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arXiv · 2505.09563

Trace Estimation of Quantum State Powers: Sample Complexity and Computational Hardness

Kean Chen·Yupan Liu·Qisheng Wang
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Abstract

As often emerges in various basic quantum properties such as R\'enyi and Tsallis entropies, the trace of quantum state powers $\text{tr}(\rho^q)$ has attracted a lot of attention. The recent work of Liu and Wang (SODA 2025) showed that, even for (possibly) non-integer $q>1$, $\text{tr}(\rho^q)$ can be estimated to within additive error $\epsilon$ using a dimension-independent (and also rank-independent) sample complexity of $\widetilde O(1/\epsilon^{3+\frac2{q-1}})$, together with a lower bound of $\Omega(1/\epsilon)$. In addition, combining this result with subsequent work of Liu (STACS 2026) shows that the corresponding promise problem is ${\sf BQP}$-complete. In this paper, we significantly improve and extend the sample complexity bounds for this problem. Furthermore, we show that for $0 2$, we settle the sample complexity with matching upper and lower bounds $\widetilde\Theta(1/\epsilon^2)$. - For $1 1$. Technically, our upper bounds are obtained by (non-plug-in) quantum estimators based on weak Schur sampling, in sharp contrast to the prior approach based on quantum singular value transformation and samplizer.

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Kean Chen, Yupan Liu, Qisheng Wang. 2025-05-14. Trace Estimation of Quantum State Powers: Sample Complexity and Computational Hardness. https://arxiv.org/abs/2505.09563

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