◦ N (x | yi = ω) = z] Pr[N (x) = ω] i ω∈{±1} Pr[τω ◦ Nω (x) = z] Pr[N (x) = ω] = Pr h E [τω ◦ Nω ](x) = z i ω∼µ as desired. Clearly τω is strictly shorter than τ , and Nω is an NSC by Corollary 54. Case ii), B(i) 6= ⊥. Let τ ∗ be the one-vertex tree consisting of the root vertex of τ relabeled with xB(i) and let τ1 , τ−1 be the left and right subtrees of τ respectively. Observe that N {i} (x) = N {i} (xB(i) ) is a univariate channel. Hence it can be decomposed as a convex combination  N {i}        1 1 0 0 1 0 0 1 (xB(i) ) = a(1,1)  + a(−1,−1)  + a(1,−1)  + a(−1,1)  , 0 0 1 1 0 1 1 0 where only three of a(L,R) are nonzero. Let µ = {(L, R) w.p. a(L,R) }. Then we claim h  i (1) (−1) τ ◦N = E Tτ ∗ τL ◦ NL , τR ◦ NR , (8.3.3) (L,R)∼µ 2 where for b, c ∈ {±1} , Nc(b) (x) = N (x|xB(i) = b, yi = c). We check Eq. (8.3.3) pointwise. First consider an x with xB(i) = 1. We condition on the value of yi , rearrange, and then “complete the tree”: Pr[τ ◦ N (x) = z] = X Pr[τ ◦ N (x) = z | Ni (x) = L] Pr[Ni (x) = L] L∈{±1} = X Pr[τ ◦ N (x | yi = L) = z](a(L,1) + a(L,−1) ) L∈{±1} = X Pr[τL ◦ N (x | xB(i) = 1, yi = L) = z] P R∈{±1} a(L,R) L∈{±1} (1) X = a(L,R) Pr[τL ◦ NL (x) = z] L,R∈{±1} (1) X = (−1) a(L,R) Pr[Tτ ∗ (τL ◦ NL , τR ◦ NR )(x) = z] L,R∈{±1}  = Pr E (L,R)∼µ (1) (−1) [Tτ ∗ (τL ◦ NL , τR ◦ NR )](x) = z  , as desired. A similar argument goes through for xB(i) = −1 by expanding over R instead of L.  107 Proof of Theorem 53. Let τ be a depth-optimal decision tree for f . Construct the trivial hybrid tree T with no internal nodes and a single leaf with label τ ◦ N . Put Γ = {T w.p. 1}. We will recursively break apart leaves of T into distributions of hybrid trees, which are then combined with the parent tree to become distributions over hybrid trees of greater depth. This is done by repeated application of the following sequence of steps. Suppose Tτ (G1 , . . . , G` ) is some hybrid tree and Gι = τ 0 ◦ N for some nontrivial DT τ 0 and (potentially conditioned) NSC N . Then depending on the case in Claim 3 we either have Tτ (. . . , τ| ◦{zN}, . . .) = Tτ (. . . , E(L,R)∼µ [Tτ ∗ (τωL ◦ NωL , τωR ◦ NωR )], . . .) index ι (Claim 6.i) = E h (ωL ,ωR )∼µ i Tτ (. . . , Tτ ∗ (τωL ◦ NωL , τωR ◦ NωR ), . . .) (Claim 4) = E (ωL ,ωR )∼µ h i Tτ ◦ι τ ∗ (. . . , τωL ◦ NωL , τωR ◦ NωR , . . .) , (Claim 5) where τ ∗ has depth 1 and depth(τωL ), depth(τωR ) ≤ depth(τ 0 ) − 1, or we have Tτ (. . . , τ| ◦{zN}, . . .) = Tτ (. . . , Eω∼µ [τω ◦ Nω ], . . .) (Claim 6.ii) index ι h i = ω∼µ E Tτ (. . . , τω ◦ Nω , . . .) , (Claim 4) where depth(τω ) ≤ depth(τ 0 ) − 1. If we repeatedly make these transformations on the elements of Γ, we will eventually be left with a distribution over hybrid decision trees (τ, G, . . .) where each channel G = τ 0 ◦ N is in the base case of Claim 6. Such a hybrid tree is equal to a deterministic channel. Hence we are left with a distribution over deterministic channels that is trivially equivalent to a distribution of standard, deterministic decision trees. Further, it’s easy to see that once done, the longest path in any tree of Supp(Γ) is bounded by the longest path in the original tree τ . 8.4 Discussion We have seen several pieces of evidence for Conjecture 3, as well as highlighted new connections between quantum complexity theory, nonlocal games, and approximate degree. 108 If Conjecture 1 is ultimately proved true, we may wish to reach for a stronger no-advantage theorem closer to that of Beals et al. [Bea+01] from query complexity. A natural expression of AC0 ◦ QNC0 non-advantage might use the language of Fourier decay. Question 3. Does AC0 ◦ QNC0 exhibit LMN-like Fourier decay? To make this precise for the randomized function f ◦ C, consider the expectation over the randomness in C to get a function F : {0, 1}n → [−1, 1]. Then we ask, is W≥t [F ] ∈ O(exp(−t))? As mentioned in the introduction, a similar result is known depth-d QAC0 circuits with at most O(n1/d ) ancillas [NPVY23]. Finally, one may consider any number of variations on the theme of precomposing a Boolean function with QNC0 . It is natural to ask: Question 4. View a QNC0 circuit C as a map from (randomized) Boolean functions to randomized Boolean functions: C f 7−→ f ◦ C . By how much can this map increase influence, sensitivity, or other complexity measures of f ? Theorem 53 gives the answer “not at all” to a variant Question 4 where QNC0 is replaced by nonlocal channels, and the complexity measure is randomized decision tree complexity. 8.5 Acknowledgements We are grateful to Chris Umans and Thomas Vidick for numerous valuable discussions and for the opportunity to share the contents of this chapter with Henry Yuen and the quantum group at Columbia University in the fall of 2022. We are also grateful to Atul Singh Arora, discussions with whom inspired this project. Finally we thank the anonymous ITCS 2024 reviewers for their generous and meticulous feedback on an earlier draft. 109 Chapter 9 Testing classical properties from quantum data In property testing we consider a subset P of the set of all Boolean functions f : {0, 1}n → {0, 1} and aim to find fast algorithms for deciding (with high probability) whether an unknown function f has property P or is ε-far from having property P; that is, we wish to decide between Case (i) f ∈ P or Case (ii) min kf − gk1 ≥ ε , g∈P promised one of these is the case. Here kf − gk1 = Prx∼{0,1}n [f (x) 6= g(x)] is the L1 distance. Property testing began in the context of program checking [BLR90; RS96], where it was shown that only O(1) queries to f are needed to determine (with high probability) whether f is linear or is Θ(1)-far from linear—which compares very favorably to the Ω(n) query lower bound for learning linear functions. The extreme query efficiency of property testing algorithms soon after played a critical role in interactive proofs and PCP theorems [AS98; Aro+98; Din07]. Since then property testing has developed into a rich landscape of access models, complexity regimes, and separations [Fis04; Rub07; Ron09; Sud10; Gol17]. One of the promises of this broad view of property testing, identified very early on [GGR98], is its potential in data analysis and machine learning: one could run inexpensive property testing algorithms to guide the choice of which long-running learning algorithm to use. But there is an unfortunate catch: the dramatic complexity advantage of testing over learning typically disappears in the natural access model for data analysis and machine learning, where fresh queries to f cannot be made and only a limited dataset {(xj , f (xj )}j of random samples from f is available. This setting is known as passive or sample-based testing [GGR98]. Indeed, many results in passive testing are lower bounds that grow with n, unlike the algorithms available in query-based testing: compare among the “Classical” columns in Table 9.1. In fact, Blais and Yoshida [BY19] showed that if a Boolean property can be tested from O(1) random samples, then the property is of a rather restricted kind.1 1 In particular, such a property is only a function of the conditional expected values 110 Quantum Property k-Juntas Linearity F2 degree-d Monotonicity Symmetry Triangle-freeness Queries √ e O( k) [ABRW15] O(1) Classical Examples O(k) Samples k/2 Ω(2 + k log n) [AS07] Θ(1) [AHW16] n + Θ(1) [BV97] O(2d ) O(nd−1 ) [AHW16] Θ(nd ) [ABDY23] e 1/4 ) O(n [BB15] O(1) Õ(n2 ) [AHW16] [Bla24] Θ(n1/4 ) O(1) O(1) [Theorem 66] [AHW16] Ω(n) via [AHW16] Learning (from queries) Ω(2k + k log n) [AHW16] [Bla09; CG04] Θ(1) n + Θ(1) [BLR90] [Alo+03; Bha+10] e √n) O( 2Ω( n) [Theorem 60] e Θ(k) Θ(2d ) √ [Theorem 63] O(1) Queries [AHW16] Θ(nd ) [AHW16] √ Ω( n) 2 [BBL98] [KMS18] O(1) Θ(n1/2 ) [BWY15] O(1) [BWY15] Table 9.1: Upper and lower bounds for testing and learning in various access models. All bounds are given for (a sufficiently small) constant ε > 0. Bounds that are given without a reference follow trivially from other bounds in the table. Remark 1. This is not to say that the classical passive testing model is uninteresting; there are many exciting positive results for the model, falling under the umbrella of sublinear algorithms. For example, the line of work [FLV15; GR16; DGL23] showed that the existence of certain constant-query testers implies sample-based algorithms with sublinear dependence on n. But passive testers still cannot compete with query-based testing for many important problems, as the lower bounds in Table 9.1 attest. How could we recover large testing speedups in the context of passive testing from data? In the present chapter we advocate for quantum computing (and “quantum datasets”) as an answer. Viewed from the right perspective, early results in quantum complexity theory actually demonstrate that quantum data—in the form quantum examples, or copies of the function state |f i := P 2−n/2 x |x, f (x)i—can sometimes suffice for highly efficient property testing. For example, the Bernstein-Vazirani algorithm, usually understood as an O(1) Ex [f (x)|x ∈ Sj ] of f for sets Sj forming a constant-cardinality partition of the hypercube, S1 t · · · t SO(1) = {0, 1}n . [AHW16] – 111 quantum query algorithm, really only needs O(1) function states to test for linearity [BV97] (vs. Ω(n) classical samples), and the quantum k-junta tester of Atıcı and Servedio [AS07] also requires only O(k) quantum examples (c.f. the lower bound of Ω(2k/2 + k log n) classical samples). The present chapter seeks to establish passive quantum testing as a fundamental model of property testing by making progress on the question: What is the extent of quantum advantage in testing classical properties from data? Before the contributions of this chapter it was not fully clear whether quantum data in the form of quantum examples can lead to testing speedups beyond linearity and k-junta-like properties (such as low Fourier degree): both the Bernstein–Vazirani algorithm and the Atıcı–Servedio junta tester rely only on quantum Fourier sampling [BV97], a quantum subroutine which, given copies of |f i, returns the label S ⊆ [n] of a Fourier character with probability fb(S)2 . Despite the success of quantum Fourier sampling, its utility is restricted to properties that are “plainly legible” from the Fourier spectrum.2 In this chapter we expand the list of properties with efficient passive quantum testers, including one which provably requires a non-Fourier sampling approach. We also compare the power of quantum data to that of classical queries, finding that they are (essentially) maximally incomparable as resources for testing. Finally, we begin a study of lower bounds for testing monotonicity from quantum data by showing that the ensembles leading to exponential lower bounds for classical sample-based testing yield no nontrivial lower bounds for quantum data-based testing. In the remainder of the introduction we explore each of these points in greater detail. Remark 2 (Where might quantum data appear?). While from the perspective of complexity theory quantum data leads to a natural counterpart to classical passive testing, and demonstrates a “data-based” quantum advantage, the reader may still feel it is not entirely natural from a practical or “physical” standpoint. To the contrary, we contend that quantum data may be a useful 2 As an example of a property not detectable from the Fourier spectrum, consider the task of testing if f is a quadratic F2 polynomial. It is well-known (see, e.g., [HHL+19, Claim 2.4]) that degree-2 F2 polynomials can have Fourier coefficients with uniformly exponentially-small magnitudes, so Fourier sampling is not directly useful for this task. Our Theorem 55 below serves as another example. 112 component of emerging quantum technologies. We briefly list some scenarios where quantum data may be a natural object. • Suppose a researcher has time-limited query access to a data-generating process, but does not yet know what questions about the process she will eventually ask. She may prefer to store data in quantum memory rather than classical, to broaden the range of questions that can be answered post hoc. • In high-latency and bandwidth-limited scenarios, back-and-forth (adaptive, query-based) interaction is not feasible, for example in space exploration. If a space probe departing Earth shared some entanglement with a ground station, it could later in its journey encode observations into quantum data and teleport the resulting states back to Earth. In such a scenario, the advantages of quantum data could lead to significant speedups in research and analysis. • Rather than sharing the source code for a program f , a company may prefer to share a quantum data encoding of it as a form of copy protection— provided the function state is sufficient for the intended application. Quantum advantage in testing from data: an emerging theme Our first contribution is to expand the list of properties exhibiting quantum advantage in testing from data. Our algorithms work by finding new quantum ways to exploit insights from prior work in classical testing. See Section 9.1 for proofs. Symmetry testing. A Boolean function is symmetric if f (x) = f (y) when x is a permutation of y. We confirm that projecting |f i onto the symmetric subspace suffices for an O(1)-copy quantum test. For comparison, classical passive symmetry testing requires Ω(n1/4 ) samples [AHW16]. Monotonicity testing. A Boolean function f is monotone if f (x) ≤ f (y) when x ≺ y in the standard partial order ≺ on the hypercube. Monotonicity has been of central importance in the classical property testing literature [Gol+00; BB15; KMS18]. We give a quantum algorithm that tests monotonicity with Õ(n2 ) copies of the function state for f , in comparison to the lower bound of √ 2Ω( n) samples for classical passive testing [Bla24]. 