Operator-Level Quantum Acceleration of Non-Logconcave Sampling

TL;DR

This paper presents the first quantum algorithms that significantly accelerate sampling from complex, non-logconcave distributions, including Langevin dynamics and replica exchange methods, achieving up to quartic speedups over classical approaches.

Contribution

The authors develop novel quantum algorithms that encode Gibbs measures into quantum states and accelerate Langevin-based sampling methods for non-logconcave distributions.

Findings

Achieves up to quartic quantum speedup over classical Langevin methods.

Introduces quantum encoding of Gibbs measures via the Witten Laplacian.

Develops quantum acceleration for replica exchange Langevin diffusion.

Abstract

Sampling from probability distributions of the form $\sigma \propto e^{-\beta V}$, where $V$ is a continuous potential, is a fundamental task across physics, chemistry, biology, computer science, and statistics. However, when $V$ is non-convex, the resulting distribution becomes non-logconcave, and classical methods such as Langevin dynamics often exhibit poor performance. We introduce the first quantum algorithm that provably accelerates a broad class of continuous-time sampling dynamics. For Langevin dynamics, our method encodes the target Gibbs measure into the amplitudes of a quantum state, identified as the kernel of a block matrix derived from a factorization of the Witten Laplacian operator. This connection enables Gibbs sampling via singular value thresholding and yields up to a quartic quantum speedup over best-known classical Langevin-based methods in the non-logconcave setting. Building on this framework, we further develop the first quantum algorithm that accelerates replica exchange Langevin diffusion, a widely used method for sampling from complex, rugged energy landscapes.