Quantum Metropolis Sampling via Weak Measurement

TL;DR

This paper introduces a simple, provably correct quantum Gibbs sampler using weak measurement and Boosted QPE, avoiding complex techniques like Marriott-Watrous rewinding, and providing an alternative to existing methods based on Davies generators.

Contribution

It develops a new quantum Gibbs sampling algorithm that simplifies previous approaches by removing the need for shift-invariance and rewinding techniques, using weak measurement and Boosted QPE.

Findings

Provides a provably correct quantum Gibbs sampler.

Avoids complex techniques like Marriott-Watrous rewinding.

Uses weak measurement and median-based Boosted QPE.

Abstract

Gibbs sampling is a crucial computational technique used in physics, statistics, and many other scientific fields. For classical Hamiltonians, the most commonly used Gibbs sampler is the Metropolis algorithm, known for having the Gibbs state as its unique fixed point. For quantum Hamiltonians, designing provably correct Gibbs samplers has been more challenging. [TOV+11] introduced a novel method that uses quantum phase estimation (QPE) and the Marriot-Watrous rewinding technique to mimic the classical Metropolis algorithm for quantum Hamiltonians. The analysis of their algorithm relies upon the use of a boosted and shift-invariant version of QPE which may not exist [CKBG23]. Recent efforts to design quantum Gibbs samplers take a very different approach and are based on simulating Davies generators [CKBG23,CKG23,RWW23,DLL24]. Currently, these are the only provably correct Gibbs samplers for quantum Hamiltonians. We revisit the inspiration for the Metropolis-style algorithm of [TOV+11] and incorporate weak measurement to design a conceptually simple and provably correct quantum Gibbs sampler, with the Gibbs state as its approximate unique fixed point. Our method uses a Boosted QPE which takes the median of multiple runs of QPE, but we do not require the shift-invariant property. In addition, we do not use the Marriott-Watrous rewinding technique which simplifies the algorithm significantly.