113 The algorithm appeals to a characterization of monotonicity in terms of the Fourier spectrum of f . In particular, let ε be the L1 distance between a Boolean function f and the set of all monotone functions. Then we may relate ε to the Fourier spectrum of f via 2ε ≤ I[f ] − P b f ({i}) ≤ 4εn . i Here I[f ] is the total influence of f and is equal to the expected size of a subset S ⊆ [n] sampled according to the Fourier distribution of f . I[f ] can thus be easily estimated with Fourier sampling, and the Fourier coefficients fb({i}) estimated with classical samples. The bounds above follow from a reinterpretation of the “pair tester” characterization of monotonicity [Gol+00], which was not originally Fourier-based. Triangle-freeness. A Boolean function f is triangle-free if there are no x, y such that (x, y, x + y) form a triangle: f (x) = f (y) = f (x + y) = 1. We give a passive quantum triangle-freeness tester that uses only O(1) copies of |f i, in contrast with the Ω(n) samples required classically.3 It is known that to test triangle-freeness, it suffices to estimate the probability that (x, y, x + y) forms a triangle for uniformly random x, y [Fox11; HST16]. Our test estimates this probability by repeating the following subroutine. First, measuring copies of |f i in the computational basis allows us to find a uniformly random y ∈ f −1 (1). Then by measuring the output register of copies of |f i, we obtain copies of the entire 1-preimage state E f −1 (1) ∝ X |xi . x∈{0,1}n , f (x)=1 Applying the unitary transformation Uy |xi = |x + yi then allows us to transform copies of |f −1 (1)i into copies of |f −1 (1) + yi ∝ X |xi . x∈{0,1}n , f (x+y)=1 The overlap | hf −1 (1)|f −1 (1) + yi | = Prx∼{0,1}n [f (x) = f (x + y) = 1] can now be estimated with a SWAP test [BCWD01]. 3 This classical lower bound can be seen by an argument via linear independence similar to that used in the lower bound proof of [AHW16, Theorem 10]. 114 Fourier sampling does not suffice Given that Fourier sampling is sufficient to test linearity [BV97], k-juntas [AS07; ABRW15], and (as shown above) monotonicity, one might wonder whether Fourier sampling is “all that quantum data is good for” in the context of property testing Boolean functions. To the contrary, we exhibit a property for which a Fourier sampling-based approach requires super-polynomially more data than the optimal passive quantum tester. Theorem 55. There is a property P of Boolean functions on 2n bits such that: √ (i) There is no algorithm for testing P that uses 2o( n) classical samples and any number of Fourier samples. (ii) There is an efficient quantum algorithm for testing P from O(1) copies of |f i. Theorem 55 is proved in Section 9.2 as Theorems 68 and 69. The property P is the Maiorana-McFarland (MM) class of bent functions, which take the form f (x, y) = hx, yi + h(x) for h any n-bit Boolean function (see e.g., [CM16] for more). To prove (i), we show a special subset Fyes of MM functions with far-fromconstant h are indistinguishable from the set Fno of their “duals,” defined by replacing h(x) with h(y). Every function in both these sets is bent—i.e., all Fourier coefficients have equal magnitude—so Fourier samples cannot help. It thus suffices to lower bound the number of classical samples needed to solve the distinguishing problem. The set Fyes is chosen so that for a uniformly√ random hx, yi + h(x) from Fyes , the distribution of truth tables of h is 2c n -wise √ independent. This means that for any number of samples less than 2c n , except in the very unlikely event that there is a collision among the sampled points {(x(i) , y (i) )}i , the distribution of values f (x(i) , y (i) ) will look uniformly random, regardless of whether f is sampled uniformly from Fyes or Fno —and so distinguishing is impossible. The truth tables for h are constructed from certain affine shifts of Reed–Muller codewords. As for item (ii), the passive quantum tester for this property first applies the unitary U defined by |x, y, bi 7→ |x, y, b ⊕ hx, yii to |f i. If f is a MM function the result should be h, a function depending only on the first n variables, while if |f i is far from MM functions, it will have noticeable dependence on coordinates 115 n + 1, . . . , 2n. This dependence can be measured by Fourier-sampling the transformed state. Comparing access models Quantum data is always at least as good as classical samples, and from Table 9.1 we see that for a growing list of properties, testing from quantum data is competitive with testing from classical queries. In fact, quantum data can be vastly more powerful than classical queries for testing. An extremal example of this is the Forrelation problem, which can be tested from O(1) function state copies but requires Ω(2n/2 ) classical queries [AA15]. Conversely, one may wonder to what extent classical queries may outperform quantum data for property testing. An answer is not so obvious. Although classical queries enable direct access to f (x) at any point x of the algorithm’s choosing—a powerful advantage over quantum data—it is not so clear whether this can lead to a separation for property testing. Recall that for a property testing problem, yes and no instances must be Ω(1)-far in L1 distance. So to create a hard property for quantum data-based testers, one must find two sets of functions which pairwise differ on a constant fraction of the locations in their truth tables, yet still remain hard to distinguish by a quantum algorithm operating on copies of their function states. We succeed in “hiding” these large differences and identify a testing problem for which classical queries have a dramatic advantage over quantum data. Theorem 56. There exists a testing task (3-fold intersection detection) that can be accomplished with O(1) classical queries but requires Ω(2n/2 ) copies for quantum testing from data. Combined with the Forrelation separation of [AA15], Theorem 56 entails that quantum data and classical queries are (essentially) maximally incomparable. See Chapter 9 for a full picture of resource inequalities for testing. Theorem 56 is proved in Section 9.3 as Theorem 70. Given a function f : {0, 1, 2} × {0, 1}n → {0, 1} that indicates three subsets of the hypercube A, B, C ⊆ {0, 1}n , the 3-fold intersection detection task is to determine if the fractional 3-fold intersection |A ∩ B ∩ C|/2n is 0 or Ω(1)-far from 0. This property is readily tested from queries by computing the probability x ∈ A ∧ x ∈ B ∧ x ∈ C for uniformly-random x. To prove the quantum passive testing lower bound, we show the indistinguishability of two ensembles 116 Classical Queries ≶ ≥ ≤ Quantum Function States ≥ ≤ Quantum Queries Classical Samples Figure 9.1: Property testing resource inequalities. The figure illustrates the connections between four different data access models in property testing, namely classical/quantum example/query access. Here, “resource A ≥ resource B” means that access to resource B can be simulated from access to resource A without any overhead. (For example, a single classical query can be used to simulate a single classical sample.) As a consequence of Theorem 56 and [BV97; Sim97; AA15], the only two among these access models that are not trivially comparable are in fact very incomparable. of function states encoding set triples n o (A, B, C) iid A,B,C ∼ P{0,1}n vs. n (A, B, A∆B)} iid A,B ∼ P{0,1}n . Here ∆ denotes symmetric difference, P denotes the power set, and the samples are uniform. Note the first ensemble has mutual intersection of Ω(1) density with high probability, while the ensemble always has zero intersection. To obtain the lower bound, the main observation is that the t-copy versions of the two associated function state ensembles are equal when projected onto the so-called distinct subspace (i.e., the subspace spanned by basis states for which the t input registers are distinct). This projection moves the state ensembles at most O(t/2n/2 ) in trace distance, so we conclude that for any t = o(2n/2 ), the two ensembles cannot be distinguished using t function state copies. A challenge: lower bounds for quantum monotonicity testing We also begin the project of finding lower bounds for the passive quantum testing model. Our main contribution is to establish lower bounds for monotonicity as an important first open problem. In particular, we show the ensembles that entail strong lower bounds for classical passive testing are wholly inadequate for quantum passive testing. 117 Theorem 57. The ensembles in Goldreich et al. [Gol+00] and Black [Bla24] can be distinguished by a quantum algorithm with O(1/ε) copies of the corresponding function states. This theorem says that the best lower bound such ensembles could imply for quantum passive testing is Ω(1/ε). But that is no better than the lower bound that exists generically for every (non-trivial) property.4 To see that Ω(1/ε) holds generically, it suffices to consider only two functions, fyes and fno , that are exactly ε-far apart. This is equivalent to hfyes |fno i = 1 − ε, so the trace q √ ⊗t ⊗t distance between |fyes i and |fno i is 1 − (1 − ε)2t ≤ 2tε. Therefore, distinguishing between fyes and fno with success probability ≥ 2/3 requires t ≥ Ω(1/ε) copies of the respective function state. We prove Theorem 57 via a combinatorial analysis of the spectrum of the matrix A := E ψ ⊗t − E φ⊗t , ψ∼E0 φ∼E1 where E0 and E1 are the “yes” and “no” ensembles from [Gol+00] (or, later, from [Bla24]). As neither of our ensembles is close to Haar-random, we cannot directly draw on the rich recent literature on quantum pseudorandomness [JLS18; BS19; GB23; JMW24; MPSY24; Che+24a; SHH24; MH24]. Instead, we notice that our function state is unitarily equivalent to a phase state for a closely-related Boolean function. An intricate index rearrangement reveals A to be block-diagonal, with each block interpretable as the adjacency matrix for a complete bipartite graph. We then determine the spectrum of each block, with eigenvalues and their multiplicities given as functions of certain combinatorial quantities. Exponential generating function techniques lead to explicit formulas for these quantities, and finally the asymptotics can be understood by taking a probabilistic perspective on the counting formulas. Concentration arguments finish the proof and allow us to conclude that kAk1 ≥ Ω(1) (and thus the two ensembles are distinguishable) as soon as t = Ω(ε−1 ). This argument is presented in detail in Section 9.4. A final remark for this section: for certain regimes of ε, the Ω(1/ε) lower bound on the number of function state copies already separates passive quantum testing from quantum query-based testing. For example, (adaptive) quantum query complexity upper bounds of Õ(n1/4 /ε1/2 ) for monotonicity testing [BB15] 4 A classical query complexity lower bound of Ω(1/ε) also holds for testing any non-trivial property [Fis24]. 118 and of Õ((k/ε)1/2 ) for k-junta testing [ABRW15] are known. However, to the best of our knowledge, the “correct” ε-scaling for quantum property testing of classical functions is far from understood; prior works such as [AS07; BFNR08; CFMW10; AA15; ABRW15; MW16] seem to establish quantum query complexity lower bounds only for constant ε. Outlook and future directions Our results, (most of them) summarized in Table 9.2, highlight passive quantum property testing as a rich testing model deserving of concerted study. We grow the list of properties with efficient passive quantum testers, introduce new techniques for testing, show that the abilities of passive quantum testing extend beyond the reach of Fourier sampling, and highlight subtleties in comparing classical and quantum resources for property testing. Quantum function states Monotonicity testing Õ(n2 /ε2 ) Theorem 63 Symmetry testing Õ(1/ε2 ) Theorem 60 Triangle-freeness testing Õ    m6  l Tower C · log 1ε Theorem 66   Classical samples    √ exp Ω min{ n/ε, n} [Gol+00; Bla24] Θ(n1/4 ) [AHW16] Classical queries  n o √ Õ min n/ε, n/ε2 [Gol+00; KMS18] O(1/ε) [BWY15] Ω(n) via [AHW16] O Tower C · log 1ε via [Fox11; HST16]   l  m   O(1) Theorem 56   Θ̃ 2n/2 [AA15] 3-fold intersection estimation Ω 2n/2 Theorem 56 Ω 2n/2 via Theorem 56 Forrelation O(1) [AA15] Ω̃ 2n/2 [AA15]  Table 9.2: Our bounds in context. The table contrasts our results on property testing from quantum function states with results from the literature (in gray). Where the ε-dependence is not shown explicitly, we have set ε to some suitably small positive constant value. For monotonicity, symmetry, and triangle-freeness, passive quantum testing from function states is (at least) exponentially easier than passive classical testing from samples and at most polynomially harder than classical testing from queries. The testing problem derived from 3-fold intersection estimation is complementary to the Forrelation problem in that quantum function/phase states and classical queries swap roles in the exponential separation.  