AI 中文总结
该研究针对多分布均值相似性检验问题，在查询与采样模型下设计量子算法实现二次加速，并证明算法在ε依赖上的最优性。
AI 中文摘要
分布的性质检验是信息论、学习理论与统计学的核心研究主题。已知量子算法可为单个或一对分布的性质检验提供显著加速，但量子算法能否为m（m≥3）个分布的性质检验提供加速尚不明确。本研究聚焦于两类模型下检验m个分布是否具有相似均值，或在均值相似性上为ε-远的量子算法：查询模型中算法可选择从哪个分布采样，采样模型中则均匀选择分布。我们设计的量子算法在查询模型下复杂度为Õ(1/ε)（Õ记号隐藏了多对数因子），采样模型下为Õ(√m/ε)，分别实现了对经典对应算法的二次加速。我们还为查询模型和采样模型建立了量子下界Ω(1/ε)和Ω(m^(1/3)+m^(1/4)/ε)，证明了所提量子算法在对ε的依赖关系上，除对数因子外是最优的。
英文摘要
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 $ε$-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 $\tilde{O}(1/ε)$ (the $\tilde{O}$ notation hides poly-logarithmic factors) and $\tilde{O}(\sqrt{m}/ε)$ in the query and sampling models, respectively, achieving quadratic speedups over the classical counterparts. We further establish quantum lower bounds of $Ω\left(1/ε\right)$ and $Ω\rbra{m^{1/3}+\frac{m^{1/4}}ε}$ for the query model and the sampling model, demonstrating the optimality of our quantum algorithms in terms of the dependence on $ε$ up to logarithmic factors.
