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 the 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：本文研究了估计$n$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$q$，我们提出了一种计算$\text{S}_q(\rho)$的量子估计器，其时间复杂度为$\text{poly}(n)$，相较于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）先前提出的$\exp(n)$最优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均匀逼近引入量子奇异值变换实现的。我们的量子算法揭示了量子$q$-Tsallis熵差问题（TsallisQED$_q$）计算复杂度中$q=1$与常数$q>1$情形间的急剧相变，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，鉴于$\text{S}_q(\rho) \leq \text{S}(\rho)$，这导致逼近冯·诺依曼熵的困难性，前提是$\mathsf{BQP} \subsetneq \mathsf{QSZK}$成立。这些困难性结果源于基于量子$q$-Jensen-(Shannon-)Tsallis散度（其中$1\leq q \leq 2$）新不等式的归约推导，这些不等式本身具有独立的研究价值。