We investigate the computational complexity of estimating the trace of quantum state powers $\text{tr}(\rho^q)$ for an $n$-qubit mixed quantum state $\rho$, given its state-preparation circuit of size $\text{poly}(n)$. This quantity is closely related to and often interchangeable with the Tsallis entropy $\text{S}_q(\rho) = \frac{1-\text{tr}(\rho^q)}{q-1}$, where $q = 1$ corresponds to the von Neumann entropy. For any non-integer $q \geq 1 + \Omega(1)$, we provide a quantum estimator for $\text{S}_q(\rho)$ with time complexity $\text{poly}(n)$, exponentially improving the prior best results of $\exp(n)$ due to Acharya, Issa, Shende, and Wagner (ISIT 2019), Wang, Guan, Liu, Zhang, and Ying (TIT 2024), and Wang, Zhang, and Li (TIT 2024), and Wang and Zhang (ESA 2024). Our speedup is achieved by introducing efficiently computable uniform approximations of positive power functions into quantum singular value transformation. Our quantum algorithm reveals a sharp phase transition between the case of $q=1$ and constant $q>1$ in the computational complexity of the Quantum $q$-Tsallis Entropy Difference Problem (TsallisQED$_q$), particularly deciding whether the difference $\text{S}_q(\rho_0) - \text{S}_q(\rho_1)$ is at least $0.001$ or at most $-0.001$: - For any $1+\Omega(1) \leq q \leq 2$, TsallisQED$_q$ is $\mathsf{BQP}$-complete, which implies that Purity Estimation is also $\mathsf{BQP}$-complete. - For any $1 \leq q \leq 1 + \frac{1}{n-1}$, TsallisQED$_q$ is $\mathsf{QSZK}$-hard, leading to hardness of approximating von Neumann entropy because $\text{S}_q(\rho) \leq \text{S}(\rho)$, as long as $\mathsf{BQP} \subsetneq \mathsf{QSZK}$. The hardness results are derived from reductions based on new inequalities for the quantum $q$-Jensen-(Shannon-)Tsallis divergence with $1\leq q \leq 2$, which are of independent interest.

翻译：我们研究了估计$n$量子比特混合量子态$\rho$的幂次迹$\text{tr}(\rho^q)$的计算复杂度，给定其规模为$\text{poly}(n)$的态制备电路。该量与Tsallis熵$\text{S}_q(\rho) = \frac{1-\text{tr}(\rho^q)}{q-1}$密切相关且通常可互换，其中$q = 1$对应冯·诺依曼熵。对于任意满足$q \geq 1 + \Omega(1)$的非整数$q$，我们提出了一个时间复杂度为$\text{poly}(n)$的$\text{S}_q(\rho)$量子估计器，相对于Acharya、Issa、Shende和Wagner（ISIT 2019）、Wang、Guan、Liu、Zhang和Ying（TIT 2024）、Wang、Zhang和Li（TIT 2024）以及Wang和Zhang（ESA 2024）先前$\exp(n)$的最佳结果实现了指数级加速。我们的加速是通过将正幂函数的高效可计算一致逼近引入量子奇异值变换实现的。我们的量子算法揭示了在计算量子$q$-Tsallis熵差问题（TsallisQED$_q$）的复杂度中，$q=1$情形与常数$q>1$情形之间存在急剧相变，特别是判定差值$\text{S}_q(\rho_0) - \text{S}_q(\rho_1)$是否至少为$0.001$或至多为$-0.001$时：- 对于任意满足$1+\Omega(1) \leq q \leq 2$的$q$，TsallisQED$_q$是$\mathsf{BQP}$完全的，这意味着纯度估计也是$\mathsf{BQP}$完全的。- 对于任意满足$1 \leq q \leq 1 + \frac{1}{n-1}$的$q$，TsallisQED$_q$是$\mathsf{QSZK}$难的，只要$\mathsf{BQP} \subsetneq \mathsf{QSZK}$成立，这将导致冯·诺依曼熵逼近的困难性，因为$\text{S}_q(\rho) \leq \text{S}(\rho)$。这些困难性结果源于基于量子$q$-Jensen-(Shannon-)Tsallis散度（其中$1\leq q \leq 2$）的新不等式的归约，这些不等式本身具有独立的研究价值。