Quantum speedup of Monte Carlo methods

TL;DR

This paper introduces a quantum algorithm that significantly accelerates Monte Carlo methods, including Markov chain techniques for partition functions, achieving near-quadratic speedups over classical algorithms.

Contribution

It presents a general quantum algorithm for estimating expected values of randomised or quantum subroutines with bounded variance, improving classical Monte Carlo efficiency.

Findings

Achieves near-quadratic speedup over classical algorithms

Provides rigorous performance bounds for quantum-enhanced partition function computation

Enables efficient estimation of total variation distance between distributions

Abstract

Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomised or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.