Optimal single-copy estimation of quantum state moments: why $\operatorname{Tr}(\rho^3)$ and $\operatorname{Tr}(\rho^4)$ are equally hard

2026-10-19 · arXiv Quantum Physics

arXiv:2609.37781v2 Announce Type: replace

Abstract: 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 $\operatorname{Tr}(\rho^t)$, allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity $\Theta(\sqrt d)$ 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 \[ \Theta\!\left(d^{\frac{\lceil\log_2 t\rceil}{1+\lceil\log_2 t\rceil}}\right) \] with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer $h\geq1$, all moment orders $2^{h-1}

原文链接：https://arxiv.org/abs/2609.37781

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