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 converging 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 ${\rm S}^{\tt T}_q(ρ)$, respectively, is at least $τ_{\tt Y}$ or at most $τ_{\tt N}$, where $τ_{\tt Y} - τ_{\tt N}$ 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 orders, the rank-$2$ variants Rank2RényiQEA$_α$ and Rank2TsallisQEA$_q$ are ${\sf BQP}$-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), our results imply: - For all real orders $α> 0$ and $0 < q \leq 1$, LowRankRényiQEA$_α$ and LowRankTsallisQEA$_q$ are ${\sf BQP}$-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 ${\sf BQP}$-complete. Our hardness results stem from reductions based on new inequalities relating the $α$-Rényi or $q$-Tsallis binary entropies of different orders, where the reductions differ substantially from previous approaches, and the inequalities are also of independent interest.

翻译：本文研究了量子$α$-Rényi熵${\rm S}^{\tt R}_α(ρ) = \frac{\ln {\rm Tr}(ρ^α)}{1-α}$与量子$q$-Tsallis熵${\rm S}^{\tt T}_q(ρ) = \frac{1-{\rm Tr}(ρ^q)}{q-1}$的计算复杂性估计问题，二者在阶数趋近于$1$时均收敛于冯·诺依曼熵。承诺问题“量子$α$-Rényi熵逼近”（RényiQEA$_α$）与“量子$q$-Tsallis熵逼近”（TsallisQEA$_q$）分别判定${\rm S}^{\tt R}_α(ρ)$或${\rm S}^{\tt T}_q(ρ)$是否至少为$τ_{\tt Y}$或至多为$τ_{\tt N}$，其中$τ_{\tt Y} - τ_{\tt N}$通常为一正常数。先前的复杂性结果仅覆盖了冯·诺依曼熵（阶数$1$）及部分量子$q$-Tsallis熵的情形，而现有方法难以推广至其他阶数。我们证明：对于所有正实数阶数，其秩为$2$的变体Rank2RényiQEA$_α$与Rank2TsallisQEA$_q$均为${\sf BQP}$-困难问题。结合Wang、Guan、Liu、Zhang与Ying（TIT 2024）、Wang、Zhang与Li（TIT 2024）以及Liu与Wang（SODA 2025）中已有的（秩相关）量子查询算法，我们的结果意味着：- 对于所有实数阶$α>0$与$0<q≤1$，限制于多项式秩密度矩阵$ρ$的RényiQEA$_α$与TsallisQEA$_q$的受限版本LowRankRényiQEA$_α$与LowRankTsallisQEA$_q$均为${\sf BQP}$-完全问题。- 对于所有实数阶$q>1$，TsallisQEA$_q$为${\sf BQP}$-完全问题。我们的困难性证明源于基于不同阶$α$-Rényi或$q$-Tsallis二元熵之间新不等式的归约，这些归约与先前方法有本质差异，且相关不等式本身亦具有独立的理论价值。