Toroidal Gibbs sampling using quantum and classical techniques

Abstract

Sampling from a Gibbs distribution for some energy function at some inverse temperature is a fundamental task in statistical physics, chemistry, and machine learning. However, it is often slow in practice for multimodal or non-convex energy landscapes. Gibbs sampling has been widely explored for discrete variable problems in statistical mechanics and machine learning, and also for continuous variable problems, where it is typically implemented by simulating Langevin dynamics, a stochastic differential equation (SDE) whose stationary distribution is the target Gibbs density. Langevin dynamics can be equivalently represented by a partial differential equation called the Fokker-Planck equation. This thesis presents two approaches to accelerate Gibbs sampling on a continuous domain on a hypertorus, using a classical technique inspired by Langevin dynamics and a quantum algorithm which solves the Fokker-Planck equation.

First, we introduce diffusion-warmed MCMC, a method that uses a diffusion model trained on small lattices of the two-dimensional XY model to generate warm-start samples for larger lattices. We show that combining the diffusion generated samples with a small number of Wolff MCMC steps reduces thermalization time by approximately an order of magnitude compared to standard MCMC, across a range of system sizes and temperatures spanning the phase transition.

Secondly, we develop a quantum algorithm to do Gibbs sampling (by solving the Fokker-Planck equation) for some class of Gibbs distributions on a hypertorus based on Quantum Singular Value Transformation (QSVT), extending previous work that required a warm start. We remove the warm start by constructing a temperature annealing schedule such that the full algorithm has constant success probability. We then establish a quantum-classical separation result for Gibbs sampling on the hypertorus under suitable regularity conditions, by showing that the upper bound query complexity of the quantum algorithm is lower than that of any possible classical algorithm, using a lower bound in query complexity proved by our collaborators. This establishes a quantum-classical separation result for continuous Gibbs sampling, which is, to the best of our knowledge, the first such result for this setting and has broad implications in quantum algorithms.