Mixing time of quantum Gibbs sampling for random sparse Hamiltonians

TL;DR

This paper proves a polylogarithmic upper bound on the mixing time of a quantum Gibbs sampling algorithm for certain random sparse Hamiltonians, advancing understanding of quantum state preparation efficiency.

Contribution

It establishes a rigorous polylog(n) bound on the mixing time for quantum Gibbs sampling of random sparse Hamiltonians at any constant temperature.

Findings

Polylogarithmic mixing time bound for random sparse Hamiltonians.

Analysis of how spectral properties affect mixing time.

Comparison showing efficiency similar to other quantum state preparation algorithms.

Abstract

Providing evidence that quantum computers can efficiently prepare low-energy or thermal states of physically relevant interacting quantum systems is a major challenge in quantum information science. A newly developed quantum Gibbs sampling algorithm by Chen, Kastoryano, and Gily\'en provides an efficient simulation of the detailed-balanced dissipative dynamics of non-commutative quantum systems. The running time of this algorithm depends on the mixing time of the corresponding quantum Markov chain, which has not been rigorously bounded except in the high-temperature regime. In this work, we establish a polylog(n) upper bound on its mixing time for various families of random n by n sparse Hamiltonians at any constant temperature. We further analyze how the choice of the jump operators for the algorithm and the spectral properties of these sparse Hamiltonians influence the mixing time. Our result places this method for Gibbs sampling on par with other efficient algorithms for preparing low-energy states of quantumly easy Hamiltonians.