119 In fact, it seems passive quantum testing can make good on the promise of testing from data where classical passive testing cannot. With passive quantum testing, it is possible to generate a dataset about a Boolean function without foreknowledge of the property one would eventually like to test, and still be assured (for a growing list of properties) that testing will be very efficient. In particular, these results suggest that quantum data, rather than classical data, could enable the application to machine learning imagined in [GGR98]: as an inexpensive preprocessing procedure that informs the choice of suitable, more data-intensive learning algorithms. Here we lay out some directions for future work. More and improved bounds for passive quantum property testing. We have established upper bounds for passive quantum testing of monotonicity, symmetry, and triangle-freeness from function states. These three properties together with linearity testing [BV97] and junta testing [AS07; ABRW15] already demonstrate the power of quantum data for testing a variety of quite different properties, and it seems important to explore quantum datasets in the context of other testing problems. As highlighted in Table 9.1, quantum low-degree testing of Boolean functions is a natural next challenge, with the more general class of locally characterized affine-invariant properties [Bha+13] as a longer-term goal. One may aim to tighten our bounds to precisely pin down the power of quantum data for these testing tasks. Here, having established that the constructions from classical passive monotonicity testing lower bounds are inadequate for the quantum case, we consider it especially interesting to obtain a n-dependent lower bound for passive quantum monotonicity testing in the constant ε regime. Settling the n-dependence of the quantum sample complexity for passive quantum monotonicity testing is a tantalizing question for future work. Characterizing properties with constant-complexity passive quantum testers In the classical case, [BY19] gave a complete characterization of those properties that can be tested with a constant number of samples. Achieving an analogous characterization for properties that can be tested from constantlymany function state copies would help demarcate the boundary of quantum advantage for this model. 120 Intriguingly, the quantum case raises a further question about the constant complexity regime: For properties that admit a constant-copy passive quantum testers, can this always be achieved by algorithms that do not use entangled multi-copy measurements? The role of single- versus multi-copy quantum processing has recently been explored in the literature on learning and testing for quantum objects (see, e.g., [CCHL22; Hua+22; Car24; HH24]) and in quantum computational learning theory [ABDY23], but the picture is far from clear for properties of function states (and of pure states more generally). Concretely, while our testers for monotonicity and symmetry are single-copy algorithms, our triangle-freeness tester uses two-copy SWAP tests and there does not seem to be an immediate way of replacing this by single-copy quantum processing. One may also ask about the necessity of auxiliary quantum systems in quantum sample-based testers with constant sample complexity. (For example, our symmetry tester relied on auxiliary systems to implement the symmetric subspace projector.) The number of available auxiliary systems is already known to play an important role in, for instance, Pauli channel learning [CZSJ22; Che+24b; CG24], and exploring its relevance for constant-complexity passive quantum testing may shed new light on how these quantum testers achieve their better-than-classical performance. Other quantum datasets for classical properties We have considered only one kind of quantum representation of classical functions: coherent superpositions of evaluations of f (as function states). Already these are enough to gain major advantages over testing from classical data, but one could ask for more. Are there other, better quantum datasets that lead to even faster testers or extend quantum advantage to more properties? To keep this question interesting, one would require that the dataset be not too tailored to any property. In fact, this question may be best phrased as a sort of “compression game”: we are first given a very long list of questions that we might be asked regarding some black-box function f . We then have T (n) time to interact with an oracle for f , during which period we generate whatever data we would like. What is the best quantum dataset to generate, so that we are best prepared to answer a random (or perhaps worst-case) question from the list? 121 Passive quantum testing for quantum properties. Recently there has been a growing interest in property testing for quantum objects, such as states [HM13; OW15; HLM17; CHST17; BO20; GNW21; SW22; CCHL22; GIKL23; AD24; CGYZ24; ABD24; BDH24], unitaries [DGRT22; CNY23; SY23], channels [CCHL22; ACQ22; BY23], and Hamiltonians [LW22; BCO24; ADG24]. It is an interesting challenge to design datasets to enable passive versions of these tasks. Just as in the above, we would want quantum datasets that are mostly agnostic to the property to be tested. In fact, some existing work can be viewed as advocating for quantum datasets. When restricting ourselves to collecting classical data, classical shadows [HKP20; HCP23] serve as a useful representation, but place restrictions on the properties that can be tested after-the-fact. Shadow tomography procedures [Aar18; BO21; Car24] can remove such restrictions but use multicopy measurements that depend on the properties of interest, and thus in general seem to require quantum data storage to enable passivity.5 The relevance of data storage in a quantum memory for certain quantum process learning tasks has also been explored in [Bis+10; BDPS11; SBZ19; LKPP22; LK24]. In this context, the contents of this chapter can be viewed as investigating the power of quantum data, stored in quantum memory, for testing properties of diagonal unitary processes arising from classical Boolean functions. We hope that this will inspire future attempts at using quantum data as a resource for passively quantumly testing properties of more general quantum processes. Related work Passive classical property testing.6 Passive (or sample-based) property testing goes back to [GGR98] (see also [KR98]), who introduced it as a testing counterpart to Valiant’s model of probably approximately correct (PAC) learning [Val84]. In particular, [GGR98, Proposition 3.1.1] observes that PAC 5 (Non-adaptive) Pauli shadow tomography [HKP21; Car24; KGKB24; CGY24] in some sense interpolates between the (dis-)advantages of classical shadows and shadow tomography for the current discussion: When promised in advance that the properties in question are characterized by expectation values of arbitrary Pauli observables, some of the relevant data can be collected and stored classically in advance, without knowing which specific Pauli observables matter. However, part of the quantum processing still requires knowing the specific Paulis of interest, so to achieve passivity, it seems that some data still has to be stored quantumly. 6 Due to the vastness of the area of property testing, even when restricting the focus to passive testing, this paragraph is intended to provide context for this chapter rather than an exhaustive bibliography for the field. 122 learners give rise to passive testers (see also [Ron08, Proposition 2.1]). Later, [BBBY12] proposed active testing as a model interpolating between sampleand query-based testing. Both for passive and active testing, and for a variety of problems, several works have established lower bounds separating them from the more standard query-based testing model. Some notable examples of tasks with such separations include (k-)linearity [BBBY12; AHW16], k-juntaness [AHW16], (partial) symmetry [BWY15; AHW16], low-degreeness [AHW16], and monotonicity [Gol+00; Bla24]. We present these results and how they compare to quantum testing in Table 9.1. [BY19] gave a full characterization of properties of Boolean-valued functions that admit passive testing with a constant (i.e., independent of domain size) number of uniformly random samples, demonstrating that this is indeed a relatively restricted type of properties. While the works mentioned so far have focused on the case of uniformly random data points (or, in the case of active learning, uniformly random sets of admissible query points), more recently there has been renewed interest in passive distribution-free testing, see for instance [HK07; BFH21]. Finally, the framework of passive testing has also been explored for objects other than Boolean functions, especially for testing geometric properties [MORS10; BBBY12; KNOW14; Nee14; CFSS17; BMR19b; BMR19a]. Quantum property testing. The focus here is on quantumly testing properties of classical functions. This topic, considered for example in [AS07; CFMW10; HA11; AA15; ABRW15], is one of the main directions in quantum property testing, an area that goes back to [BFNR08] and is surveyed in [MW16]. However, quantum property testing also considers quantum algorithms that test properties of other classical objects from quantum data access. Notable examples of other objects to quantumly test include probability distributions [BHH11; CFMW10; GL20], graphs [ACL11; CFMW10], and groups [FSMS09; IL11]. Finally, recently there have also been significant insights in quantum property testing for quantum objects, notably states [HM13; OW15; HLM17; CHST17; BO20; GNW21; SW22; CCHL22; GIKL23; HH24; AD24; CGYZ24; ABD24; BDH24; MT24], unitaries [DGRT22; CNY23; SY23], channels [CCHL22; ACQ22; BY23], and Hamiltonians [LW22; BCO24; ADG24]. 123 9.1 Passive quantum testing upper bounds Defining passive quantum property testing As outlined in the introduction, passive property testing considers testing from (non-adaptively chosen) data that does not depend on the property to be tested. We propose a quantum version of this model by considering quantum testing algorithms that have access to copies of a quantum data state. Here, we consider the following form of quantum data encoding for a Boolean function f : {0, 1}n → {0, 1}: We work with function states X 1 |f i = |Ψf i = √ |x, f (x)i . 2n x∈{0,1}n (9.1.1) When the function f is clear from context, we will also use the notation |Ψi = |Ψf i. Natural variations of this notation, e.g., |Ψ0 i = |Ψf 0 i, will also be used. With this, we can now formally define the notion of passive quantum property testing for Boolean functions. Definition 4 (Passive quantum property testing). Let Pn ⊆ {0, 1}{0,1} be some property of Boolean functions on n bits, let δ, ε ∈ (0, 1). A quantum algorithm is a passive quantum tester with accuracy/distance parameter ε and confidence parameter δ for Pn from m = m(ε, δ) function state copies if the following holds: When given m copies of |Ψf i, the quantum algorithm correctly decides, with success probability ≥ 1 − δ, whether n (i) f ∈ Pn , or (ii) Prx∼{0,1}n [f (x) 6= g(x)] ≥ ε holds for all g ∈ Pn , promised that f satisfies either (i) or (ii). This chapter explores Definition 4 for different properties. Passive quantum symmetry testing A function f : {0, 1}n → {0, 1} is called symmetric if f ◦ π = f holds for all permutations π ∈ Sn . Here, π ∈ Sn acts on n-bit strings by permuting coordinates. That is, π(x1 . . . xn ) = xπ−1 (1) . . . xπ−1 (n) . This gives rise to the following classical testing problem. 124 Problem 58 (Classical symmetry testing). Given query access to an unknown function f : {0, 1}n → {0, 1} and an accuracy parameter ε ∈ (0, 1), decide with success probability ≥ 2/3 whether (i) f is symmetric, or (ii) f is ε-far from all symmetric functions, that is, we have Prx∼{0,1}n [f (x) 6= g(x)] ≥ ε for all symmetric functions g : {0, 1}n → {0, 1}, promised that f satisfies either (i) or (ii). Symmetry allows for a (trivial) reformulation in terms of (in general nonlocal) pairwise comparisons: Proposition 1. A function f : {0, 1}n → {0, 1} is symmetric if and only if for all x ∈ {0, 1}n and for all π ∈ Sn , the equality f (x) = f (π(x)) (9.1.2) holds. This characterization becomes important for testing because of the following result. Theorem 59 (Soundness of symmetry testing (compare [BWY15, Lemma 3.3])). If f : {0, 1}n → {0, 1} is exactly ε-far from all symmetric functions, then ε≤ Prn [f (x) 6= f (π(x))] ≤ 2ε . (9.1.3) x∼{0,1} ,π∼Sn Theorem 59 implies that we can classically test symmetry from query access simply by sampling a random permutation π and a random input x and then comparing the function values f (x) and f (π(x)). Here, ∼ 1/ε many queries suffice to achieve success probability ≥ 2/3 in symmetry testing. We now describe how to make use of Theorem 59 to build a passive quantum symmetry tester. Theorem 60 (Passive quantum symmetry testing). There is an efficient   quantum algorithm that uses O log(1/δ) many copies of the function state ε2 P 1 |Ψi = √2n x∈{0,1}n |x, f (x)i to decide, with success probability ≥ 1−δ, whether f is symmetric or ε-far from all symmetric functions. 