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

TL;DR

This paper advances the understanding of quantum trace estimation by providing tight bounds on sample complexity for different ranges of q, revealing computational hardness for certain cases, and introducing new quantum estimators.

Contribution

It significantly improves sample complexity bounds for estimating quantum state power traces and establishes hardness results for specific q ranges, using novel quantum estimators.

Findings

For q>2, sample complexity is tightly bounded at Θ(1/ε²).

For 1<q<2, upper and lower bounds are established for dimension-independent estimators.

For 0<q<1, the problem is shown to be computationally hard, with bounds depending on state dimension.

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 $0<q<1$, the problem does not admit an efficient estimator unless ${\sf BQP}={\sf NIQSZK}$, which is considered highly unlikely. In particular, we have the following results. - For $q>2$, we settle the sample complexity with matching upper and lower bounds $\widetilde\Theta(1/\epsilon^2)$. - For $1<q<2$, we obtain an upper bound of $\widetilde O(1/\epsilon^{\frac2{q-1}})$, with a lower bound of $\Omega(1/\epsilon^{\max\{\frac1{q-1},2\}})$ for dimension-independent (in fact, rank-independent) estimators. - For $0<q<1$, we obtain an upper bound of $O((d/\epsilon)^{\frac2{q}})$, with a lower bound of $\Omega((d/\epsilon)^{\frac1{q}})$ for $d$-dimensional states (in fact, both bounds can be naturally refined to depend on the rank rather than the dimension). Accordingly, the corresponding promise problem is ${\sf NIQSZK}$-hard, which is in sharp contrast to the case of $q>1$. 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.