We explore potential quantum speedups for the fundamental problem of testing the properties of closeness and $k$-wise uniformity of probability distributions. \textit{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 $O\rbra{\sqrt{n}/\varepsilon}$ and $O\rbra{1/\varepsilon}$, respectively, both of which achieve optimal dependence on $\varepsilon$, improving the prior best results of \hyperlink{cite.gilyen2019distributional}{Gily{\'e}n and Li~(2019)}. \textit{$k$-wise uniformity testing} is the problem of distinguishing whether a distribution over $\cbra{0, 1}^n$ 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 $O\rbra{\sqrt{n^k}/\varepsilon}$, achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexity $O\rbra{n^k/\varepsilon^2}$ by \hyperlink{cite.o2018closeness}{O'Donnell and Zhao (2018)}. Moreover, when $k = 2$ our quantum algorithm outperforms any classical one because of the classical lower bound $\Omega\rbra{n/\varepsilon^2}$. All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.
Comment: We have added the proof of lower bounds and have polished the language