This paper reveals a conceptually new connection from information theory to approximation theory via quantum algorithms for entropy estimation. Specifically, we provide an information-theoretic lower bound $Ω(\sqrt{n})$ on the approximate degree of the monomial $x^n$, compared to the analytic lower bounds shown in Newman and Rivlin (Aequ. Math. 1976) via Fourier analysis and in Sachdeva and Vishnoi (Found. Trends Theor. Comput. Sci. 2014) via the Markov brothers' inequality. This is done by relating the polynomial approximation of monomials to quantum Tsallis entropy estimation. This further implies a quantum algorithm that estimates to within additive error $\varepsilon$ the Tsallis entropy of integer order $q \geq 2$ of an unknown probability distribution $p$ or an unknown quantum state $ρ$, using $\widetilde Θ(\frac{1}{\sqrt{q}\varepsilon})$ queries to the quantum oracle that produces a sample from $p$ or prepares a copy of $ρ$, improving the prior best $O(\frac{1}{\varepsilon})$ via the Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki and Kwek (Phys. Rev. Lett. 2002). To the best of our knowledge, this is the first quantum entropy estimator with optimal query complexity (up to polylogarithmic factors) for all parameters simultaneously.
This paper reveals a conceptually new connection from information theory to approximation theory via quantum algorithms for entropy estimation. Specifically, we provide an information-theoretic lower bound Ω(n) on the approximate degree of the monomial xn, compared to the analytic lower bounds shown in Newman and Rivlin (Aequ. Math. 1976) via Fourier analysis and in Sachdeva and Vishnoi (Found. Trends Theor. Comput. Sci. 2014) via the Markov brothers' inequality. This is done by relating the polynomial approximation of monomials to quantum Tsallis entropy estimation. This further implies a quantum algorithm that estimates to within additive error ε the Tsallis entropy of integer order q≥2 of an unknown probability distribution p or an unknown quantum state ρ, using Θ(qε1) queries to the quantum oracle that produces a sample from p or prepares a copy of ρ, improving the prior best O(ε1) via the Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki and Kwek (Phys. Rev. Lett. 2002). To the best of our knowledge, this is the first quantum entropy estimator with optimal query complexity (up to polylogarithmic factors) for all parameters simultaneously.
Estimating statistical properties of probability distributions is a fundamental problem in statistics and theoretical computer science. We develop quantum algorithms for estimating the q-Tsallis entropy HqTsa(P)=1−q∑i=1npiq−1 when 0<q<1, given purified quantum query access to a distribution P on [n]. When 0<q≤21, we provide a quantum algorithm with query complexity $\widetilde{O}\rbra{\left(\frac{n}{\varepsilon}\right)^{\frac{1}{2q}}}$, which is optimal up to polylogarithmic factors. When 21<q<1, we introduce a two-stage simultaneous probability estimation procedure, yielding a quantum algorithm with query complexity O(ε2q1n21+2q1+εn1−q), which improves the result given by Chen et al. (ICALP 2026) when $\varepsilon=O\rbra{n^{1-2q}}$.
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.
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 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.
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.