Quantum Gibbs Sampling Using Szegedy Operators

TL;DR

This paper introduces a quantum algorithm for Gibbs sampling that leverages Szegedy operators and phase estimation, achieving quadratic speedup over classical methods in terms of convergence rate.

Contribution

The paper presents a novel quantum Gibbs sampling algorithm combining phase estimation and Grover's algorithm, improving efficiency over classical approaches.

Findings

Quantum Gibbs sampling runs in ${ m O}(1/\sqrt{\delta})$ steps.

Achieves ${ m O}(\epsilon)$ precision in sampling.

Provides quadratic speedup compared to classical Gibbs sampling.

Abstract

We present an algorithm for doing Gibbs sampling on a quantum computer. The algorithm combines phase estimation for a Szegedy operator, and Grover's algorithm. For any $\epsilon>0$, the algorithm will sample a probability distribution in ${\cal O}(\frac{1}{\sqrt{\delta}})$ steps with precision ${\cal O}(\epsilon)$. Here $\delta$ is the distance between the two largest eigenvalue magnitudes of the transition matrix of the Gibbs Markov chain used in the algorithm. It takes ${\cal O}(\frac{1}{\delta})$ steps to achieve the same precision if one does Gibbs sampling on a classical computer.