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.
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.
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.
We investigate the computational complexity of estimating the operator norm distance T∞(ρ0,ρ1), defined via the operator norm ∥A∥∞=σmax(A), given poly(n)-size state-preparation circuits of n-qubit quantum states ρ0 and ρ1. We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states:
1. When one state is pure, we establish an optimal quantum estimator using Θ(1/ε) queries to the state-preparation circuits. Consequently, for constant additive error, say ε=1/5, our estimator runs in poly(n) time. Since the operator norm distance T∞(∣ψ⟩⟨ψ∣,ρ) is exactly half of the trace distance T(∣ψ⟩⟨ψ∣,ρ), our result also gives rank-independent query complexity for estimating both quantities, whereas the approaches due to van Apeldoorn, Cornelissen, Gily{é}n, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with rank(ρ), which can be exp(n) in general.
2. For general quantum states, we also provide a quantum estimator using O(1/ε3/2) queries to the state-preparation circuits, which shows that the corresponding promise problem is BQP-complete and improves the QMA upper bound sketched by Liu and Wang (ESA 2025). Together with an Ω(1/ε) quantum query complexity lower bound, this leaves only square-root room for improvement.
The key intuition behind our estimators is that, when one state is pure, the pure state ∣ψ⟩ has overlap at least 1/2 with the top unit eigenvector of ∣ψ⟩⟨ψ∣−ρ, reflecting a structural feature specific to the operator norm distance.
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 entropyS(ρ) and Rényi entropySα(ρ) of anN-dimensional quantum state ρ, given access to independent samples of ρ. Specifically, we provide the following quantum estimators. • A quantum estimator forS(ρ) with time complexity Õ (N2),1improving the prior best time complexity Õ(N6) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). • A quantum estimator forSα(ρ) with time complexity Õ (N4/α−2) for 0N4−2/α) for α > 1, improving the prior best time complexity Õ(N6/α) for 0N6) 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 Ω(max{N/ε,N1/α−1/ε1/α}) for estimatingSα(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 calledsamplizer, which can “samplize” a quantum query algorithm to a quantum algorithm with similar behavior using only samples of quantum states; this suggests a new framework for estimating quantum entropies. Specifically, when a quantum oracleUblockencodes a mixed quantum state ρ, any quantum query algorithm usingQqueries toUcan 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.
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 $ {\rm S}^ {\tt R}_α(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.
Estimating the trace of quantum state powers, Tr ( ρ k ) , for k identical quantum states is a fundamental task with numerous applications in quantum information processing, including nonlinear function estimation of quantum states and entanglement detection. On near-term quantum devices, reducing the required quantum circuit depth, the number of multi-qubit quantum operations, and the copies of the quantum state needed for such computations is crucial. In this work, inspired by the Newton-Girard method, we significantly improve upon existing results by introducing an algorithm that requires only O ( r ~ ) qubits and O ( r ~ ) multi-qubit gates, where r ~ = min { rank ( ρ ) , ⌈ ln ( 2 k / ϵ ) ⌉ } . This approach is efficient, as it employs the r ~ -entangled copy measurement instead of the conventional k -entangled copy measurement, while asymptotically preserving the known sample complexity upper bound. Furthermore, we prove that estimating { Tr ( ρ i ) } i = 1 r ~ is sufficient to approximate Tr ( ρ k ) even for large integers k > r ~ . This leads to a rank-dependent complexity for solving the problem, providing an efficient algorithm for low-rank quantum states while also improving existing methods when the rank is unknown or when the state is not low-rank. Building upon these advantages, we extend our algorithm to the estimation of Tr ( M ρ k ) for arbitrary observables and Tr ( ρ k σ l ) for multiple quantum states.
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).