125 Proof. For a permutation π ∈ Sn , write P (π) for the representation of that perP mutation on (C2 )⊗n given as P (π) = x∈{0,1}n |π(x)i hx|. Then, the orthogonal projector onto the symmetric subspace of (C2 )⊗n can be written as n Psym = 1 X P (π) . n! π∈Sn Notice that, if |Ψi is the function state for f : {0, 1}n → {0, 1}, then n hΨ| (Psym ⊗ 12 ) |Ψi = Pr x∼{0,1}n ,π∼Sn [f (x) 6= f (π(x))] . So, by a Chernoff-Hoeffding bound, we can, with success probability ≥ 1 − δ, obtain a (ε/3)-accurate estimate of the probability in Equation (9.1.3) by n independently performing the two-outcome projective measurement {Psym ⊗ ⊗(n+1) n 2 12 , 12 − Psym ⊗ 12 } on m = O(log(1/δ)/ε ) many single copies of |Ψi and then taking the empirical average of the observed outcomes (with outcome 1 ⊗(n+1) n n associated to Psym ⊗12 ). As the two-outcome measurement {Psym ⊗12 , 12 − n 2 Psym ⊗ 12 } can be implemented efficiently using O(n ) auxiliary qubits and O(n2 ) controlled-SWAP gates [Bar+97; LW22], our quantum symmetry tester is also computationally efficient. In contrast to the classical sample complexity of Θ(n1/4 ) for classical passive symmetry testing [AHW16], our passive quantum symmetry tester in Theorem 60 achieves an n-independent quantum sample complexity. Thus, we have an unbounded separation between classical and quantum for this passive testing task. Finally, let us comment on two extensions. Firstly, relying on the second inequality in Theorem 59, we can modify the proof of Theorem 60 to obtain   an efficient tolerant quantum passive symmetry tester that uses O (εlog(1/δ) 2 2 −ε1 ) copies of the unknown function state to decide whether f is ε1 -close to or ε2 -far from symmetric, assuming that ε2 > 2Cε1 holds with C > 1 some constant. Secondly, as Theorem 59 can be extended to so-called partial symmetric functions (compare again [BWY15, Lemma 3.3]), also our passive quantum symmetry tester can be modified to test for partial symmetry. Passive quantum monotonicity testing We define the natural partial order  on the Boolean hypercube {0, 1}n via x  y :⇔ (xi ≤ yi holds for all 1 ≤ i ≤ n). A function f : {0, 1}n → {0, 1} is called monotone if f (x) ≤ f (y) holds for all x, y ∈ {0, 1}n with x  y. The associated classical testing problem can be formulated as follows. 126 Problem 61 (Classical monotonicity testing). Given query access to an unknown function f : {0, 1}n → {0, 1} and an accuracy parameter ε ∈ (0, 1), decide with success probability ≥ 2/3 whether (i) f is monotone, or (ii) f is ε-far from all monotone functions, that is, we have Prx∼{0,1}n [f (x) 6= g(x)] ≥ ε for all monotone functions g : {0, 1}n → {0, 1}, promised that f satisfies either (i) or (ii). Here, as well as in our other property testing tasks below, will think of (i) as the accept case and of (ii) as the reject case. This then allows us to speak of completeness (for getting case (i) right) and soundness (for getting case (ii) right). Here, the chosen success probability of 2/3 is an arbitrary constant > 1/2, it can be boosted arbitrarily close to 1 through repetition and majority voting. As introduced above, monotonicity is a global property of a function. However, there is a straightforward equivalent local formulation: Proposition 2 (Local characterization of monotonicity). A function f : {0, 1}n → {0, 1} is monotone if and only if for all x ∈ {0, 1}n and for all i ∈ [n], the following holds: ¬((xi = 0 ∧ f (x) = 1 ∧ f (x + ei ) = 0) ∨ (xi = 1 ∧ f (x) = 0 ∧ f (x + ei ) = 1)) , (9.1.4) th where ei denotes the i standard basis vector. It turns out that functions far from the set of all monotone functions necessarily violate Equation (9.1.4) on a non-negligible fraction of all possible x and i. This makes it possible to test for monotonicity by checking Equation (9.1.4) on a small number of randomly chosen x and i. Theorem 62 (Soundness of monotonicity testing (compare [Gol+00])). If f : {0, 1}n → {0, 1} is exactly ε-far from all monotone functions, then ε ≤ Pr [(xi = 0 ∧ f (x) = 1 ∧ f (x + ei ) = 0) ∨ (xi = 1 ∧ f (x) = 0 ∧ f (x + ei ) = 1)] ≤ 2ε . n x∼{0,1}n ,i∼[n] (9.1.5) Therefore, we can solve Problem 61 from only O(n/ε) many queries to the unknown function, which was exactly the celebrated conclusion of [Gol+00]. 127 While this query complexity does depend on n, the dependence is only logarithmic in the size of the function domain, and it in particular is exponentially better than the n-dependence in the query complexity of learning monotone functions [BBL98]. Our passive quantum monotonicity tester also crucially relies on Theorem 62. Here, we first reinterpret the probability appearing in Equation (9.1.5) in terms of Fourier-analytic quantities, which we then estimate based on quantum Fourier sampling. Our procedure is summarized in Algorithm 1, and our next theorem establishes that it is both complete and sound. Algorithm 1 Monotonicity testing from quantum examples Input: δ ∈ (0, 1);  2accuracy  parameter ε ∈ (0, 1); confidence parameter n log(1/δ) 1 P √ Õ many copies of a function state |f i = 2n x∈{0,1}n |x, f (x)i. ε2 Output: “accept” or “reject”. ε δ Initialization: ε2 = 3ε , ε5 = 3n , δ =  δ2 = 3δ , δ5 = 3n , m1 =  1 max{3m2 , d18 ln(2/δ1 )e}, m2 = n2 ln(2/δ2 ) 2ε22 , m4 = m5 = 4 ln(2/δ5 ) ε25 m1 many copies of |f i to produce m2 many Fourier samples S1 , . . . , Sm2 ⊆ [n] from g = (−1)f . Pm2 2: Take Î = m1 `=1 |S` |. 2 3: Use m4 many copies of |f i to generate m5 many classical samples (x1 , f (x1 )), . . . , (xm5 , f (xm5 ) from f . 4: for 1 ≤ i ≤ n do P 5 xk ·ei +f (xk ) 5: Take g̃i = m15 m . k=1 (−1) 6: end for 1 1 Pn 7: Set p̂ = 2n Î − 2n i=1 g̃i . 8: If p̂ ≤ ε/3n, conclude that f is monotone and accept. If p̂ ≥ 2ε/3n, conclude that f is ε-far from all monotone functions and reject. 1: Use Theorem 63 (Passive quantum monotonicity testing). Algorithm 1 is an effi 2  P cient quantum algorithm that uses Õ n log(1/δ) copies of |f i = √12n x∈{0,1}n |x, f (x)i ε2 to decide, with success probability ≥ 1 − δ, whether f is monotone or ε-far from monotone. In particular, Theorem 63 shows that passive quantum testers can exponentially outperform the classical passive monotonicity testing lower bound of    √ exp Ω min{ n/ε, n} [Gol+00; Bla24]. Proof. We begin with a useful rewriting of the probability from Equation (9.1.5). To this end, as is commonly done in the analysis of Boolean functions, consider the induced function g : {−1, 1}n → {−1, 1} obtained from f via the relabeling 128 0 ↔ 1 and 1 ↔ −1. Next, we recall the definition of the ith derivative in Boolean analysis (compare, e.g., [ODo21, Definition 2.16]): for i ∈ [n], Di g(x) := g(x(i7→1) ) − g(x(i7→−1) ) , 2 where we used the notation x(i7→b) to denote the n-bit string obtained from x by replacing the ith bit with b. Consequently, we can compute   1 if g(x(i7→1) ) = −1 ∧ g(x(i7→−1) ) = 1 (Di g(x)) − Di g(x) =  2 0 otherwise 2 =   1 if f (x(i7→0) ) = 1 ∧ f (x(i7→1) ) = 0  0 otherwise . Therefore, we can now rewrite our probability of interest as Pr x∼{0,1}n ,i∼[n] [(xi = 0 ∧ f (x) = 1 ∧ f (x + ei ) = 0) ∨ (xi = 1 ∧ f (x) = 0 ∧ f (x + ei ) = 1)] " # (Di g(x))2 − Di g(x) = Ex∼{0,1}n ,i∼[n] 2 h i 1 1 = Ei∼[n] Ex∼{0,1}n (Di g(x))2 − Ei∼[n] Ex∼{0,1}n [Di g(x)] 2 2 n 1 1 X = Ei∼[n] [Inf i [g]] − gb({i}) 2 2n i=1 = n 1 1 X I[g] − gb({i}) , 2n 2n i=1 where the second-to-last step used the definition of the ith influence (compare [ODo21, Definition 2.17]) as well as [ODo21, Proposition 2.19], and where the last step used the definition of the total influence (compare [ODo21, Definition 2.27]). With this rewriting established, let us first analyze the probabilities that the different steps of Algorithm 1 succeed and discuss what this implies for the estimator p̂. Then, we will see how this gives rise to completeness and soundness. We have the following: • Using the procedure of [BV97], one copy of |f i suffices to produce one Fourier sample from g = (−1)f —that is, an n-bit string sampled from the probability distribution {|ĝ(S)|2 }S⊆[n] —with success probability 1/2. Additionally, one knows whether the sampling attempt was successful.7 To see this, note that the procedure works as follows: Apply H ⊗(n+1) ; measure the last qubit; abort if that produces a 0, continue if produces a 1; measure the first n qubits to produce an n-bit string. 7 129 So, by simply repeating the above m1 many times, we see that Step 1 succeeds in producing m2 Fourier samples with success probability ≥ 1 − δ1 . • By a standard Chernoff-Hoeffding bound, we have |Î − ES∼Sg [|S|]| ≤ ε2 with success probability ≥ 1−δ2 . Here, Sg denotes the Fourier distribution of g, defined via Sg (S) = |gb(S)|2 . • For any 1 ≤ i ≤ n, by a standard Chernoff-Hoeffding bound, Step 5 produces a ε5 -accurate estimate g̃i of gb({i}) with probability ≥ 1 − δ5 . Therefore, by a union bound, we have that, with overall success probability ≥ 1 − (δ1 + δ2 + nδ5 ) = 1 − δ, the estimates Î and g̃i simultaneously satisfy |Î − ES∼Sg [|S|]| ≤ ε2 and |g̃i − gb({i})| ≤ ε5 for all 1 ≤ i ≤ n. We condition on this high probability success event for the rest of the proof. In this event, our rewriting of the probability of interest from the beginning of the proof implies: Pr x∼{0,1}n ,i∼[n] ≤ [(xi = 0 ∧ f (x) = 1 ∧ f (x + ei ) = 0) ∨ (xi = 1 ∧ f (x) = 0 ∧ f (x + ei ) = 1)] − p̂ n 1 1 X I[g] − Î + |gb({i}) − g̃i | 2n 2n i=1 n 1 1 X ES∼Sg [|S|] − Î + |gb({i}) − g̃i | 2n 2n i=1 ε2 ε5 ≤ + 2n 2 ε ≤ , 3n = where we used the identity I[g] = ES∼Sg [|S|] (compare [ODo21, Theorem 2.38]). So, we see that p̂ is a (ε/3n)-accurate estimate for our probability of interest. To prove completeness of Algorithm 1, assume f to be monotone. Then, Proposition 2 and Theorem 62 together with the above inequality imply that p̂ ≤ ε/3n ≤ ε/2n, and thus the final step in Algorithm 1 correctly concludes that f is monotone and accepts. To prove soundness, assume f to be ε-far from monotone. Then, the lower bound in Theorem 62 together with the above inequality implies that p̂ ≥ 2ε/3n, and thus the final step in Algorithm 1 correctly concludes that f is ε-far from monotone and rejects. 130 The overall number of copies of |f i used by the algorithm is m1 + m4 . Plugging in the choices of the different mi , we see that & ' 4 ln(2/δ5 ) m1 + m4 ≤ max{3m2 , d18 ln(2/δ1 )e} + ε25 ( & ' ) & ' n2 ln(2/δ2 ) 36n2 ln(6n/δ) ≤ max 3 , d18 ln(2/δ1 )e + 2ε22 ε2 ( & ' ) & ' 9n2 ln(6/δ) 36n2 ln(6n/δ) ≤ max 3 , d18 ln(6/δ)e + 2ε2 ε2 ! n2 log(1/δ) ≤ Õ , ε2 where the Õ hides a logarithmic dependence on n. The quantum computational efficiency of Algorithm 1 follows immediately from the efficiency of quantum Fourier sampling. The classical computational efficiency is immediately apparent from our sample complexity bounds and the fact that the classical computation is dominated by the complexity of computing the empirical averages in Steps 2 and 4. We further note that because of the second inequality in Theorem 62, the above procedure and proofs can be modified to quantumly efficiently solve the tolerant version (as defined in [PRR06]) of the monotonicity testing problem— i.e., distinguishing between f being ε1 -close or ε2 -far from monotone—using  2  Õ n(εlog(1/δ) copies of a quantum function state, assuming that ε2 > Cnε1 2 2 −ε1 ) holds with C > 1 some constant.8 Because of this restrictive assumption on how ε1 and ε2 relate, this still falls short of a general tolerant passive quantum monotonicity tester. Let us also note room for a qualitative improvement in our passive quantum monotonicity tester. Classical query-based testers typically enjoy perfect completeness, i.e., they accept monotone functions with unit probability. In contrast, our tester can be made to accept monotone functions with probability arbitrarily close but not equal to 1. We leave as an open question whether our passive quantum monotonicity tester can be modified to achieve perfect completeness, while enjoying similar guarantees on the quantum sample and time complexity of the procedure. 8 In more generality, one can see: If an inequality like Equation (9.1.5) holds with lower bound `n (ε) and upper bound un (ε), satisfying `n (0) = 0 = un (0), then estimating the relevant probability to accuracy ∼ `n (ε2 − ε1 ) suffices for tolerant property testing in the parameter range where there is a constant c ∈ (0, 1/2) such that `n (ε2 ) − c · `n (ε2 − ε1 ) > un (ε1 ) + c · `n (ε2 − ε1 ). 131 Passive quantum triangle-freeness testing For x, y ∈ {0, 1}n and for a Boolean function f : {0, 1}n → {0, 1}, we say that (x, y, x + y) is a triangle in f if f (x) = f (y) = f (x + y) = 1. Accordingly, we call the function f triangle-free if no triple (x, y, x + y) is a triangle in f . Testing for triangle-freeness thus becomes the following problem. Problem 64 (Classical triangle-freeness testing). Given query access to an unknown function f : {0, 1}n → {0, 1} and an accuracy parameter ε ∈ (0, 1), decide with success probability ≥ 2/3 whether (i) f is triangle-free, or (ii) f is ε-far from all triangle-free functions, that is, we have Prx∼{0,1}n [f (x) 6= g(x)] ≥ ε for all triangle-free functions g : {0, 1}n → {0, 1}, promised that f satisfies either (i) or (ii). The natural approach towards testing for triangle-freeness from query access is to choose x, y ∈ {0, 1}n at random and check whether (x, y, x+y) is a triangle in f , and to repeat this sufficiently often. Bounding the number of repetitions needed to succeed with this approach is non-trivial, connecting to Szemerédi’s regularity lemma [Sze76] and the triangle removal lemma [RS78]. In our next result, we recall the to our knowledge best known corresponding bounds. Theorem 65 (Soundness of triangle-freeness testing [Fox11; HST16]). If f : {0, 1}n → {0, 1} is ε-far from all triangle-free functions, then Pr x,y∼{0,1} [f (x) = f (y) = f (x + y) = 1] ≥ n 1  l  m , Tower C · log 1ε (9.1.6) where C > 0 is a universal integer constant. Here, Tower(i) denotes a tower of 2’s of height i. That is, we define the tower function Tower : N → N inductively via Tower(0) = 1 and Tower(i + 1) = 2Tower(i) . In a way familiar by now from the two previous subsections,  l  m Theorem 65 can be used to show that ∼ Tower C · log 1ε many queries suffice for the the simple query-based triangle-freeness tester mentioned above to achieve success probability 2/3. We now use Theorem 65 to develop a passive quantum triangle-freeness tester. 