Aspects of Topology and Measurement in Quantum Lattice Systems

Author: Artymowicz, Adam Markus

Year: 2025

Degree: Dissertation (Ph.D.)

Advisor: Kapustin, Anton N.

Committee Members: Kitaev, Alexei; Kapustin, Anton N.; Marcolli, Matilde; Ranard, Daniel

Option: Mathematics

DOI: 10.7907/jacd-t049

Abstract

In the first part of this thesis, topological invariants of gapped phases on the lattice are studied. These include the Berry curvature, Thouless pump, the Hall conductance, and their higher-dimensional analogs. These invariants are proven to obstruct the promotion of a global symmetry to a gauge symmetry. Two of these invariants, the 1d higher Berry curvature and the 2d higher Thouless pump, are studied in detail. First, it is shown that they are related by a relation involving flux insertion, which can be interpreted physically as identifying the higher Thouless pump invariant with the excess Berry curvature of a fluxon. Second, it is proven that these two invariants take on quantized values in an invertible state.

In the second part of this thesis, an algorithm is presented for learning Hamiltonian parameters from local expectation values of its Gibbs state via a local free-energy variational principle. The algorithm is benchmarked on the problem of black-box learning of a nearest-neighbour Hamiltonian in a 100-qubit spin chain, giving evidence of favourable scaling with system size. The theoretical analysis is then extended to incorporate measurement noise, as well as equipping the algorithm with certified a posteriori lower and upper error bounds on the inferred parameters. For commuting Hamiltonians, a priori convergence guarantees are also established.

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