Quantum algorithms for multivariate Monte Carlo estimation

TL;DR

This paper develops quantum algorithms that significantly speed up multivariate Monte Carlo estimations, achieving quadratic improvements in precision but with potential dimension-related slowdowns, impacting applications like machine learning.

Contribution

The paper introduces quantum algorithms for multivariate Monte Carlo estimation that are asymptotically optimal and demonstrate a quadratic speed-up in precision over classical methods.

Findings

Quadratic speed-up in query complexity for precision

Optimal asymptotic complexity of the quantum algorithms

Trade-off between speed-up and dimension-related slowdown

Abstract

We consider the problem of estimating the expected outcomes of Monte Carlo processes whose outputs are described by multidimensional random variables. We tightly characterize the quantum query complexity of this problem for various choices of oracle access to the Monte Carlo process and various normalization conditions on its output variables and the estimates to be made. More specifically, we design quantum algorithms that provide a quadratic speed-up in query complexity with respect to the precision of the resulting estimates and show that these complexities are asymptotically optimal in each of our scenarios. Interestingly, we find that these speed-ups with respect to the precision come at the cost of a (up to exponential) slowdown with respect to the dimension of the Monte Carlo random variables, for most of the parameter settings we consider. We also describe several applications of our algorithms, notably in the field of machine learning, and discuss the implications of our results.