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 epsilon -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(root n/epsilon) and O(1/epsilon) , respectively, both of which achieve optimal dependence on epsilon , improving the prior best results of Gily & eacute;n 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 epsilon -far from any such distribution. We propose the first quantum algorithm for this problem with query complexity O(root n(k)/epsilon) , achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexity O(n(k)/epsilon(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/epsilon(2)) . All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.