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arXiv:2412.01696 · Communications Physics

Sample-efficient estimation of nonlinear quantum state functions

Hongshun Yao, Yingjian Liu, Tengxiang Lin, Xin Wang

Published Jun 6, 2026• 1 citation
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Abstract

Estimating nonlinear functions of quantum states is essential for quantum information processing tasks such as entropy quantification, fidelity estimation, and entanglement spectroscopy. Existing approaches based on quantum state tomography incur exponential resource overhead, while methods relying on purified query access impose significant practical constraints. Here, we introduce the quantum state function framework, which extends the swap test via a linear combination of unitaries and parameterized quantum circuits. The framework estimates any normalized degree-n polynomial function of a quantum state with precision ε using $${{{\mathcal{O}}}}(n/{\varepsilon }^{2})$$ copies, operating directly on identical copies without block-encoding or purified queries. Applied to von Neumann entropy, quantum relative entropy, and state fidelity, it achieves sample complexity $$\widetilde{{{{\mathcal{O}}}}}({\gamma }^{2}/({\varepsilon }^{2}\kappa ))$$, where κ is the minimal nonzero eigenvalue and γ a normalization factor. This framework establishes a unified paradigm for nonlinear quantum state processing, broadening the toolkit for practical quantum data analysis. Estimating nonlinear functions of quantum states is essential for quantum information processing but often computationally expensive. The quantum state function framework estimates a degree-n polynomial of a quantum state with $${{{\mathcal{O}}}}(n/{\varepsilon }^{2})$$ copies without purified query access, enabling practical entropy, fidelity, and eigenvalue estimation.

DOI
10.1038/s42005-026-02710-8

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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.
The von Neumann and quantum R\'enyi entropies characterize fundamental properties of quantum systems and lead to many theoretical and practical applications. Quantum algorithms using a purified quantum query model can speed up quantum entropy estimation, while little is known about the complexity of using identical copies of the quantum state. This paper presents quantum entropy estimation algorithms with a cost of copies scaling polynomially in the rank of the state. In contrast to current methods that depend on the dimension of the system, our methods could provide exponential resource savings in the scenario of low-rank states. Furthermore, we show how to construct quantum circuits using primitive single-qubit or two-qubit gates efficiently and thus provide practical methods for estimating quantum entropies of quantum systems. We also conduct simulation experiments to show the effectiveness and noise robustness of our algorithms.
A fundamental task in quantum information science is to measure nonlinear functionals of quantum states, such as Tr(ρ^{k}O). Intuitively, one expects that computing a kth order quantity generally requires O(k) copies of the state ρ, and we rigorously establish this lower bound under sample access to ρ. Surprisingly, this limitation can be overcome when one has purified access via a unitary that prepares a purification of ρ, a scenario naturally arising in quantum simulation and computation. In this setting, we find a different lower bound of Θ(sqrt[k]) and present a quantum algorithm that achieves this bound, demonstrating a quadratic advantage over sample-based methods. The key technical innovation lies in a designed quantum algorithm and optimal polynomial approximation theory-specifically, Chebyshev polynomial approximations tailored to the boundary behavior of power functions. Our results unveil a fundamental distinction between sample and purified access to quantum states, with broad implications for estimating quantum entropies and quantum Fisher information, realizing quantum virtual distillation and cooling and evaluating other multiple nonlinear quantum observables with classical shadows.
Direct fidelity estimation (DFE) is a famous tool for estimating the fidelity with a target pure state. However, such a method generally requires exponentially many sampling copies due to the large magic of the target state. This work proposes a sample- and gate-efficient fidelity estimation algorithm that is affordable within feasible quantum devices. We show that the fidelity estimation with pure states close to the structure of phase states, for which sample-efficient DFE is limited by their strong entanglement and magic, can be done by using O(poly(n))\mathcal{O}({\rm poly}(n))O(poly(n)) sampling copies, with a single nnn-qubit fan-out gate. As the target state becomes a phase state, the sampling complexity reaches O(1)\mathcal{O}(1)O(1). Such a drastic improvement stems from a crucial step in our scheme, the so-called phase stripping, which can significantly reduce the target-state magic. Furthermore, we convert a complex diagonal gate resource, which is needed to design a phase-stripping-adapted algorithm, into nonlinear classical post-processing of Pauli measurements so that we only require a single fan-out gate. Additionally, as another variant using the nonlinear post-processing, we propose a nonlinear extension of the conventional DFE scheme. Here, the sampling reduction compared to DFE is also guaranteed, while preserving the Pauli measurement as the only circuit resource. We expect our work to contribute to establishing noise-resilient quantum algorithms by enabling a significant reduction in sampling overhead for fidelity estimation under the restricted gate resources, and ultimately to clarifying a fundamental gap between the resource overhead required to understand complex physical properties and that required to generate them.
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 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.