Concise and Efficient Quantum Algorithms for Distribution Closeness Testing

TL;DR

This paper introduces the most efficient quantum algorithms to test whether two unknown distributions are close or far apart, improving on previous methods in complexity and simplicity.

Contribution

It presents new quantum algorithms for distribution closeness testing that outperform prior algorithms in complexity and conciseness, using advanced quantum techniques.

Findings

Lower complexity than previous algorithms

More concise algorithms based on quantum singular value transformation

Effective for both $l^1$-distance and $l^2$-distance metrics

Abstract

We study the impact of quantum computation on the fundamental problem of testing the property of distributions. In particular, we focus on testing whether two unknown classical distributions are close or far enough, and propose the currently best quantum algorithms for this problem under the metrics of $l^1$-distance and $l^2$-distance. Compared with the latest results given in \cite{gilyen2019distributional} which relied on the technique of quantum singular value transformation (QSVT), our algorithms not only have lower complexity, but also are more concise.