发表机构
School of Computer Science, Shanghai Jiao Tong University（上海交通大学计算机学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出了一个量子泛函估计的统一下界框架，推导了多种量子属性测试问题的近紧下界，证明了20多种量子算法的最优性。
AI 中文摘要
我们开发了一个统一框架，用于证明d维量子状态泛函估计的下界：$\mathcal L_\phi(\rho)=\frac1d\sum_{i=1}^d\phi(d\lambda_i(\rho))$。该框架结合了Haar随机矩编码过程、矩匹配和最优多项式逼近。我们提供了一个主定理，可根据$\phi$的性质推导$\mathcal L_\phi$估计的样本复杂度下界。利用该框架，我们通过为多种量子属性测试问题建立近紧下界解决了多个开放问题，包括Uhlmann保真度估计、迹距离估计、冯·诺依曼/雷尼/ Tsallis熵估计、谱估计和秩测试。此外，通过量子样本到查询的提升，这些样本下界也暗示了量子查询下界。这些样本/查询下界表明，自2015年以来的20多种量子算法是最优的。
英文摘要
We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form $\mathcal{L}_ϕ(ρ) = \operatorname{tr}(ϕ(dρ))/d$, where $ρ$ is an unknown $d$-dimensional quantum state and $ϕ$ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating $\mathcal{L}_ϕ(ρ)$ from properties of $ϕ$, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields nearly tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). Combined with known upper bounds, these results resolve several open problems and establish the optimality of 31 quantum algorithms since 2015, up to polylogarithmic factors.
Comments46 pages, 2 tables. Title changed and more references added; added Schmidt rank testing and matrix product state (MPS) testing; restructured the Abstract, Introduction, and the master theorem; improved the precision-regime for Renyi and Tsallis entropy estimation; refined the proofs
