Succinct quantum testers for closeness and $k$-wise uniformity of probability distributions

TL;DR

This paper presents new quantum algorithms that significantly speed up the testing of distribution closeness and $k$-wise uniformity, achieving optimal or near-optimal query complexities and outperforming classical methods.

Contribution

The authors develop the first quantum algorithms for $k$-wise uniformity testing with quadratic speedup and improve quantum bounds for closeness testing, using simple, efficient quantum subroutines.

Findings

Quantum algorithms for $ ext{l}^1$- and $ ext{l}^2$-closeness testing with optimal $ ilde{O}(rac{ ext{sqrt}(n)}{ ext{epsilon}})$ and $ ilde{O}(rac{1}{ ext{epsilon}})$ complexities.

First quantum algorithm for $k$-wise uniformity testing with $ ilde{O}(rac{ ext{sqrt}(n^k)}{ ext{epsilon}})$ complexity, outperforming classical sample complexity.

Quantum algorithms are simple, using basic subroutines like amplitude estimation, and achieve significant speedups over classical methods.

Abstract

We explore potential quantum speedups for the fundamental problem of testing the properties of closeness and $k$-wise uniformity of probability distributions. Closeness testing is the problem of distinguishing whether two $n$-dimensional distributions are identical or at least $\varepsilon$-far in $\ell^1$- or $\ell^2$-distance. We show that the quantum query complexities for $\ell^1$- and $\ell^2$-closeness testing are $O(\sqrt{n}/\varepsilon)$ and $O(1/\varepsilon)$, respectively, both of which achieve optimal dependence on $\varepsilon$, improving the prior best results of Gily\'en and Li (2020). $k$-wise uniformity testing is the problem of distinguishing whether a distribution over $\{0, 1\}^n$ is uniform when restricted to any $k$ coordinates or $\varepsilon$-far from any such distributions. We propose the first quantum algorithm for this problem with query complexity $O(\sqrt{n^k}/\varepsilon)$, achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexity $O(n^k/\varepsilon^2)$ by O'Donnell and Zhao (2018). Moreover, when $k = 2$ our quantum algorithm outperforms any classical one because of the classical lower bound $\Omega(n/\varepsilon^2)$. All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.