Quantum Speedups for Testing Similar Means

Property testing of distributions is a central topic in information theory, learning theory, and statistics. While quantum algorithms are known to offer significant speedups for property testing of a single distribution or a pair of distributions, it is unclear whether quantum algorithms provide speedups for property testing of $m$ ($m\geq 3$) distributions. This work focuses on quantum algorithms for testing whether $m$ distributions have similar means or are $\epsilon$-far from mean similarity under two models. In the query model, the algorithm can choose which distribution to sample from, whereas in the sampling model, the distributions are selected uniformly. We design quantum algorithms with complexities $\tilde{O}(1/\epsilon)$ (the $\tilde{O}$ notation hides poly-logarithmic factors) and $\tilde{O}(\sqrt{m}/\epsilon)$ in the query and sampling models, respectively, achieving quadratic speedups over the classical counterparts. We further establish quantum lower bounds of $\Omega\left(1/\epsilon\right)$ and $\Omega\rbra{m^{1/3}+\frac{m^{1/4}}{\epsilon}}$ for the query model and the sampling model, demonstrating the optimality of our quantum algorithms in terms of the dependence on $\epsilon$ up to logarithmic factors.