132 Theorem 66 (Passive quantum triangle-freeness testing). There is an effi    m6 l cient quantum algorithm that uses Õ ln(1/δ) Tower C · log 1ε many copies of the function state |Ψi = √12n x∈{0,1}n |x, f (x)i to decide, with success probability ≥ 1 − δ, whether f is triangle-free or ε-far from all triangle-free functions. P Proof. Our passive quantum triangle-freeness tester first sets confidence and ac  l  m−1 curacy parameters δ̃ = δ/(5m) and ε̃ = Tower C · log 1ε , respectively. Then, it repeats the following for 1 ≤ i ≤ m = l 18 ln(10/δ) ε̃2 m : δ̃) 1. Take ln(m/ many copies of |Ψi and, for each of them, measure the last ε qubit in the computational basis. If none of these measurements produces outcome 1, abort this iteration, set µ̂i = 0, and go to the next iteration. Otherwise, take any one of the post-measurement states for which 1 was observed, measure the first n qubits, let the outcome be yi . l 4 m  ln(2162 ln(6/δ̃)(6/ε̃)4 /δ̃)  2. Run the procedure from Lemma 83 on 2 162 ln(6/δ̃)(6/ε̃) · l many copies of |Ψi to obtain 2 162 ln(6/δ̃)(6/ε̃) post-measurement state 4 |Ψ1 i = (|{x ∈ {0, 1}n : f (x) = 1}|)−1/2 m ε many copies of the X |xi , x∈{0,1}n :f (x)=1 where we threw away the last qubit. If the procedure from Lemma 83 outputs FAIL, abort this iteration, set µ̂i = 0, and go to the next iteration. 3. Consider the n-qubit unitary Uyi acting as Uyi |xi = |x + yi i. Run the  l m procedure from Lemma 83 on 2 162 ln(6/δ̃)(6/ε̃)4 · ln(2 162 ln(6/δ̃)(6/ε̃)4 /δ̃) ε l m many copies of (Uyi ⊗ 12 ) |Ψi to obtain 2 162 ln(6/δ̃)(6/ε̃)4 many copies of the post-measurement state |Ψyi ,1 i = (|{x ∈ {0, 1}n : f (x + yi ) = 1}|)−1/2 X |xi , x∈{0,1}n :f (x+yi )=1 where we threw away the last qubit. If the procedure from Lemma 83 outputs FAIL, abort this iteration, set µ̂i = 0, and go to the next iteration. l m 4. Run the procedure from Corollary 85 on 162 ln(6/δ̃)(6/ε̃)4 copies of l m each of |Ψi, (Uyi ⊗ 12 ) |Ψi and on 2 162 ln(6/δ̃)(6/ε̃)4 copies of each of |Ψ1 i, |Ψyi ,1 i to produce an (ε̃/6)-accurate estimate µ̂i of the probability Prx∼{0,1}n [f (x) = 1 = f (x + yi )]. 133 1 Pm Finally, the tester makes a decision as follows: if m i=1 µ̂i ≤ ε̃/3, output “triangle-free”. Otherwise, output “ε-far from triangle-free”. Let us analyze the completeness and soundness of this tester. First, we consider completeness. So, assume that f is triangle-free. Note the first step above failing in any iteration only ever decreases the empirical average evaluated by the tester in the end, and thus cannot increase the probability of falsely rejecting a triangle-free function. Thus, we can condition on the first step succeeding in all m iterations. In particular, we can assume that Pry∼{0,1}n [f (y) = 1] > 0. By a similar argument, we can also condition on the second and third step succeeding in all iterations. And given these successes, the fourth step will, by Corollary 85, produce, with probability ≥ 1 − δ̃, a µ̂i satisfying |µ̂i − Prx∼{0,1}n [f (x) = 1 = f (x + yi )]| ≤ ε̃/6. By a union bound, P this means that, with probability ≥ 1 − δ/5, the empirical average m1 m i=1 µ̂i is P a (ε̃/6)-accurate estimate of m1 m i=1 Prx∼{0,1}n [f (x) = 1 = f (x + yi )]. We can consider the Prx∼{0,1}n [f (x) = 1 = f (x + yi )] for 1 ≤ i ≤ m as i.i.d. random variables taking values in [0, 1]. Hence, by a Chernoff-Hoeffding concentration bound and our choice of m, with probability ≥ 1 − δ/5, we have h i Ey∼{0,1}n :f (y)=1 Prx∼{0,1}n [f (x) = 1 = f (x + y)] = Prx,y∼{0,1}n [f (x)=f (y)=f (x+y)=1] = 0, Pry∼{0,1}n [f (y)=1] 1 Pm i=1 Prx∼{0,1}n [f (x) = 1 = f (x + yi )] − Ey∼{0,1}n :f (y)=1 Prx∼{0,1}n [f (x) = 1 = f (x + y)] m Noticing that h i (9.1.7) since f was assumed to be triangle-free, we conclude (after one more union P bound) that m1 m i=1 µ̂i ≤ ε̃/3 holds with probability ≥ 1 − 2δ/5 ≥ 1 − δ, and in this case the tester outputs “triangle-free”, thus proving completeness. Next, we consider soundness. So, assume that f is ε-far from trianglefree. This in particular implies that Prx∼{0,1}n [f (x) = 1] ≥ ε, since the zero-function is triangle-free. Hence, when measuring the last qubit of |Ψi in the computational basis, outcome 1 is observed with probability ≥ ε. Therefore, in any iteration i, the first step in our sketched procedure above will succeed and produce some yi with f (yi ) = 1 with probability ≥ 1 − δ̃. We condition on this high-probability event E1 for the rest of the soundness analysis. By an analogous reasoning, the assumption of Lemma 83 is satisfied in the scenario of steps 2 and 3—with S = {0, 1}n , η = ε and the function either given directly by f or by f (· + yi )—so in any iteration i, the second and third step each will ≤ ε̃/6 . 134 succeed with probability ≥ 1 − δ̃. We now further condition on these success events E2 and E3 . At this point, by the same reasoning as in the completeness case, we know that the fourth step, with success probability ≥ 1 − 2δ/5 overall, produces estimates µ̂i such that m 1 X µ̂i − Pr n [f (x) = f (y) = f (x + y) = 1] ≤ ε̃/3 . x,y∼{0,1} m i=1 By Theorem 65, since we assumed f to be ε-far from triangle-free, we have Prx,y∼{0,1}n [f (x) = f (y) = f (x + y) = 1] ≥ ε̃. Thus, by the first equality from Equation (9.1.7), we have " Ey∼{0,1}n :f (y)=1 # Pr [f (x) = 1 = f (x + y)] ≥ ε̃ . x∼{0,1}n So, the above implies the inequality m1 m i=1 ≥ 2ε̃/3. Hence, the tester will in this case correctly output “ε-far from triangle-free”. A final union bound shows that this occurs with probability ≥ 1 − δ, which proves soundness. The quantum sample  complexity of a single iteration is given by 2 ·  P l m 2 162 ln(6/δ̃)(6/ε̃)4 · ln(2 162 ln(6/δ̃)(6/ε̃)4 /δ̃) ε l m   δ̃) +2 162 ln(6/δ̃)(6/ε̃)4 ≤ Õ ln(1/ . ε̃4    2  δ̃) δ̃) Thus, the overall quantum sample complexity is ≤ m · Õ ln(1/ ≤ Õ ln ε̃(1/ . 6 ε̃4    l  m6  Plugging in the chosen values for δ̃ and ε̃ yields an upper bound of Õ ln2 (1/δ) Tower C · log 1ε on the number of quantum copies used by the tester. To achieve the claimed linear dependence on ln(1/δ), one can simply run the protocol described above for a constant confidence parameter (say, δ = 1/3), and then amplify the success probability through majority votes. Finally, we have to argue that the tester is quantumly computationally efficient. This, however, follows immediately from the efficiency of the procedures from Lemma 83 and Corollary 85, and from the fact that every Uyi can be implemented by at most n Pauli-X gates. Classically, passive triangle-freeness testing requires at least Ω(n) samples.9 Therefore, just like in symmetry testing, there is an unbounded separation between a classical n-dependent and a quantum n-independent passive testing sample complexity. 9 This can be seen by an argument via linear independence similar to that used in the lower bound proof of [AHW16, Theorem 10]. 135 9.2 Fourier sampling does not suffice The passive quantum testers for symmetry and triangle-freeness given in Section 9.1 notably do not use quantum Fourier sampling. One might ask if this is really necessary, given that Fourier sampling (sometimes augmented by classical samples) suffices for so many other learning and testing tasks. This section presents a class of functions, Maiorana–McFarland (bent) functions, which can be tested with O(1) function state copies, but any algorithm relying solely on Fourier samples and classical samples requires super-polynomial classical samples to succeed. The Maiorana–McFarland functions [CM16, Section 6.1] on 2n bits, denoted MMn , are given by fh : Fn2 × Fn2 → F2 (x, y) 7→ hx, yi + h(x), where h ranges over all functions Fn2 → F2 . Maiorana–McFarland functions are a subset of the class of bent Boolean functions g : {0, 1}m → {0, 1}, which are those with gb(S)2 = 1/2m for all S ⊂ [m] (so they are maximally-far from any Fn2 -linear function). We begin by proving hardness of testing MMn using only classical samples   x, f (x) and Fourier samples. The proof follows swiftly once we establish the existence of k-wise independent distributions supported only on moderatelybiased strings. Lemma 67. For sufficiently large n, there is a probability distribution on n {0, 1}2 that is (i) 20.1n -wise independent and (ii) supported on strings x with fractional hamming weight |x|/2n bounded as |x| 1 − ≤ 2−n/3 . 2n 2 Proof. Let c > 0 be a small universal constant to be chosen later. For simplicity we assume cn is an integer, otherwise one may round to a nearest integer without issue. For a function f : Fn2 → F2 let eval(f ) denote the truth table (viz. a 2n -bit string) of f . Let Pcn−1 denote the set of n-variate F2 polynomials of degree at most cn − 1. We claim there exists a Boolean function f such that the ensemble n o eval(p ⊕ f ) p∼Pcn−1 satisfies properties (i) and (ii) above. 136 Consider the Reed–Muller code C = RM(n − cn, n). (See [ASSY23] for the definition and properties of Reed–Muller codes.) It is well-known (e.g., Section 24.2.2 in [GRS23]) that uniformly random codewords from the dual code C ⊥ n form a k-wise independent distribution on {0, 1}2 for k := dist(C)−1 = 2cn −1, and the codewords of C ⊥ = RM(cn − 1, n) are exactly {eval(p) : p ∈ Pcn−1 }. Invariance of the uniform distribution under addition in Fk2 implies that for any Boolean function f the uniform distribution over strings eval(p ⊕ f ) = eval(p) + eval(f ) is also k-wise independent. Property (i) follows by decreasing c slightly. To see (ii), we will argue that a random f suffices with high probability.10 For any F2 polynomial p and f a uniformly random Boolean function, we have from Chernoff that  Pr f h E (−1) x f (x) | p(x) (−1) {z i −n/3 ≥2  ≤ exp(−const · 2n/3 ) . (9.2.1) } ∗ Union bounding over the ≤ exp(23cn )-many p ∈ Pcn−1 we find  h Pr ∃p ∈ Pcn−1 , E (−1) f (x) x f p(x) , (−1) i −n/3 ≥2    ≤ exp −const · 2n/3 + 23cn , which goes to 0 for c = 0.11, for example. So for large-enough n there exists an f with absolute correlation at most 2−n/3 with all p ∈ P0.11n−1 . But the correlation (∗) in Equation (9.2.1) is nothing but the bias of the function p ⊕ f . Thus for all p ∈ P0.1n ⊆ P0.11n−1 , the fractional Hamming weight of eval(p ⊕ f ) is at most 2−n/3 away from 1/2. Theorem 68. Suppose a tester for MMn using exclusively classical samples and Fourier samples succeeds with probability at least 1/2 + 2−0.7n for the accuracy parameter ε = 1/2 − 2−0.6n . Then the tester uses at least 20.1n classical samples. Proof. Let H be the distribution on n-bit functions with truth tables distributed as in Lemma 67. Consider the two ensembles of Boolean functions Fn2 ×Fn2 → F2 n o F1 = (x, y) 7→ hx, yi+h(x) h∼H and n o F2 = (x, y) 7→ hx, yi+h(y) h∼H Note that for any f = hx, yi+h(x) ∈ MMn and any g = hx, yi+m(y) ∈ suppF2 , E (−1)f (x,y) (−1)g(x,y) = x,y E (−1)h(x) (−1)m(y) = |bias(h)bias(m)| ≤ 2−2n/3 x,y 10 If one desires explicit hard instances one can use the correlation bounds of Smolensky [Smo87b; Smo93] for degree-d F2 polynomials against √ the Majority function, but they are quadratically worse (correlation bounded by ≤ O(d/ n)). , 137 by property (ii) in Lemma 67. Thus all g ∈ suppF2 are at least (1/2−22n/3 )-far from MMn ⊇ suppF1 . Suppose a testing algorithm using R ≤ 20.1n classical samples succeeds with probability δ. That implies that, given access to a function from F1 or from F2 with equal probability the algorithm can guess which of the two ensembles the function was drawn from with success probability δ. Let b ∼ {1, 2} and f ∼ Fb . With probability at least 1−R2 /2n , all x(r) ’s and all y (r) ’s in the R-many samples are distinct (collision bound and union bound). Call this event D. Conditioned on D, for all f ∈ suppF1 and all f ∈ suppF2 ,   the distribution of observed values f (x(1) , y (1) , . . . , f (x(R) , y (R) ) is uniformly random because h ∼ H is 20.1n -wise independent (property (i) in Lemma 67) and we assumed R ≤ 20.1n . Thus conditioned on D, the data observed is independent of b. Moreover, all functions in suppF1 and suppF2 are bent, so Fourier sampling provides no information whatsoever. The distinguishing probability is thus bounded by δ ≤ Pr[D] · 1 1 R2 1 + Pr[Dc ] · 1 ≤ + n ≤ + 2−0.8n . 2 2 2 2 Having established that the Maiorana–McFarland class is hard to test from classical samples and Fourier samples alone, we now give a very efficient passive quantum tester for MMn . While this tester still uses quantum Fourier sampling at the end of the algorithm, it crucially preprocesses the function state in superposition before applying performing Fourier sampling. Theorem 69. There is an efficient quantum algorithm that uses O(1) copies P of the function state |f i = 21n x,y∈{0,1}n |x, y, f (x, y)i to decide, with success probability ≥ 2/3, whether f is in MMn or (1/3)-far from MMn . Proof. Let U denote the (2n+1)-qubit unitary acting as |x, y, bi 7→ |x, y, b ⊕ hx, yii. Note that U can be implemented by a quantum circuit with O(n) many 2-qubit gates and depth O(log n). Moreover, notice that U |f i = |f˜i for f˜(x, y) := f (x, y) ⊕ hx, yi. The quantum algorithm works as follows. Recall that one function state copy suffices to obtain one Fourier sample with success probability 1/2, using only 2n many single-qubit gates. Applying this Fourier sampling subroutine to O(1) many copies of U |f i thus suffices to obtain, with success probability ≥ 5/6, m ≥ C many Fourier samples S1 , . . . , Sm ⊆ [2n] of the function f˜, where 138 C > 0 is a universal constant to be chosen later. Let J = {n + 1, n + 2, . . . , n} and compute o 1 n p̂ = 1 ≤ k ≤ m : J ∩ Sk 6= ∅ . m If p̂ ≤ 1/9, output “f ∈ MMn ”. Otherwise, output “f is (1/3)-far from MMn .” First, let us show completeness of the protocol. So, suppose f ∈ MMn . Then there is a function h : Fn2 → F2 such that U |f i = 1 X |x, y, h(x)i = |hi , 2n x,y where we abused notation by using h to denote the function h : Fn2 × Fn2 → F2 defined as h(x, y) = h(x). As h(x, y) depends only on the first n variables, we have X p := |ĥ(S)|2 = 0 . S⊆[2n], J∩S6=∅ The constant C can be chosen such that, conditioned on the high probability event that we obtained at least C many Fourier samples, we have |p − p̂| ≤ 1/9 with probability ≥ 5/6 (by Chernoff-Hoeffding). So p̂ ≤ 1/9, and our tester correctly outputs “f ∈ MMn ” with probability ≥ 2/3. Next, we analyze soundness. So, suppose f is (1/3)-far from MMn . Equivalently, f˜(x, y) = f (x, y) ⊕ hx, yi is (1/3)-far from any Boolean function h that depends only on the first n variables, h(x, y) = h(x), where we again abused notation. Consider the function g defined as g(x, y) = X fˆ˜(S)χS (x, y) . S⊆[2n]:J∩S=∅ Notice that g(x, y) depends only on x, but g is in general not Boolean. Define g̃(x, y) = 1g(x,y)≥1/2 . Notice that g̃ is a Boolean function and that g̃(x, y) depends only on x. Then (compare [AS07, Fact II.2]) we have P 1 ≤ Px1 ,x2 [f˜(x1 , x2 ) 6= g̃2 (x2 )] ≤ Ex1 ,x2 [(f˜(x1 , x2 ) − g(x1 , x2 ))2 ] = S⊂[2n]:J∩S6=∅ fˆ˜(S) 3 2 Again, conditioned on having produced at least C many Fourier samples, with probability ≥ 5/6, we have |p − p̂| ≤ 1/9 and thus p̂ ≥ 2/9. So, our tester correctly outputs “f is (1/3)-far from MMn ” with probability ≥ 2/3. 