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