Succinct Quantum Testers for Closeness and <italic>k</italic>-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 <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>-far in <inline-formula> <tex-math notation="LaTeX">$\ell ^{1}$ </tex-math></inline-formula>- or <inline-formula> <tex-math notation="LaTeX">$\ell ^{2}$ </tex-math></inline-formula>-distance. We show that the quantum query complexities for <inline-formula> <tex-math notation="LaTeX">$\ell ^{1}$ </tex-math></inline-formula>- and <inline-formula> <tex-math notation="LaTeX">$\ell ^{2}$ </tex-math></inline-formula>-closeness testing are <inline-formula> <tex-math notation="LaTeX">$O( \sqrt {n}/\varepsilon ) $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$O( 1/\varepsilon ) $ </tex-math></inline-formula>, respectively, both of which achieve optimal dependence on <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>, improving the prior best results of Gily&#x00E9;n and Li (2019) and 2) k-wise uniformity testing is the problem of distinguishing whether a distribution over <inline-formula> <tex-math notation="LaTeX">$\{0, 1\}^{n}$ </tex-math></inline-formula> is uniform when restricted to any k coordinates or <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>-far from any such distribution. We propose the first quantum algorithm for this problem with query complexity <inline-formula> <tex-math notation="LaTeX">$O( \sqrt {n^{k}}/\varepsilon ) $ </tex-math></inline-formula>, achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexity <inline-formula> <tex-math notation="LaTeX">$O( n^{k}/\varepsilon ^{2}) $ </tex-math></inline-formula> by O&#x2019;Donnell and Zhao (2018). Moreover, when <inline-formula> <tex-math notation="LaTeX">$k = 2$ </tex-math></inline-formula> our quantum algorithm outperforms any classical one because of the classical lower bound <inline-formula> <tex-math notation="LaTeX">$\Omega ( n/\varepsilon ^{2}) $ </tex-math></inline-formula>. All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.