发表机构
Paul G. Allen School of Computer Science & Engineering, University of Washington; Institute of Software, Chinese Academy of Sciences; University of Chinese Academy of Sciences; School of Computer Science, Shanghai Jiao Tong University（华盛顿大学保罗·G·艾伦计算机科学与工程学系; 中国科学院软件研究所; 中国科学院大学; 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
该研究针对未知d维量子态的冯·诺依曼熵估计问题，突破了此前Ω(d²)样本复杂度的二次壁垒，提出首个亚二次样本估计量，复杂度为o(d²)，并引入新的夹挤不等式及两类特征值估计方法。
AI 中文摘要
我们研究未知d维量子态的冯·诺依曼熵估计的样本复杂度。所有已知估计量都需要Ω(d²)个样本，且插件估计量存在二次壁垒。我们提出首个亚二次样本估计量：对于加性误差ε，该估计量使用O( d² log²(log(d)) log(1/ε) / (ε² log²(d)) + log²(d/ε) / ε² )个样本。特别地，当ε为常数时，复杂度为O_ε(d² log²(log(d))/log²(d))=o(d²)。分析引入了新的夹挤不等式，用于界定空间直和分解下的熵损失，同时结合了针对大特征值的偏差校正估计量和针对小特征值的新型有界系数多项式估计量。
英文摘要
We show that the von Neumann entropy of an unknown $d$-dimensional quantum state $ρ$ can be estimated to within additive error $\varepsilon$ using \[ Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)(1+\varepsilon\log^2(d))} + \frac{\log^2(d)}{\varepsilon^2}\right) \] samples of $ρ$, matching the lower bound due to Wang (2026). Our estimator is based on a high-degree approximation polynomial and the main technical challenge is bounding its variance despite the large polynomial degree. We overcome this using a quantum Hoeffding decomposition, which preserves cancellations between different polynomial terms and reduces the analysis to derivative bounds for the approximation polynomial.
Comments22 pages, 1 table, 1 algorithm. Improved the sample complexity to optimal
