Quasi-optimal sampling from Gibbs states via non-commutative optimal transport metrics

TL;DR

This paper introduces a novel approach using non-commutative optimal transport metrics to efficiently sample and prepare quantum Gibbs states of local commuting Hamiltonians, extending previous results to broader interaction ranges.

Contribution

It is the first to apply a non-commutative transport metric to quantum dynamics and links clustering conditions to quasi-rapid mixing in quantum systems.

Findings

Quantum Gibbs states satisfying MCMI decay can be quasi-optimally prepared.

The approach extends rapid mixing results to interactions beyond nearest neighbors.

Quantum CSS codes at high temperature satisfy MCMI decay and are efficiently preparable.

Abstract

We study the problem of sampling from and preparing quantum Gibbs states of local commuting Hamiltonians on hypercubic lattices of arbitrary dimension. We prove that any such Gibbs state which satisfies a clustering condition that we coin decay of matrix-valued quantum conditional mutual information (MCMI) can be quasi-optimally prepared on a quantum computer. We do this by controlling the mixing time of the corresponding Davies evolution in a normalized quantum Wasserstein distance of order one. To the best of our knowledge, this is the first time that such a non-commutative transport metric has been used in the study of quantum dynamics, and the first time quasi-rapid mixing is implied by solely an explicit clustering condition. Our result is based on a weak approximate tensorization and a weak modified logarithmic Sobolev inequality for such systems, as well as a new general weak transport cost inequality. If we furthermore assume a constraint on the local gap of the thermalizing dynamics, we obtain rapid mixing in trace distance for interactions beyond the range of two, thereby extending the state-of-the-art results that only cover the nearest neighbor case. We conclude by showing that systems that admit effective local Hamiltonians, like quantum CSS codes at high temperature, satisfy this MCMI decay and can thus be efficiently prepared and sampled from.