Using Gaussian Boson Samplers to Approximate Gaussian Expectation Problems

TL;DR

This paper demonstrates that Gaussian Boson Sampling can be used to efficiently approximate Gaussian expectation problems, offering exponential speedups over traditional Monte Carlo methods.

Contribution

The study introduces two GBS-based estimators for Gaussian expectations, proving their exponential speedup over Monte Carlo in certain problem spaces.

Findings

GBS estimators outperform Monte Carlo in sample efficiency

Exponential speedup achieved for specific Gaussian expectation problems

Identifies a subset of problems where GBS provides computational advantage

Abstract

Gaussian Boson Sampling (GBS) have shown advantages over classical methods for performing some specific sampling tasks. To fully harness the computational power of GBS, there has been great interest in identifying their practical applications. In this study, we explore the use of GBS samples for computing a numerical approximation to the Gaussian expectation problem, that is to integrate a multivariate function against a Gaussian distribution. We propose two estimators using GBS samples, and show that they both can bring an exponential speedup over the plain Monte Carlo (MC) estimator. Precisely speaking, the exponential speedup is defined in terms of the guaranteed sample size for these estimators to reach the same level of accuracy $\epsilon$ and the same success probability $\delta$ in the $(\epsilon, \delta)$ multiplicative error approximation scheme. We prove that there is an open and nonempty subset of the Gaussian expectation problem space for such computational advantage.