As often emerges in various basic quantum properties such as Rényi and Tsallis entropies, the trace of quantum state powers $\text{tr}(ρ^q)$ has attracted a lot of attention. The recent work of Liu and Wang (SODA 2025) showed that, even for (possibly) non-integer $q>1$, $\text{tr}(ρ^q)$ can be estimated to within additive error $ε$ using a dimension-independent (and also rank-independent) sample complexity of $\widetilde O(1/ε^{3+\frac2{q-1}})$, together with a lower bound of $Ω(1/ε)$. 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 $\widetildeΘ(1/ε^2)$. - For $1<q<2$, we obtain an upper bound of $\widetilde O(1/ε^{\frac2{q-1}})$, with a lower bound of $Ω(1/ε^{\max\{\frac1{q-1},2\}})$ for dimension-independent (in fact, rank-independent) estimators. - For $0<q<1$, we obtain an upper bound of $O((d/ε)^{\frac2{q}})$, with a lower bound of $Ω((d/ε)^{\frac1{q}})$ 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.

翻译：随着诸如Rényi熵和Tsallis熵等基本量子性质中频繁出现，量子态幂的迹$\text{tr}(ρ^q)$引起了广泛关注。Liu和Wang的最新工作（SODA 2025）表明，即使对于（可能）非整数的$q>1$，$\text{tr}(ρ^q)$也可在加性误差$ε$内，通过维度独立（且秩独立）的样本复杂度$\widetilde O(1/ε^{3+\frac2{q-1}})$进行估计，并附带一个下界$Ω(1/ε)$。此外，将此结果与Liu的后续工作（STACS 2026）相结合，表明相应的承诺问题是${\sf BQP}$-完全的。在本文中，我们显著改进并扩展了该问题的样本复杂度边界。进一步，我们证明对于$0<q<1$，除非${\sf BQP}={\sf NIQSZK}$（这被认为极不可能），否则该问题不存在有效估计器。具体而言，我们得到以下结果： - 对于$q>2$，我们通过匹配的上下界$\widetildeΘ(1/ε^2)$确定了样本复杂度。 - 对于$1<q<2$，我们得到上界$\widetilde O(1/ε^{\frac2{q-1}})$，以及针对维度独立（实际上是秩独立）估计器的下界$Ω(1/ε^{\max\{\frac1{q-1},2\}})$。 - 对于$0<q<1$，我们得到上界$O((d/ε)^{\frac2{q}})$，以及针对$d$维量子态的下界$Ω((d/ε)^{\frac1{q}})$（实际上，两个边界均可自然地改进为依赖于秩而非维度）。相应地，相应的承诺问题是${\sf NIQSZK}$-困难的，这与$q>1$的情况形成鲜明对比。技术上，我们的上界通过基于弱Schur采样的（非插入式）量子估计器获得，这与之前基于量子奇异值变换和采样器的方法截然不同。