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.
Journal reference
Proceedings of the 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026), pp. 66:1-66:23, 2026
39 pages, 3 tables. v2: Added the BQP-completeness result for the α=infinity case; corrected a calculation error in the BQP-hardness proof of PureInfidelity (Lemma 2.8) and the corresponding threshold parameters in the related BQP-hardness results; corrected calculation errors in the proof of Lemma 3.12; and made other minor changes. v1: Appeared in STACS 2026
We investigate the computational hardness of estimating the quantum α-Rényi entropy SαR(ρ)=1−αlnTr(ρα) and the quantum q-Tsallis entropy SqT(ρ)=q−11−Tr(ρq), both of which converge to the von Neumann entropy as the order approaches 1. The promise problems Quantum α-Rényi Entropy Approximation (RényiQEAα) and Quantum q-Tsallis Entropy Approximation (TsallisQEAq) ask whether SαR(ρ) or SqT(ρ), is at least τ1 or at most τ2, where τ1−τ2 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=∞, the rank-2 variants Rank2RényiQEAα and Rank2TsallisQEAq 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=∞, and for all real orders 0<q≤1, LowRankRényiQEAα and LowRankTsallisQEAq are BQP-complete, where both are restricted versions of RényiQEAα and TsallisQEAq with ρ of polynomial rank.
For all real order q>1, TsallisQEAq 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.
In this paper, we provide estimators for quantum Rényi and Tsallis entropies with nearly optimal sample complexity. Specifically, for order α, dimension d, and additive error ε,
1. For 0<α<1, the sample complexity is O(d1+1/α/ε1/α+d1/α−1/ε2) for Rényi entropy and O(d1+1/α/ε1/α+d2−2α/ε2) for Tsallis entropy. In particular, for 0<α≤1/2, the sample complexity for both entropies is O(d1+1/α/ε1/α).
2. For non-integer α>1, the sample complexity is O(d2/ε1/α+d1−1/α/ε2) for Rényi entropy.
Our upper bounds improve the quantum Rényi entropy estimators due to Acharya, Issa, Shende, and Wagner (2017) and the quantum Tsallis entropy estimators due to Chen, Liu, and Wang (2026), and match the lower bounds recently established by Wang (2026).
We investigate the computational complexity of estimating the trace of quantum state powers tr(ρq) for annqubit mixed quantum state ρ, given its state-preparation circuit of size poly(n). This quantity is closely related to and often interchangeable with the Tsallis entropy Sq(ρ) = 1—tr(ρq)/q—1, whereq= 1 corresponds to the von Neumann entropy. For any non-integerq≥ 1 + Ω(1), we provide a quantum estimator for Sq(ρ) with time complexity 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)| Wang, Zhang, and Li (TIT 2024), and Wang and Zhang (TIT 2025). Our speedup is achieved by introducing efficiently computableuniform approximationsof positive power functions into quantum singular value transformation. Our quantum algorithm reveals a sharp phase transition between the case ofq= 1 and constantq> 1 in the computational complexity of the Quantumq-Tsallis Entropy Difference Problem (TsallisQEDq), particularly deciding whether the difference Sq(ρ0)−Sq(ρ1) is at least 0.001 or at most −0.001: • For any 1 + Ω (1) ≤q≤ 2, TsallisQEDqis BQP-complete, which implies that Purity Estimation is also BQP-complete. • For any 1≤q≤1+1/n— 1, TsallisQED q is QSZK-hard, leading to hardness of approximating the von Neumann entropy because Sq(ρ) ≤ S (ρ), as long as BQPQSZK. The hardness results are derived from reductions based on new inequalities for the quantumq-Jensen–(Shannon–)Tsallis divergence with 1 ≤q≤ 2, which are of independent interest.
The von Neumann, Tsallis, and Rényi entropies are fundamental measures of quantum information. A common estimation strategy multiplies the worst-case costs of spectral transformation and statistical readout, leaving gaps to query lower bounds. We reduce this overhead with multi-level algorithms that use local normalization and precision allocation, while variable-time estimation accounts for the probability of reaching expensive spectral tests. With controlled purified access and its inverse, we obtain bounds for additive error ε, rank upper bound R, and fixed order α. Tsallis entropy estimation has near-optimal rank-independent query complexity O(ε−1/(α−1)) for 1<α<2, when the allowed dimension and rank accommodate the lower-bound instances, and O(1/ε) for α≥2, where O suppresses logarithmic factors. The first saves a factor 1/ε; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from O(R/ε2) to O(R/ε) and obtain O(R/ε) for noninteger Rényi orders 1<α<3, with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using O(R/ε) queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.
Entropy is a measure of the randomness of a system. Estimating the entropy of a quantum state is a basic problem in quantum information. In this paper, we introduce a time-efficient quantum approach to estimating the von Neumann entropy S(ρ) and Rényi entropy Sα(ρ) of an N-dimensional quantum state ρ, given access to independent samples of ρ. Specifically, we provide the following: 1. A quantum estimator for S(ρ) with time complexity O~(N2), improving the prior best time complexity O~(N6) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). 2. A quantum estimator for Sα(ρ) with time complexity O~(N4/α−2) for 0<α<1 and O~(N4−2/α) for α>1, improving the prior best time complexity O~(N6/α) for 0<α<1 and O~(N6) for α>1 by Acharya, Issa, Shende, and Wagner (2020), though at a cost of a slightly larger sample complexity. Moreover, these estimators are naturally extensible to the low-rank case. We also provide a sample lower bound for estimating Sα(ρ). Technically, our method is quite different from the previous ones that are based on weak Schur sampling and Young diagrams. At the heart of our construction, is a novel tool called samplizer, which can "samplize" a quantum query algorithm to a quantum algorithm with similar behavior using only samples of quantum states; this suggests a unified framework for estimating quantum entropies. Specifically, when a quantum oracle U block-encodes a mixed quantum state ρ, any quantum query algorithm using Q queries to U can be samplized to a δ-close (in the diamond norm) quantum algorithm using Θ~(Q2/δ) samples of ρ. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.
Entropy estimation of discrete probability distributions is a fundamental problem in statistical inference and quantum property estimation. In this paper, we study the problem of estimating Rényi entropy Hα(p) for any distribution p=(pi)i=1n given its purified quantum query-access. For non-integer α>1, our algorithm estimates Hα(p) to within additive error ε using only O(nmax{1/2,1−1/α}/ε) queries, which improves the best known upper bound for dependence on both n and 1/ε. For 0<α<1, our algorithm has query complexity O(n1/(2α)/εmax{1,1/(2α)}), which improves the dependence on 1/ε of the current best upper bound without incresing its n dependence. Our approach adapts the multi-level estimation framework to relative approximation of the power sums and leverage binary search to remove the requirement of prior knowledge of them. We also strengthen the precision-dependent lower bound to Ω(1/εmax{1,1/(2α)}) for 0<α<1, which identifies the optimal precision dependence of our algorithm.
We investigate the computational complexity of estimating the trace of quantum state powers tr(ρq ) for an n-qubit mixed quantum state ρ, given its state-preparation circuit of size poly(n ). This quantity is closely related to and often interchangeable with the Tsallis entropywhere q = 1 corresponds to the von Neumann entropy. For any non-integer q ≥ 1 + Ω(1), we provide a quantum estimator for Sq (ρ ) with time complexity 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), 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.