We investigate the computational hardness of estimating the quantum $α$-Rényi entropy ${\rm S}^{\tt R}_α(ρ) = \frac{\ln {\rm Tr}(ρ^α)}{1-α}$ and the quantum $q$-Tsallis entropy ${\rm S}^{\tt T}_q(ρ) = \frac{1-{\rm Tr}(ρ^q)}{q-1}$, both of which converge to the von Neumann entropy as the order approaches $1$. The promise problems Quantum $α$-Rényi Entropy Approximation (RényiQEA$_α$) and Quantum $q$-Tsallis Entropy Approximation (TsallisQEA$_q$) ask whether $ {\rm S}^ {\tt R}_α(ρ)$ or ${\rm S}^{\tt T}_q(ρ)$, is at least $τ_1$ or at most $τ_2$, where $τ_1 - τ_2$ is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order $1$) and some cases of the quantum $q$-Tsallis entropy, while existing approaches do not readily extend to other orders. We establish that for all positive real $α$ and $q$, and also for $α=\infty$, the rank-$2$ variants Rank2RényiQEA$_α$ and Rank2TsallisQEA$_q$ are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), as well as the one derived from O'Donnell and Wright (STOC 2016), our results imply: - For all real orders $α> 0$ or $α=\infty$, and for all real orders $0 < q \leq 1$, LowRankRényiQEA$_α$ and LowRankTsallisQEA$_q$ are BQP-complete, where both are restricted versions of RényiQEA$_α$ and TsallisQEA$_q$ with $ρ$ of polynomial rank. - For all real order $q>1$, TsallisQEA$_q$ is BQP-complete. Our hardness results stem from reductions based on new inequalities relating the $α$-Rényi or $q$-Tsallis binary entropies of different orders. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.

翻译：我们研究了量子$α$-Rényi熵${\rm S}^{\tt R}_α(ρ)=\frac{\ln {\rm Tr}(ρ^α)}{1-α}$和量子$q$-Tsallis熵${\rm S}^{\tt T}_q(ρ)=\frac{1-{\rm Tr}(ρ^q)}{q-1}$的计算难度问题，两者在阶数趋近于$1$时均收敛于冯·诺依曼熵。承诺问题量子$α$-Rényi熵近似（RényiQEA$_α$）和量子$q$-Tsallis熵近似（TsallisQEA$_q$）询问${\rm S}^{\tt R}_α(ρ)$或${\rm S}^{\tt T}_q(ρ)$是否至少为$τ_1$或至多为$τ_2$，其中$τ_1-τ_2$通常为正常数。先前的难度结果仅覆盖冯·诺依曼熵（阶数$1$）和量子$q$-Tsallis熵的某些情形，而现有方法难以直接推广至其他阶数。我们证明：对所有正实数$α$和$q$，以及$α=\infty$，其秩-$2$变体Rank2RényiQEA$_α$和Rank2TsallisQEA$_q$均为BQP困难问题。结合Wang、Guan、Liu、Zhang和Ying（TIT 2024）、Wang、Zhang和Li（TIT 2024）、Liu和Wang（SODA 2025）中的先验（秩依赖）量子查询算法，以及源自O'Donnell和Wright（STOC 2016）的算法，我们的结果意味着：- 对所有实数阶$α>0$或$α=\infty$，以及所有实数阶$0<q\leq1$，LowRankRényiQEA$_α$和LowRankTsallisQEA$_q$（二者均为RényiQEA$_α$和TsallisQEA$_q$在$ρ$具有多项式秩时的限制版本）是BQP完全的。- 对所有实数阶$q>1$，TsallisQEA$_q$是BQP完全的。我们的难度结果源于基于新不等式的归约，这些不等式关联了不同阶数下的$α$-Rényi或$q$-Tsallis二进制熵。这些归约与先前方法存在本质差异，且该不等式具有独立的研究价值。