9.3 Separating passive quantum from query-based classical property testing In this section we give a property for which classical queries have exponential advantage over quantum testing from function states. This property is closely = p. 139 related to the inability of quantum computers to measure the intersection of three subset states, where for a subset S ⊆ Fn2 , the corresponding subset state P is defined as |Si = p1 x∈S |xi. We explain this connection at the end of the |S| section. The main result of this section is the following theorem. Theorem 70. There exist two sets of Boolean functions F0 , F1 such that min f0 ∈F0 ,f1 ∈F1 kf0 − f1 k1 ≥ 1 64 and such that: • Any passive quantum tester requires Ω(2n/2 ) copies of a function state to distinguish F0 and F1 with constant probability 2/3. • F0 and F1 may be distinguished with probability 2/3 from O(1) classical queries. The families F0 and F1 arise from certain encodings of triples of subsets A, B, C ⊆ {0, 1}n . Consider the class of Boolean functions f(A,B,C) : {0, 1}n+2 → {0, 1} on n + 2 bits parameterized by subsets A, B, C ⊆ {0, 1}n and defined as follows: f(A,B,C) (x, a) =    1x∈A       1x∈B    1x∈C      0 a = 00 a = 01 . a = 10 a = 11 With A, B, C drawn uniformly from subsets of {0, 1}n , we define two function state ensembles {|f(A,B,C) i}A,B,C and {|f(A,B,A∆B) i}A,B , with their mixed state average over t-copy states given by E0 = E A,B,C  f(A,B,C) ED f(A,B,C) ⊗t  , E1 = E A,B  f(A,B,A∆B) ED f(A,B,A∆B) ⊗t  . We now show that these two mixed state averages over t-copy states are close in trace distance unless t scales exponentially in n. This means that exponentially-in-n many copies are needed to distinguish between the two function state ensembles. Theorem 71. kE0 − E1 k1 ≤ O(t/2n/2 ). 140 E Proof. It will help to reinterpret f(A,B,C) as a subset state via the rewriting X |x, ai |f(A,B,C) (x, a)i = 1x∈Sa,b |x, (a, b)i , (9.3.1) b∈{0,1} where Sa,b denotes A, B, C, or ∅ according to a when b = 1, or the respective complements if b = 0. Using r to represent the concatenation of a and b we may then write |f(A,B,C) i = √ 1 X X 4N r∈{0,1}3 x∈{0,1}n 1x∈Sr |x, ri , where N := 2n and, like before, Sr denotes one of A, B, C, or ∅ or the complements thereof. With this notation let us consider the basis for the space of t copies of function states given by n o |x1 , r1 , . . . , xt , rt i : xj ∈ {0, 1}n , rj ∈ {0, 1}3 , j = 1, . . . , t . Let Π denote the projector onto the subspace spanned by those |x1 , r1 , . . . , xt , rt i for which all xj are distinct. First, we claim t kE0 − ΠE0 Πk1 , kE1 − ΠE1 Πk1 ≤ O √ N ! (9.3.2) . These bounds follow from applying the triangle inequality to the following estimate: for any fixed A, B, C, we have f(A,B,C) ED f(A,B,C) ⊗t − Π f(A,B,C) ED f(A,B,C) ⊗t Π 1 v u u =t 4t N t 1+3 (4N )t ! 4t N t 1− (4N )t (9.3.3) 2t ≤√ , N (9.3.4) where (x)t = x(x − 1) . . . (x − t + 1) denotes falling factorial, and where in the second step we applied the bound 4t N t t ≥ 1− t (4N ) N  t ≥1− t2 . N To see Equation (9.3.3), note that M := f(A,B,C) ED f(A,B,C) ⊗t − Π f(A,B,C) ED f(A,B,C) ⊗t Π ! 141 has the following block form after reordering columns:   0    1   M= (4N )t     | This is because f(A,B,C) ED 1 0   4t N t 4t (N t −N t ) 1 1 0 0 0 0        {z } {z 4t N t }| }| {z . (8N )t −(4N )t 4t (N t −N t ) (8N )t −(4N )t f(A,B,C) ⊗t is an all-zeros matrix except for the prin  cipal submatrix associated to indices (x1 , r1 ), . . . , (xt , rt ) where xj ∈ Srj for all j, and here it is equal to (4N )−t . There are (4N )t such entries. Moreover, ED ⊗t Π f(A,B,C) f(A,B,C) Π is an all-zeros matrix except for the principal subma  trix associated to indices (x1 , r1 ), . . . , (xt , rt ) where xj ∈ Srj for all j and xj 6= xk for j 6= k and here it is also equal to (4N )−t —and there are 4t N t of these entries. M thus has rank 2 and its spectrum is easily determined, leading to the estimate in Equation (9.3.3). Now we claim that in fact ΠE0 Π = ΠE1 Π . (9.3.5) Let us consider a specific entry in ΠE0 Π with row and column indices r = (. . . , (xj , rj ), . . .), s = (. . . , (yj , sj ), . . .) . It will be useful to write Sz = Sz (r, s) for the set types that appear with a particular string z ∈ {0, 1} in r and s. That is, for any z ∈ {0, 1}n define n o Sz = Sz (r, s) = q ∈ {0, 1}3 : (z, q) ∈ r or (z, q) ∈ s . Then Q hr| ΠE0 Π |si = (4N )−t E A,B,C = (4N ) −t j 1xj ∈Srj Q E Q Y z∈{xj }j ∪{yj }j A,B,C j 1yj ∈Ssj  q∈Sz 1z∈Sq . It follows from the definition of Π that |Sz | ≤ 2 for any z ∈ {xj }j ∪ {yj }j : there is at most one contribution to Sz from each of r and s. As a result we have for any z that E iid A,B,C ∼P{0,1}n Q q∈Sz 1z∈Sq = iid E A,B ∼P{0,1}n C=A∆B Q q∈Sz 1z∈Sq . 142 This follows from mild case analysis, the most important part of which is to note that for any S 6= T ∈ {A, B, C, A∆B}, (1x∈S , 1x∈T ) ∼ (b1 , b2 ), where b1 and b2 are i.i.d. Bernoulli 1/2 random variables. So we see hr| ΠE0 Π |si = hr| ΠE1 Π |si and Equation (9.3.5) is satisfied. Combining the triangle inequality with Equation (9.3.2) and Equation (9.3.5) gives the result. Proof of Theorem 70. Consider n F0 = f(A,B,C) : A, B, C ⊆ {0, 1}n , 2−n |A ∩ B ∩ C| ≥ 1/16 n o o and F1 = f(A,B,A∆B) : A, B ⊆ {0, 1}n , First we prove the minimum distance between F0 and F1 . For any f0 ∈ F0 , there are 2n /16 strings x ∈ {0, 1}n such that f0 (x00) = f0 (x01) = f0 (x10) = 1. On the other hand, for all f1 ∈ F1 , by definition there are no strings x with this property. Thus the minimum L1 distance between F0 and F1 is at least 2n /16 1 = . n 4·2 64 Now define the state ensembles E00 = E |f ihf |⊗t f ∼F0 and E1 = E |f ihf |⊗t . f ∼F1 E1 here is exactly E1 from Theorem 71. To compare E00 and E0 from Theorem 71, iid note that for A, B, C ∼ P{0, 1}n , any string x is in A ∩ B ∩ C with probability 1/8 and so from Chernoff we have  Pr |A ∩ B ∩ C| < 1 2n 2n · ≤ exp − . 2 8 64    This dramatic concentration, together with Theorem 71 implies kE00 − E1 k1 ≤ kE00 − E0 k1 + kE1 − E0 k1 ≤ O(t/2n/2 ) . To test this property with classical queries, given an unknown f = f(A,B,C) one may simply choose a random x ∈ {0, 1}n and check if f (x00) = f (x01) = f (x10) = 1. This test accepts with probability 1/8 when f ∈ F0 and accepts with probability 0 when f ∈ F1 . 143 k-fold intersection is “unfeelable” for k ≥ 3 In this subsection, we reinterpret Theorem 70 in the context of subset states. Given access to copies of k different subset states |S1 i , . . . , |Sk i, it is natural to ask how many copies of each are required to estimate the fractional size of the mutual intersection, |S1 ∩ · · · ∩ Sk | . 2n When k = 2, this can be readily accomplished using ideas similar to our algorithms presented above. In the case of intersection estimation with k = 2, we have the identity |S1 ∩ S2 | = hSall |S1 i hS1 |S2 i hS2 |Sall i , 2n where Sall := {0, 1}n denotes the full hypercube. The quantities on the righthand side are easily estimated via swap tests, so it takes O(1) copies of |S1 i , |S2 i to estimate the quantity of interest to any constant additive error. In contrast, it is a consequence of Theorem 70 that the same question for k = 3 has a very different answer: it requires Ω(2n/2 ) copies to achieve constant additive error. To see this, note that from any |f(A,B,C) i one may obtain each of |Ai, |Bi, and |Ci with constant probability by measuring the a and f (x, a) registers, provided that the minimum among |A|, |B|, and |C| is at least a constant fraction of 2n —and this condition is satisfied by the overwhelming majority of functions in the families F0 and F1 of Theorem 70. From F0 and F1 we obtain: Corollary 72. There are two families S0 and S1 of triples of subsets of {0, 1}n such that and ∀(A0 , B0 , C0 ) ∈ S0 , |A0 ∩ B0 ∩ C0 |/2n ≥ 1/16 ∀(A1 , B1 , C1 ) ∈ S1 , |A1 ∩ B1 ∩ C1 |/2n = 0 , and yet any quantum algorithm distinguishing the two families via their subset states requires Ω(2n/2 ) copies of |Ai , |Bi , or |Ci. 9.4 A challenge: lower bounds for monotonicity testing Here we show that the ensembles used in [Gol+00] to establish strong lower bounds on monotonicity testing from samples do not improve upon the basic Ω(1/ε) sample complexity lower bound in the quantum case. To prove this, we 144 consider the pair of distributions over functions from [Gol+00], constructed such that one is supported entirely on monotone functions, and the other with high probability on functions that are ε-far from monotone; we show that the associated t-copy quantum function state ensembles become distinguishable with constant success probability as soon as t = Ω(1/ε). At the end of the section, we discuss how to extend our reasoning to the ensembles used in [Bla24]. Distinguishability of twin ensembles For the proof, it will be useful to also consider phase states, which are given by X 1 |Ψph (−1)f (x) |xi . f i = √ n 2 x∈{0,1}n The proof reduces to the distinguishability of phase state ensembles encoding the following classical sets of functions taken from [Gol+00]. Definition 5 (Twin ensembles). Let M = {(ui , vi )}m i=1 be a set of pairs of n elements in {0, 1} such that all u1 , v1 , . . . , um , vm are distinct. Let ∪M := ∪(u,v)∈M {u, v} be the complete set of elements in the matching. Fix a function g : {0, 1}n \(∪M ) → {0, 1}. We now define the twin ensembles associated to M and g, which are two sets F0 , F1 of functions on {0, 1}n . For any bipartition of M , M = A t B, define the following two functions. (0) 1. fA,B is defined as follows: (0) (0) (0) (0) • For (u, v) ∈ A, we set fA,B (u) = fA,B (v) = 1. • For (u, v) ∈ B, we set fA,B (u) = fA,B (v) = 0. (0) • If x 6∈ ∪M , then define fA,B (x) = g(x). (1) 2. fA,B is defined as follows: (1) (1) (1) (1) • For (u, v) ∈ A, we set fA,B (u) = 1 and fA,B (v) = 0. • For (u, v) ∈ B, we set fA,B (u) = 0 and fA,B (v) = 1. (1) • If x 6∈ ∪M , then define fA,B (x) = g(x). (0) Then the twin ensembles associated to M and g are F0 = {fA,B }AtB=M and (1) F1 = {fA,B }AtB=M . 145 Let us recall the reasoning from [Gol+00] that connects these ensembles to monotonicity testing. Take k = dn/2e and consider the k th and (k − 1)th layer of the Boolean hypercube with respect to the standard partial ordering on strings . These layers we denote by Lk and Lk−1 respectively; i.e., Li = {x ∈ {0, 1}n−1 : |x| = i}, where |x| denotes the Hamming weight √ of x. Stirling’s formula gives that |Lk |, |Lk−1 | = Ω(2n / n). As argued in [Gol+00], we can find a matching M = {(ui ≺ vi )}m i=1 ⊂ Lk−1 × Lk such that (i) there is no i 6= j such that ui and vj are comparable, (ii) |M | is even, and (iii) m := |M | = ε · 2n , for any ε = ε(n) with 0 < ε ≤ O(n−3/2 ). Now define g : {0, 1}n \∪M → {0, 1} x 7→ 1|x|≥n/2 . The choices of g and M define twin ensembles F0 and F1 . (0) (1) Clearly, every fA,B is a monotone function. Let F1far ⊂ F1 be the set of fA,B functions for which |B| ≥ m/4. Then all functions in F1far are at least Ω(ε)-far from monotone [Gol+00]. We thus wish to bound the distinguishability of F0 from F1far , which in the quantum case is determined by the 1-norm E  (0) fA,B ∼F0 (0) (0) ⊗t |fA,B ihfA,B | − (1) E  fA,B ∼F1far (1) ⊗t (1) |fA,B ihfA,B | . 1 From a standard concentration argument, it suffices to instead bound the distinguishability between F0 and F1 , which in the quantum case is determined by E (0) fA,B ∼F0  (0) (0) ⊗t |fA,B ihfA,B | − E (1) fA,B ∼F1  (1) (1) ⊗t |fA,B ihfA,B | . 1 We will show that, in contrast to the classical case analyzed in [Gol+00], the twin ensembles actually become distinguishable already for t ∼ 1/ε. Namely, much of Section 9.4 will be dedicated to proving the following theorem: Theorem 73. Define ε > 0 so that ε2n = m = |M |. Then  !⊗t  EB  |Ψph(0) ihΨph(0) | fA,B fA,B   − EB  !