Succinct Quantum Testers for Closeness and k-Wise Uniformity of Probability Distributions

We explore potential quantum speedups for the fundamental problem of testing the properties of closeness and k-wise uniformity of probability distributions: 1) 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 Gilyén and Li (2019) and 2) 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 distribution. 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.