Gibbs Sampling of Continuous Potentials on a Quantum Computer

TL;DR

This paper presents a quantum algorithm leveraging Fourier transforms to efficiently sample from continuous Gibbs distributions, especially for periodic functions, offering exponential precision improvements and polynomial speedups in certain cases.

Contribution

It introduces a novel quantum approach for Gibbs sampling of continuous functions using Fourier transforms and differential equation solvers, with conditions for efficiency based on Fourier coefficient decay.

Findings

Achieves exponential precision in sampling from Gibbs distributions.

Provides polynomial quantum speedups in mean estimation.

Identifies conditions under which the algorithm is most effective.

Abstract

Gibbs sampling from continuous real-valued functions is a challenging problem of interest in machine learning. Here we leverage quantum Fourier transforms to build a quantum algorithm for this task when the function is periodic. We use the quantum algorithms for solving linear ordinary differential equations to solve the Fokker--Planck equation and prepare a quantum state encoding the Gibbs distribution. We show that the efficiency of interpolation and differentiation of these functions on a quantum computer depends on the rate of decay of the Fourier coefficients of the Fourier transform of the function. We view this property as a concentration of measure in the Fourier domain, and also provide functional analytic conditions for it. Our algorithm makes zeroeth order queries to a quantum oracle of the function. Despite suffering from an exponentially long mixing time, this algorithm allows for exponentially improved precision in sampling, and polynomial quantum speedups in mean estimation in the general case, and particularly under geometric conditions we identify for the critical points of the energy function.