# Operator-level quantum acceleration of non-logconcave sampling

## TL;DR

The paper introduces a quantum algorithm that accelerates sampling from non-logconcave distributions, offering a significant speedup over classical methods.

## Contribution

A novel quantum framework is introduced that accelerates sampling processes without relying on classical discretization techniques.

## Key findings

- The quantum algorithm provides up to a quartic speedup over classical Langevin-based methods for non-logconcave sampling.
- A new quantum algorithm is developed for replica exchange Langevin diffusion, enabling efficient sampling from complex energy landscapes.

## Abstract

Sampling from Gibbs distri-butions under continuous, often nonconvex, potentials is a fundamental challenge, which hinders classical methods such as the Langevin dynamics. Existing quantum approaches primarily rely on quantum walks to accelerate classical sampling algorithms, which limits the design space of quantum algorithms to the choice of classical counterparts and inherits technical difficulties in error analysis due to temporal discretization. We introduce a versatile framework for accelerating general sampling processes using quantum computers. In particular, our quantum algorithm is constructed at the operator level without relying on time discretization, thereby providing a path for quantizing continuous-time processes.

Sampling from probability distributions of the form σ∝e−βV, where V is a continuous potential, is a fundamental task across physics, chemistry, biology, computer science, and statistics. However, when V is nonconvex, the resulting distribution becomes non-logconcave, and classical methods such as Langevin dynamics often exhibit poor performance. We introduce a 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.