发表机构
Quantum Research Center, Technology Innovation Institute (TII), Abu Dhabi, United Arab Emirates（量子研究中心，技术创新研究所（TII））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了任意固定阶数 t≥2 的量子态矩 Tr(ρ^t) 在自适应单拷贝测量下的最优样本复杂度为 Θ(d^{⌈log₂t⌉/(1+⌈log₂t⌉)})，揭示了三阶与四阶矩同需 Θ(d^{2/3}) 拷贝的二进层次结构，并证明了自适应协议对 t≥4 的优势。
AI 中文摘要
在实验可及性受限的条件下估计未知量子态的非线性性质是量子学习中的一个基本问题。我们研究了估计态矩 $\operatorname{Tr}(\rho^t)$ 的样本复杂度，允许任意自适应的单拷贝测量。虽然纯度估计在常数加性误差下具有最优样本复杂度 $\Theta(\sqrt d)$，但更高阶矩的最优维度依赖关系此前尚不明确。我们解决了每个固定整数 $t\ge 2$ 的这一难题：对于任意足够小的常数加性误差，最优样本复杂度为 $\Theta\\!\left(d^{\frac{\lceil\log_2 t\rceil}{1+\lceil\log_2 t\rceil}}\right)$，且成功概率为常数。因此，复杂度遵循二进层次结构：对于每个整数 $h\geq1$，所有矩阶数 $2^{h-1}<t\leq2^h$ 共享相同的指数。特别地，三阶矩和四阶矩都需要 $\Theta(d^{2/3})$ 个拷贝。我们还为任意非自适应单拷贝协议建立了 $\Omega(d^{1-1/t})$ 的下界，与已知上界匹配，并证明了对于每个 $t\geq4$ 自适应性的优势。我们的上界使用自适应态过滤将高阶矩估计简化为更小态空间中的低阶矩估计。对于下界，我们通过平滑插值直接比较矩匹配的系综，绕过了以最大混合态作为中间假设的步骤，从而在 $\sqrt d$ 之外获得了匹配的界。
英文摘要
Estimating nonlinear properties of an unknown quantum state with restrictive experimental accessibility is a fundamental problem in quantum learning. We study the sample complexity of estimating the state moments $\operatorname{Tr}(ρ^t)$, allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity $Θ(\sqrt d)$ at constant additive error, the optimal dimension dependence for higher moments has remained unclear. We resolve this question for every fixed integer $t\ge 2$: for any sufficiently small constant additive error, the optimal sample complexity is \[ Θ\!\left(d^{\frac{\lceil\log_2 t\rceil}{1+\lceil\log_2 t\rceil}}\right) \] with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer $h\geq1$, all moment orders $2^{h-1}<t\leq2^h$ share the same exponent. In particular, the third and fourth moments both require $Θ(d^{2/3})$ copies. We also establish an $Ω(d^{1-1/t})$ lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every $t\geq4$. Our upper bound uses adaptive state filtering to reduce higher-order moment estimation to lower-order moment estimation in a smaller state space. For the lower bound, we directly compare moment-matched ensembles through a smooth interpolation, bypassing the maximally mixed state as an intermediate hypothesis and yielding matching bounds beyond $\sqrt d$.
Comments55 pages, 1 figure