⊗t  |Ψph(1) ihΨph(1) | fA,B fA,B ≥ Ω(1)  (9.4.1) 1 for t = Ω(1/ε). Let us note that Theorem 73 implies the same 1-norm lower bound and thus the same distinguishability of the two ensembles also from function state 146 copies. To see this, we notice that the function state for any Boolean function f defined on n bits is unitarily equivalent to the phase state for a related Boolean function on n + 1 bits: (I ⊗n ⊗ H) |f i = √ 1 X 2n+1 |xi |0i + √ x∈{0,1}n 1 X 2n+1 x∈{0,1}n (−1)f (x) |xi |1i =: |Ψph i, e f where fe(x1 , . . . , xn , xn+1 ) = (1xn+1 =1 )f (x1 , . . . , xn ). So, in fact the function states for functions in F0 and F1 are unitarily equivalent to phase states for another set of twin ensembles Fe 0 and Fe 1 with f obtained by appending 1 to every string in M and with g e corresponding M given by ge(x) = (1xn+1 =1 ) · g(x1 , . . . , xn ). Theorem 73 implies these phase state ensembles are distinguishable with constant success probability for t ≥ Ω(1), thus the same holds for the function state ensembles for F0 and F1 . Difference matrix: the entries Here, to prove Theorem 73, we pursue a bound on the trace norm distance in Equation (9.4.1). Define the density matrices  !⊗t  A(0) = EB  |Ψph(0) ihΨph(0) | fA,B fA,B   and !⊗t  A(1) = EB  |Ψph(1) ihΨph(1) | fA,B , fA,B Call A := A(0) − A(1) the difference matrix. We now characterize the entries of the difference matrix A by evaluating A(0) and A(1) . Rows (resp. columns) of A(0) and A(1) are indexed by t-tuples x = (x1 , . . . , xt ) (resp. y = (y1 , . . . , yt )) of strings xj ∈ {0, 1}n (resp. yj ∈ {0, 1}n ), 1 ≤ j ≤ t. It turns out that the entries of A(0) and A(1) depend only on the multiset {x1 , . . . , xt , y1 , . . . , yt }, and for this we use the notation x ∪ y. In the following it will sometimes be convenient to use ↑ for exponentiation: a ↑ b := ab . Entries of the A(0) , the f (0) matrix. The (x, y) entry of A(0) takes the form  A(0) x,y =    (0) X (0) 1 Y 1 E  nt (−1)fA,B (z)  = nt E (−1) ↑ fA,B (z) . AtB=M 2 2 AtB=M z∈x∪y z∈x∪y 147 Evaluating this expectation gives     X Y X 1  (0) A(0) g(z) E (−1) ↑ fA,B (z) x,y = nt (−1) ↑ A,B 2 z∈x∪y:z∈p p∈M z∈x∪y,z6∈∪M X 1 = nt (−1) ↑ g(z) 2 z∈x∪y,z6∈∪M · Y p∈M E (−1) ↑ A,B   |{z ∈ x ∪ y : z ∈ p}| mod 2 if p ∈ A  0 . otherwise So, if |{z ∈ x ∪ y : z ∈ p}| = 0 mod 2, the expectation is always (−1)0 . On the other hand, if |{z ∈ x ∪ y : z ∈ p}| = 1 mod 2, the expression inside the expectation is (−1)1 w.p. 1/2 and (−1)0 w.p. 1/2. Define s(x ∪ y) = (−1) ↑ X g(z) . (9.4.2) z∈x∪y,z6∈∪M Then  1     nt s(x ∪ y) if ∀p ∈ M, |{z ∈ x ∪ y : z ∈ p}| = 0 mod 2   0 otherwise . 2 A(0) x,y =  Entries of A(1) , the f (1) matrix. Similarly to above we have A(1) x,y = Y X 1 (1) s(x ∪ y) E (−1) ↑ fA,B (z) . A,B 2nt z∈x∪y:x∈p p∈M To evaluate the expectation, note there are four cases for each p = (u, v) ∈ M , depending on how many times u occurs in x ∪ y, and how many times v occurs in x ∪ y. Denote these quantities mod 2 as Lp = Lp (x ∪ y) and Up = Up (x ∪ y) respectively. • If Lp = 0 and Up = 0, the sum in the exponent is always 0, yielding (−1)0 = 1 w.p. 1. • If Lp = 0 and Up = 1, the sum is either 0 or 1, each w.p. 1/2, yielding in expectation (−1)1 /2 + (−1)0 /2 = 0. • If Lp = 1 and Up = 0, we similarly get 0 for the expectation. • If Lp = 1, Up = 1, then the sum is always 1, yielding (−1)1 = −1 with probability 1. 148 In summary,  1  |{p∈M :Lp =Up =1}|   s(x ∪ y)  nt (−1) if ∀p ∈ M, |{z ∈ x ∪ y : z ∈ p}| = 0 mod 2   0 otherwise 2 A(1) x,y =  , which is equivalent to  1  |{p∈M :Lp =Up =1}|   s(x ∪ y)  nt (−1) if ∀p ∈ M, Lp = Up   0 otherwise 2 A(1) x,y =  . Putting these together, we find (1) Ax,y = A(0) x,y −Ax,y =  2   s(x ∪ y)   nt  2         0 if ∀p ∈ M, Lp = Up and |{p ∈ M : Lp = Up = 1}| = 1 mod 2 . otherwise (9.4.3) Difference matrix: the spectrum Here we conduct a fine-grained analysis of the spectrum of the difference matrix A in order to obtain a combinatorial bound on kAk1 . In Section 9.4 we then understand the asymptotics on this bound in terms of ε, t, and n. Let D be a diagonal matrix with entries s(x) for all t-tuples x. Then à := 2nt−1 D−1 AD is similar to 2nt−1 A and moreover Ãx,y =   s(x)s(x ∪ y)s(y) if ∀p ∈ M, Lp = Up and |{p ∈ M : Lp = Up = 1}| = 1  0 otherwise   1 if ∀p ∈ M, Lp = Up and |{p ∈ M : Lp = Up = 1}| = 1 0 otherwise = mod 2 . (9.4.4) Here we used that s(x ∪ y) = s(x)s(y); the other notation (for Lp and Up ) is as above. Because of the matrix similarity, we know that A and à have the same spectrum up to a scaling factor of 1/2nt−1 . It will turn out that after a certain permutation of indices, à is block-diagonal, with each block corresponding mod 2 149 to the adjacency matrix of a complete bipartite graph. Towards defining this block structure, we write xCy (read “x is compatible with y”) if for all p ∈ M, Lp = Up and |{p ∈ M : Lp = Up = 1}| = 1 mod 2. Given an index tuple x, we define some technical quantities of x that are important for combinatorics to follow. These do not depend on the order of elements in x so we will treat x as a multiset for this discussion. The multiset x may be partitioned as: x = { pairs } t { singletons } t { elements outside of ∪M } (9.4.5) To form the “pairs” multiset, we greedily take as many copies of each pair p ∈ M as we can from x. The “singletons” multiset are the remaining elements in x from ∪M that cannot be paired up, and the final part corresponds to those elements in x outside of ∪M . This partitioning is unique. For example, suppose M = {(1, 2), (3, 4), (5, 6)}, where we identify natural numbers with their n-bit binary expansions. Then, the following multiset has the partition {0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4} 7−→ {(1, 2), (1, 2), (3, 4)} t {1, 1, 3, 3, 3} t {0} . Now define E(x) to be the number of pairs (with multiplicity) in x mod 2, i.e., the cardinality mod 2 of the first part of the partition in Equation (9.4.5). Also, define the singleton set of x as sing(x) := {e ∈ “singletons” : e occurs an odd number of times in “singletons” } . So, continuing our example, E(x) = 1 and sing(x) = {3}. We also define the type of x, type(x), to be the pairs with nonzero overlap with its singleton set: type(x) = {p ∈ M : p ∩ sing(x) 6= ∅}.   Continuing our example, we have that type (3, 5, 5, 5) = {(3, 4), (5, 6)}. We will also need a quantity on pairs of tuples x, y counting the number of elements in their singleton sets that are paired up, mod 2: P (x, y) = |{p ∈ M : p ⊆ sing(x) ∪ sing(y)}|  mod 2 .  So for example, keeping M as before, P (1, 3), (1, 4) = 1. Lemma 74. xCy if and only if type(x) = type(y) and E(x) + E(y) + P (x, y) = 1 mod 2 . 150 Proof. ( =⇒ ) Suppose by way of contradiction that xCy but type(x) 6= type(y). Then there exists a pair p = (a, b) such that (WLOG) a occurs in the “singletons” partition of x an even number of times, and a or b occurs in the “singletons” partition of y an odd number of times. This implies Lp + Up = 1 mod 2. So now suppose type(x) = type(y). Because xCy, we have |{p ∈ M : Lp = Up = 1}| = 1 mod 2. But |{p ∈ M : Lp = Up = 1}| mod 2 is precisely E(x) + E(y) + P (x, y) mod 2, because both quantities count the total number of pairs (with multiplicity) occurring in x ∪ y. ( ⇐= ) If type(x) = type(y), we must have that Lp + Up = 0 mod 2 for all p. The elements in the “pairs” partition do not affect this condition. For all pairs p = (a, b), either both a, b occur an even number of times in the “singletons” partition of x and y. Or, if (WLOG) a occurs an odd number of times in x, then a or b occurs an odd number of times in y, preserving the condition. Furthermore, as before, |{p ∈ M : Lp = Up = 1}| mod 2 counts the same quantity as E(x)+E(y)+P (x, y) mod 2. Therefore, if type(x) = type(y) and E(x) + E(y) + P (x, y) = 1 mod 2, we satisfy the criteria for compatibility. To understand the spectrum of the difference matrix we will make repeated use of the following structural fact about compatibility. Lemma 75. Let x, x0 , x00 be such that xCx00 and x0 Cx00 . Then for all y, xCy if and only if x0 Cy. Proof. By symmetry we need only argue the forward direction. Clearly type(x) = type(x0 ) = type(y). Note that for any z, z0 , z00 , we have P (z, z0 ) + P (z0 , z00 ) = P (z, z00 ) (mod 2), and because xCx00 and x0 Cx00 , Lemma 74 implies E(x) + P (x, x00 ) = E(x0 ) + P (x0 , x00 ) (mod 2). Then we have the following equivalences modulo 2: 1 ≡ E(x) + P (x, y) + E(y) ≡ E(x) + P (x, x0 ) + P (x00 , y) + E(y) ≡ E(x0 ) + P (x0 , x00 ) + P (x00 , y) + E(y) ≡ E(x0 ) + P (x0 , y) + E(y) . Appealing to Lemma 74, we conclude x0 Cy. (Lemma 74) 151 This implies, for example, that the xth and x0th rows in à are equal. We can view à as describing a graph Gà with vertices ({0, 1}n )t and edge set {(x, x0 ) ∈ V × V | xCx0 } . With Lemmas 74 and 75 in hand, we are prepared to describe the structure of Gà . It will be useful to define certain combinatorial quantities first. Definition 6 (Combinatorial quantities). Let Σ = Σodd t Σeven t Σrest be an alphabet of cardinality |Σ| = 2n partitioned such that Σodd consists of p pairs, so |Σodd | = 2p, and Σeven consists of m − p pairs, so |Σeven | = 2(m − p). Define T (t, p) as the number of strings of length t over Σ such that symbols from Σodd each occur an odd number of times, symbols from Σeven each occur an even number of times, and symbols from Σrest occur any number of times. Then define min{m,t} x1 (t) = X p=0 p odd min{m,t} ! m T (t, p) p and x2 (t) = X p=0 p even ! m T (t, p) . (9.4.6) p Further, define N (t, p) as the number of strings of length t over Σ such that for each pair in Σodd , one of the two symbols occurs an odd number of times while the other occurs an even number of times, and for each pair in Σeven , either both symbols occur an odd number of times or both symbols occur an even number of times. Pmin(t,m)   m Lemma 76 (Structure of Gà ). The graph Gà has exactly k=0 conk nected components, each associated with a specific type(·) of vertex. The connected component of Gà corresponding to the unique type of cardinality 0 (the empty type) is a complete bipartite graph (U, V, E) with parts U, V such that |U | = x1 (t) and |V | = x2 (t) .   For k ≥ 1, there are m connected components, each corresponding to k a type of cardinality k. All such components are complete bipartite graphs (U, V, E) with parts U, V such that |U |, |V | = N (t, k)/2 . Proof. First, by Lemma 74, only vertices of the same type can be compatible, Pmin(t,m) m and there are a total of k=0 types. For the remainder of the proof, k we only need to consider vertices of the same type. 152 We will analyze the case with k ≥ 1 first. Here k refers to the size of the type of the vertex. Consider a tuple x such that sing(x) = {a1 , a2 , . . . , ak−1 , ak } and E(x) = 0. Consider a second tuple x0 such that sing(x0 ) = {a1 , a2 , . . . , ak−1 , bk }, E(x0 ) = 0, and (ak , bk ) are a pair in M . Two such tuples must exist since k > 0. Note that x is compatible with x0 . Furthermore, all elements of the same type as x and x0 must be compatible with exactly one of x or x0 . To see that we form a complete bipartite graph, consider two elements y, y0 such that xCy and x0 Cy0 . By Lemma 75, we can conclude that yCy0 . Since y, y0 were arbitrary, we have a complete bipartite graph. To get the size of each of the two sets in the bipartition, fix a type of size k ≥ 1 and take (a1 , b1 ), . . . , (ak , bk ) to be the pairs representing the type. By our definition of type, we know: For each i, the elements ai and bi appear with differing parities; for each of the remaining pairs in M , the two elements of the pair occur with the same parity; and each remaining element, which does not belong to any pair in M , can occur with an arbitrary parity. Thus, the number of vertices with type k is exactly N (t, k). It remains to observe that the two components in the bipartition for type k are of equal size. To see this, take any tuple x of type k and w.l.o.g. suppose that a1 occurs with odd parity in sing(x). (Otherwise, b1 occurs with odd parity in sing(x) and the remaining argument is easily modified.) If we construct a tuple x0 from x by replacing as many occurrences of a1 as are in sing(x) with b1 (while keeping the remaining elements the same), then xCx0 by Lemma 74. This provides us with a one-to-one mapping between the two components in the bipartition, thus they are of equal size. For k = 0, we instead consider two distinct vertices x and x0 with E(x) = 0 and E(x0 ) = 1 and |sing(x)| = |sing(x0 )| = 0. Note that x and x0 are compatible. By an analogous argument to the case of k > 0, we must have all vertices connected to either x and x0 , and we get a complete bipartite graph. For the sizes of the sets in the bipartition, note that the number of vertices connected to x with E(x) = 0 will be precisely x1 (t) and the number of vertices connected to x0 with E(x0 ) = 1 will be precisely x2 (t), completing our proof. As a consequence of this lemma, when suitably ordering the indexing tuples, the matrix à has a block-diagonal structure with blocks corresponding to the connected components of Gà . Thus, to determine the spectrum of Ã, it suffices to determine the spectrum of each block. 153 The adjacency matrix of a complete bipartite graph between a- and b-many √ vertices has two nonzero eigenvalues, each of magnitude ab [Bol98, Chapter VIII.2]. Instantiating this fact in the context of Lemma 76 and renormalizing by 1/2nt−1 (recall A = (DÃD−1 )/2nt−1 ), we obtain the following bound on kAk1 . Corollary 77. The 1-norm of the matrix A satisfies: kAk1 = 2 q 2nt−1 x1 (t) · x2 (t) + min{t,m} 2 ! m N (t, k) . k 2 X 2nt−1 k=1 Difference matrix: the trace norm Here we analyze the growth of kAk1 in terms of ε, t, and n. We begin by using exponential generating functions (or EGFs—see [Wil94] for background) to derive explicit expressions for T (t, p), N (t, p), and x1 (t) + x2 (t). Lemma 78. Let T (t, p) be as in Definition 6. Then  2m 2m−2p 2p X X 1 T (t, p) = 2 j=0 2m − 2p j k (−1) k=0 ! ! 