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arXiv:2401.09947 · Leibniz international proceedings in informatics

Time-Efficient Quantum Entropy Estimator via Samplizer

Qisheng Wang, Zhicheng Zhang

Published Jan 1, 2024• 3 citations
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Abstract

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 $S_α(ρ)$ 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 $\tilde O(N^2)$, improving the prior best time complexity $\tilde O(N^6)$ by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). 2. A quantum estimator for $S_α(ρ)$ with time complexity $\tilde O(N^{4/α-2})$ for $0<α<1$ and $\tilde O(N^{4-2/α})$ for $α>1$, improving the prior best time complexity $\tilde O(N^{6/α})$ for $0<α<1$ and $\tilde O(N^6)$ 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 $S_α(ρ)$. 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 $\tildeΘ(Q^2/δ)$ samples of $ρ$. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.

DOI
10.4230/lipics.esa.2024.101

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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)S(ρ)S(ρ) and Rényi entropy Sα(ρ)S_α(ρ)Sα​(ρ) of an NNN-dimensional quantum state ρρρ, given access to independent samples of ρρρ. Specifically, we provide the following: 1. A quantum estimator for S(ρ)S(ρ)S(ρ) with time complexity O~(N2)\tilde O(N^2)O~(N2), improving the prior best time complexity O~(N6)\tilde O(N^6)O~(N6) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). 2. A quantum estimator for Sα(ρ)S_α(ρ)Sα​(ρ) with time complexity O~(N4/α−2)\tilde O(N^{4/α-2})O~(N4/α−2) for 0<α<10<α<10<α<1 and O~(N4−2/α)\tilde O(N^{4-2/α})O~(N4−2/α) for α>1α>1α>1, improving the prior best time complexity O~(N6/α)\tilde O(N^{6/α})O~(N6/α) for 0<α<10<α<10<α<1 and O~(N6)\tilde O(N^6)O~(N6) for α>1α>1α>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)S_α(ρ)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 UUU block-encodes a mixed quantum state ρρρ, any quantum query algorithm using QQQ queries to UUU can be samplized to a δδδ-close (in the diamond norm) quantum algorithm using Θ~(Q2/δ)\tildeΘ(Q^2/δ)Θ~(Q2/δ) samples of ρρρ. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.
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.
In this paper, we provide estimators for quantum Rényi and Tsallis entropies with nearly optimal sample complexity. Specifically, for order ααα, dimension ddd, and additive error ε\varepsilonε, 1. For 0<α<10 < α< 10<α<1, the sample complexity is O(d1+1/α/ε1/α+d1/α−1/ε2)O(d^{1+1/α}/\varepsilon^{1/α} + d^{1/α-1}/\varepsilon^{2})O(d1+1/α/ε1/α+d1/α−1/ε2) for Rényi entropy and O(d1+1/α/ε1/α+d2−2α/ε2)O(d^{1+1/α}/\varepsilon^{1/α} + d^{2-2α}/\varepsilon^2)O(d1+1/α/ε1/α+d2−2α/ε2) for Tsallis entropy. In particular, for 0<α≤1/20 < α\leq 1/20<α≤1/2, the sample complexity for both entropies is O(d1+1/α/ε1/α)O(d^{1+1/α}/\varepsilon^{1/α})O(d1+1/α/ε1/α). 2. For non-integer α>1α> 1α>1, the sample complexity is O(d2/ε1/α+d1−1/α/ε2)O(d^2/\varepsilon^{1/α} + d^{1-1/α}/\varepsilon^2)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).
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 S_α(ρ) of an N-dimensional quantum state ρ, given access to independent samples of ρ. Specifically, we provide the following quantum estimators. - A quantum estimator for S(ρ) with time complexity Õ(N²), improving the prior best time complexity Õ(N⁶) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). - A quantum estimator for S_α(ρ) with time complexity Õ(N^{4/α-2}) for 0 < α < 1 and Õ(N^{4-2/α}) for α > 1, improving the prior best time complexity Õ(N^{6/α}) for 0 < α < 1 and Õ(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 Ω(max{N/ε, N^{1/α-1}/ε^{1/α}}) for estimating S_α(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.
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\varepsilonε, rank upper bound RRR, and fixed order ααα. Tsallis entropy estimation has near-optimal rank-independent query complexity O~(ε−1/(α−1))\widetilde O(\varepsilon^{-1/(α-1)})O(ε−1/(α−1)) for 1<α<21<α<21<α<2, when the allowed dimension and rank accommodate the lower-bound instances, and O~(1/ε)\widetilde O(1/\varepsilon)O(1/ε) for α≥2α\ge2α≥2, where O~\widetilde OO suppresses logarithmic factors. The first saves a factor 1/ε1/\varepsilon1/ε; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from O~(R/ε2)\widetilde O(R/\varepsilon^2)O(R/ε2) to O~(R/ε)\widetilde O(R/\varepsilon)O(R/ε) and obtain O~(R/ε)\widetilde O(R/\varepsilon)O(R/ε) for noninteger Rényi orders 1<α<31<α<31<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)\widetilde O(R/\varepsilon)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 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)H_α(\mathbf{p})Hα​(p) for any distribution p=(pi)i=1n\mathbf{p} = (p_i)_{i=1}^np=(pi​)i=1n​ given its purified quantum query-access. For non-integer α>1α> 1α>1, our algorithm estimates Hα(p)H_α(\mathbf{p})Hα​(p) to within additive error εεε using only O~(nmax⁡{1/2,1−1/α}/ε)\widetilde{\mathcal{O}}\big(n^{\max\{1/2,1-1/α\}}/ε\big)O(nmax{1/2,1−1/α}/ε) queries, which improves the best known upper bound for dependence on both nnn and 1/ε1/ε1/ε. For 0<α<10<α<10<α<1, our algorithm has query complexity O~(n1/(2α)/εmax⁡{1,1/(2α)})\widetilde{\mathcal{O}}\big(n^{1/(2α)}/ε^{\max\{1,1/(2α)\}}\big)O(n1/(2α)/εmax{1,1/(2α)}), which improves the dependence on 1/ε1/ε1/ε of the current best upper bound without incresing its nnn 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)})Ω\big(1/ε^{\max\{1,1/(2α)\}}\big)Ω(1/εmax{1,1/(2α)}) for 0<α<10<α<10<α<1, which identifies the optimal precision dependence of our algorithm.
We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form Lφ(ρ)=tr⁡(φ(dρ))/d\mathcal{L}_φ(ρ) = \operatorname{tr}(φ(dρ))/dLφ​(ρ)=tr(φ(dρ))/d, where ρρρ is an unknown ddd-dimensional quantum state and φφφ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating Lφ(ρ)\mathcal{L}_φ(ρ)Lφ​(ρ) from properties of φφφ, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is Θ ⁣(d2εlog⁡2(d)max⁡{1,εlog⁡2(d)}+log⁡2(d)ε2), Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)\max\{1, \varepsilon\log^2(d)\}} + \frac{\log^2(d)}{\varepsilon^2}\right), Θ(εlog2(d)max{1,εlog2(d)}d2​+ε2log2(d)​), and the sample complexity of spectrum estimation is Θ ⁣(d2ε2log⁡2(d)max⁡{1,ε2log⁡2(d)}). Θ\!\left(\frac{d^2}{\varepsilon^2\log^2(d)\max\{1, \varepsilon^2\log^2(d)\}}\right). Θ(ε2log2(d)max{1,ε2log2(d)}d2​). In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.