2p (2n − 2j − 2k)t . k Proof. For readability let us use a for the number of symbols occurring an even number of times, b for the number of symbols occurring an odd number of times and c for the rest. Then T (t, p) has EGF f (x) = ex + e−x 2 !a ex − e−x 2 !b  ex c . Rearranging, we find  a+b X a X b 1 f (x) = 2 a j j=0 k=0 ! ! b (−1)k e(a−2j+b−2k+c)x . k And we assume c > a + b, so we use that the EGF eηx corresponds to the sequence {η t }∞ t=0 , read off the relevant coefficient to derive the formula for T (t, p):  a+b X a X b 1 T (t, p) = 2 j=0 k=0 a j ! ! b (−1)k (a − 2j + b − 2k + c)t . k Substituting for a, b, c yields the result. Lemma 79. With x1 and x2 defined as in Equation (9.4.6),  m X m 1 x1 (t) + x2 (t) = 2 k=0 ! m (2n − 4k)t . k 154 Proof. The combinatorial interpretation of x1 (t)+x2 (t) is the number of strings of length t over Σ where symbols from A t B, |A t B| = 2m are paired up and within each pair, they must appear with the same parity. The EGF for strings with 2 elements appearing with same parity is ex + e−x 2 !2 ex − e−x + 2 !2 e2x + e−2x = . 2 To construct the desired strings, we combine m copies of this with 2n − m copies of the unrestricted EGF, leading to the overall EGF  2−m e2x + e−2x m n e(2 −2m)x . The result follows from simplifying this EGF and recognizing the related counting formula. Lemma 80. Let N (t, p) be as in Definition 6. Then  m X p m−p X 1 N (t, p) = 2 (−1) j=0 k=0 j p j ! ! m−p (2n − 4j − 4k)t . k Proof. The proof uses EGFs analogously to how we derived the expression for T (t, p). The EGF for N (t, p) is f (x) = e2x − e−2x 2 !p e2x + e−2x 2 !m−p  ex 2n −2m . We can now rearrange this product of sums into a sum of products and, using again that the EGF eηx corresponds to the sequence {η t }∞ t=0 , read off the expression for N (t, p). Lemma 81. With x1 and x2 defined as in Equation (9.4.6), X 1 min{t,m} m N (t, k) 1 x1 (t) + x2 (t) = 1 − . 2nt k=1 k 2 2 2nt ! ! Pmin{t,m}   m Proof. First, observe that k=0 N (t, k) = 2nt , since every one of the k overall 2nt tuples belongs to some type and we are summing over all sizes of types. Consequently, X 1 min{t,m} m N (t, k) 1 1 m N (t, 0) = − nt 2 k 2 2 2 0 2nt k=1 ! ! m 1 1 X m 4k = − m+1 1− n 2 2 2 k=0 k 1 1 x1 (t) + x2 (t) = − , 2 2 2nt ! !t 155 where the second-to-last step used Lemma 80 and the last step used Lemma 79. We now further study the trace norm of A. We will need the following technical fact. Proposition 3. Suppose m = |M | = ε2n and ε = ε(n) < 1. Let D ∈ R be a fixed constant. Then, for any t = t(n) ≤ O(2bn ) with 0 ≤ b < 14 , we have DK E 1− n K∼B(m,1/2) 2  t Dm = 1 − n2 2 !t + o(1) . In Proposition 3 and what follows, B(`, 1/2) denotes the Binomial distribution with ` trials and success probability 1/2. Proof. Notice that, for any t ∈ N, the function f : [0, B) → R given by f (x) = (1 − x)t is Lipschitz with Lipschitz constant t max0≤x≤B |1 − x|t−1 . Using that |1 − Dk/2n | ≤ 1 holds for all 0 ≤ k ≤ m for sufficiently large n, this implies  E K∼B(m,1/2) DK 1− n 2 t Dm − 1 − n2 2 !t DK D m2 ≤ E t − n K∼B(m,1/2) 2n 2 ! . From Chernoff we have that 2 DK D m2 n η Pr − ≥ η ≤ 2 exp −const · 2 · . 2n 2n ε " # ! Call the low-probability event above E. Then DK D m2 E t − n K∼B(m,1/2) 2n 2 ! ≤ t Pr[E c ]η + t Pr[E] D m2 2n ! η2 ≤ tη + 2tDε exp −const · 2n · . ε We can set η = min{1/n, 1/t2 }, then, because of our assumption on t = t(n), both summands go to 0 as n → ∞, finishing the proof.   Theorem 82. kAk1 = Ω(1) for t = Ω ε(n)−1 , assuming ε = ε(n) ≥ Ω(2−bn ) with 0 ≤ b < 14 . Proof. Let us label the expression for kAk1 from Corollary 77 as ! X 2 q 2 min{t,m} m N (t, k) kAk1 = nt−1 x1 (t) · x2 (t) + nt−1 . 2 2 k 2 k=1 | {z (∗) } (9.4.7) 156 We will lower-bound kAk1 by lower-bounding the second summand here, (∗). Lemma 79 implies  m X m 1 x1 (t) + x2 (t) = 2 k=0 ! m (2n − 4k)t = E (2n − 4K)t , K∼B(m,1/2) k (9.4.8) where B(m, 1/2) refers to the Binomial distribution. Returning to (∗) and using Lemma 81, we have 4 x1 (t) + x2 (t) (∗) = 1− 2 2nt ! 4K =2 1−E 1− n 2  t ! . Using Proposition 3, we obtain that, as long as t ≤ O(2bn ) with 0 ≤ b < 14 , 2m (∗) = 2 1 − 1 − n 2  t !   + o(1) = 2 1 − (1 − 2ε)t + o(1) . 1 Notice that our assumption on ε = ε(n) ensures that ε(n) ≤ O(2bn ) with 0 ≤ b < 14 . Therefore, we can consider t ≥ Ω(1/ε), we get 1 − (1 − 2ε)t ≥ Ω(1), and therefore kAk1 ≥ (∗) ≥ Ω(1). Remark 3. We can extend the above quantum distinguishability analysis to the ensembles from [Bla24]. The construction in [Bla24], based on Talagrands √ random DNFs [Tal96], establishes a lower bound of exp(Ω( n/ε)) for passive classical monotonicity testing via a birthday paradox argument. The construction randomly selects DNF terms of fixed width to define a partial partition of the Boolean cube into disjoint sets Uj such that any two points in different Uj are incomparable. The difference between the monotone Dyes and non-monotone Dno case lies in the function value assignments: in Dyes , values within each disjoint set Uj are structured monotonically while in Dno , values with each Uj are randomly assigned. Classically, distinguishing these √ distributions requires exp(Ω( 2n /ε)) samples, as a tester must sample at least two points from the same Uj to gain information. This leads to an exponential lower bound when parameters are chosen appropriately. Following an argument structured similarly to the one above, one may see that the difference matrix between the induced function state ensembles in the quantum setting decomposes into blocks corresponding to complete multipartite graphs. To see this, in analogy to the analysis from Section 9.4, we can define a notion of compatibility between any two index tuples. Given a collection of sets Uj and an index tuple x, we first remove duplicates from the tuple 157 and analyze its intersection pattern with each Uj . For instance, if U1 = {1, 2}, U2 = {3, 4}, and U3 = {5, 6}, and our index tuple is (1, 1, 1, 2, 3, 3, 3, 3, 5), then after removing duplicate, the corresponding subsets under the Uj sets are [1, 2], [ ], [5]. Two tuples are said to be compatible if, for every j, the number of elements from each Uj that appear in the tuple is even but not identical across the tuples after removing duplicates and decomposing. For example, the tuple (1, 1, 1, 2, 3, 3, 3, 3, 5) is compatible with (1, 1, 1, 2, 3, 3, 3, 3, 6) but not with (1, 2, 3, 3, 3, 3, 3, 3, 5), as the latter shares the same decomposition as the original. Importantly, this compatibility is transitive: if x is compatible with x0 and if x0 is compatible with x00 , then x is also compatible with x00 . This transitivity induces complete multipartite graph blocks with each block corresponding to a compatibility class. Therefore, the trace distance between the two ensembles equals the sum of the trace norms of multipartite graphs of various sizes. Bounds on the eigenvalues of such a graph in terms of the sizes of its parts can be found in, for example, [EH80; Meh23]. Also in analogy to our analysis of the [Gol+00] construction, we find the ensembles remain distinguishable when t = Ω(1/ε), and thus they achieve no improvement over the generic lower bound. Acknowledgments The authors thank Srinivasan Arunachalam, Fernando Jeronimo, Kyle Gulshen, Jiaqing Jiang, John Bostanci, Yeongwoo Hwang, Shivam Nadimpali, John Preskill, Akshar Ramkumar, Abdulrahman Sahmoud, Mehdi Soleimanifar, and Thomas Vidick for enlightening discussions. Moreover, we thank Nathan Harms for suggesting we look at monotonicity and general discussions on classical bounds for related problems. We thank Fermi Ma for helpful discussions that provided inspiration for Theorem 70. We are grateful to Francisco Escudero Gutiérrez for pointing us to the Fourier-analytic characterization of monotonicity which allowed us to improve a previous passive quantum monotonicity tester. Finally, we thank the anonymous STOC 2025 reviewers for helpful feedback. MCC was partially supported by a DAAD PRIME fellowship and by the BMBF (QPIC-1). JS is funded by Chris Umans’ Simons Foundation Investigator Grant. Parts of contributions of this chapter were completed while JS was visiting the Simons Institute for the Theory of Computing, supported by DOE QSA grant #FP00010905. 158 9.5 Appendix: Useful facts In this appendix we collect some simple lemmas that are used as subroutines in the main body. First, we make a simple observation about the possibility of post-selecting on a desired function value in a subset function state. Lemma 83. Let m ∈ N, S ⊆ {0, 1}n , f : {0, 1}n → {0, 1}, b ∈ {0, 1}, η ∈ (0, 1], and δ ∈ (0, 1). Assume that Prx∼S [f (x) = b] ≥ η. There is an efficient quantum algorithm that given md ln(m/δ) e many copies of the state η P 1 |ΨS,f i = p x∈S |x, f (x)i, outputs, with success probability ≥ 1 − δ, at least |S| m many copies of the state |ΨS,f,b i ∝ x∈S:f (x)=b |xi. Moreover, if the algorithm fails, then the algorithm explicitly outputs FAIL. P We note that via standard Chernoff-Hoeffding bounds, one can achieve the same guarantee as in Lemma 83 using max{d2m/ηe, d2 ln(1/δ)/η 2 e} many copies of the state |ΨS,f i. This improves the m-dependence, but in general comes at the cost of a worse η-dependence. δ̃) Proof. By a union bound, it suffices to show that m̃ = d ln(1/ e many copies of η |ΨS,f i suffice to quantumly efficiently obtain one copy of |ΨS,f,b i with success probability ≥ 1 − δ̃. So, let’s assume m = 1. Also here the procedure is clear: For each copy of |ΨS,f i, measure the last qubit in the computational basis. If the outcome is b ∈ {0, 1}, then the post-measurement state after discarding the last qubit is |ΨS,f,b i. If, after measuring on all the copies, outcome b has never been observed, output FAIL. Otherwise, output one copy one of the post-measurement states from rounds in which outcome b was observed. Again, the analysis of the failure probability is simple: Pr[FAIL] = Pr[outcome b never occurs] ≤ (1 − η)m̃ ≤ exp(−η m̃) ≤ δ̃ . Here, we used the assumption Prx∼S [f (x) = b] ≥ η and our choice of m̃. We also require the following standard routine for estimating the overlap of two pure quantum states. Lemma 84. Let ε, δ ∈ (0, 1). There is an efficient quantum algorithm that, given d 2 ln(2/δ) e many copies of each of two pure quantum states |ψi and |φi, ε4 159 outputs, with success probability ≥ 1 − δ, an ε-accurate estimate (in [0, 1]) of the (non-squared) overlap | hφ|ψi |. Proof. The procedure is as follows: Let m = d 2 ln(2/δ) e be the number of copies ε4 that are available for each of |ψi and |φi. For 1 ≤ i ≤ m, perform a SWAP test between one copy of |ψi and one copy of |φi, let the outcome be ôi . Define q P ô = m1 m ô and output the estimate µ̂ = 2(ô − 12 ). Let us analyze the i=1 i success probability of this procedure. First, notice that each SWAP test accepts (i.e., outputs 1) with probability 1+|hφ|ψi|2 1+|hφ|ψi|2 [BCWD01]. Thus, the ô are i.i.d. Bernoulli( ) random varii 2 2 2 ables. So, by a standard Chernoff-Hoeffding bound, we have ô − 1+|hφ|ψi| ≤ 2 2 4 ε /2 with probability ≥ 1 − 2 exp(−mε /2) ≥ 1 − δ, by our choice of m. q √ √ As | x − y| ≤ |x − y| holds for all x, y ≥ 0, this implies that also |µ̂ − | hφ|ψi || = as desired. q 2(ô − 12 ) − q 2 2( 1+|hφ|ψi| − 12 ) ≤ ε, with probability ≥ 1 − δ, 2 Finally, using the overlap estimation routine, one can start from a function state and two function subset states to estimate the probability of an input lying in both Corollary 85. Let S, S 0 ⊆ {0, 1}n , f, f 0 : {0, 1}n → {0, 1}, and b, b0 ∈ {0, 1}. Let ε, δ ∈ (0, 1). There is an efficient quantum algorithm that, given d 162 ln(6/δ) e ε4 162 ln(6/δ) 0 many copies of each of the states |Ψi, |Ψ i, and 2d ε4 e copies of each of the states |ΨS,f,b i and |ΨS 0 ,f 0 ,b0 i, outputs, with success probability ≥ 1 − δ, an ε-accurate estimate of Prx∼{0,1}n [x ∈ S ∩ S 0 , f (x) = b, f 0 (x) = b0 ]. Proof. The procedure combines the ingredients developed above. To do so, notice the following equalities: s |S ∩ f −1 (b)| , 2n s |S 0 ∩ f 0−1 (b0 )| , 2n s |S ∩ f −1 (b) ∩ S 0 ∩ f 0−1 (b0 )|2 , |S ∩ f −1 (b)| · |S 0 ∩ f 0−1 (b0 )| s |S ∩ f −1 (b) ∩ S 0 ∩ f 0−1 (b0 )|2 |S ∩ f −1 (b)| |S 0 ∩ f 0−1 (b0 )| · · . |S ∩ f −1 (b)| · |S 0 ∩ f 0−1 (b0 )| 2n 2n |hΨ|ΨS,f,b i| = hΨ|ΨS,f,b i = hΨ0 |ΨS 0 ,f 0 ,b0 i = hΨ0 |ΨS 0 ,f 0 ,b0 i = hΨS,f,b |ΨS 0 ,f 0 ,b0 i = hΨS,f,b |ΨS 0 ,f 0 ,b0 i = |S ∩ f −1 (b) ∩ S 0 ∩ f 0−1 (b0 )| = 2n So, we can estimate Prx∼{0,1}n [x ∈ S∩S 0 , f (x) = b, f 0 (x) = b0 ] = |S∩f as follows: −1 (b)∩S 0 ∩f 0−1 (b0 )| 2n 160 1. Via the procedure in Lemma 84, use d 162 ln(6/δ) e many copies of each of |Ψi ε4 and |ΨS,f,b i to output, with success probability ≥ 1− 3δ , an (ε/3)-accurate estimate α̂ of | hΨ|ΨS,f,b i |. 2. Via the procedure in Lemma 84, use d 162 ln(6/δ) e many copies of each ε4 of |Ψ0 i and |ΨS 0 ,f 0 ,b0 i to output, with success probability ≥ 1 − 3δ , an (ε/3)-accurate estimate α̂0 of | hΨ0 |ΨS 0 ,f 0 ,b0 i |. 3. Via the procedure in Lemma 84, use d 162 ln(6/δ) e many copies of each of ε4 |ΨS,f,b i and |ΨS 0 ,f 0 ,b0 i to output, with success probability ≥ 1 − 3δ , an (ε/3)-accurate estimate β̂ of | hΨS,f,b |ΨS 0 ,f 0 ,b0 i |. 4. Output the estimate γ̂ = β̂ α̂α̂0 . By a union bound, the probability that Steps 1-3 all succeed is ≥ 1 − δ. In this case, we have γ̂ − Pr x∼{0,1}n [x ∈ S ∩ S 0 , f (x) = b, f 0 (x) = b0 ] s = β̂ α̂α̂0 − s ≤ β̂ − ≤3· |S ∩ f −1 (b) ∩ S 0 ∩ f 0−1 (b0 )|2 |S ∩ f −1 (b)| |S 0 ∩ f 0−1 (b0 )| · · |S ∩ f −1 (b)| · |S 0 ∩ f 0−1 (b0 )| 2n 2n |S ∩ f −1 (b) ∩ S 0 ∩ f 0−1 (b0 )|2 + α̂ − |S ∩ f −1 (b)| · |S 0 ∩ f 0−1 (b0 )| s |S ∩ f −1 (b)| + α̂0 − 2n s |S 0 ∩ f 0−1 (b0 )| 2n ε 3 = ε. 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