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Performance Guarantees for Quantum Neural Estimation of Entropies

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Performance Guarantees for Quantum Neural Estimation of Entropies

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AbstractEstimating quantum entropies and divergences is an important problem in quantum physics, information theory, and machine learning. Quantum neural estimators (QNEs), which utilize a hybrid classical-quantum architecture, have recently emerged as an appealing computational framework for estimating these measures. Such estimators combine classical neural networks with parametrized quantum circuits, and their deployment typically entails tedious tuning of hyperparameters controlling the sample size, network architecture, and circuit topology. This work initiates the study of formal guarantees for QNEs of measured (Rényi) relative entropies in the form of non-asymptotic error risk bounds. We further establish exponential tail bounds showing that the error is sub-Gaussian and thus sharply concentrates about the ground truth value. For an appropriate sub-class of density operator pairs on a space of dimension $d$ with bounded Thompson metric, our theory establishes a copy complexity of $O(|\Theta(\mathcal{U})|d/\epsilon^2)$ for QNE with a quantum circuit parameter set $\Theta(\mathcal{U})$, which has minimax optimal dependence on the accuracy $\epsilon$. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to $O(|\Theta(\mathcal{U})|\mathrm{polylog}(d)/\epsilon^2)$. Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.Popular summaryThe paper studies how to reliably use a quantum neural estimator (QNE) to estimate a fundamental entropic quantity in quantum physics and information theory called measured Rényi relative entropy. This quantity subsumes entropy and measured relative entropy as special cases, which respectively measure how much uncertainty (or information) exists in a quantum system and how different two quantum states are. Such quantities are central objects in quantum computing, quantum cryptography, machine learning, and thermodynamics. However, calculating them exactly for large quantum systems is extremely hard. QNE learns to estimate the entropic quantity directly by training a hybrid model composed of a classical neural net and a parametrized quantum circuit, based on measurements of quantum states. Thus, it works differently from trying to reconstruct entire unknown quantum states, which can often be prohibitively expensive. The paper provides the first performance guarantees for QNE via non-asymptotic error bounds and the number of copies of quantum states required for accurate estimation. The bounds are characterized in terms of properties of the neural network, quantum circuit, and the target quantum state class. These results provide practitioners with guidance for choosing network size, circuit depth, and deciding how much quantum resources are needed for reliable estimation before running expensive quantum experiments. Additionally, we discuss how extra symmetries such as permutation invariance of quantum states can be leveraged to improve performance at a reduced resource budget. Overall, the work provides a sound theoretical grounding for QNE while also providing guidance for its implementation.► BibTeX data@article{Sreekumar2026performance, doi = {10.22331/q-2026-05-21-2113}, url = {https://doi.org/10.22331/q-2026-05-21-2113}, title = {Performance {G}uarantees for {Q}uantum {N}eural {E}stimation of {E}ntropies}, author = {Sreekumar, Sreejith and Goldfeld, Ziv and Wilde, Mark M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2113}, month = may, year = {2026} }► References [1] John von Neumann. ``Thermodynamik quantenmechanischer gesamtheiten''. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1927, 273–291 (1927). url: http://eudml.org/doc/59231. http://eudml.org/doc/59231 [2] Claude E. Shannon. ``A mathematical theory of communication''.

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Wilde, "Generative modeling using evolved quantum Boltzmann machines", arXiv:2512.02721, (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-05-21 12:57:24). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-05-21 12:57:22: Could not fetch cited-by data for 10.22331/q-2026-05-21-2113 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractEstimating quantum entropies and divergences is an important problem in quantum physics, information theory, and machine learning. Quantum neural estimators (QNEs), which utilize a hybrid classical-quantum architecture, have recently emerged as an appealing computational framework for estimating these measures. Such estimators combine classical neural networks with parametrized quantum circuits, and their deployment typically entails tedious tuning of hyperparameters controlling the sample size, network architecture, and circuit topology. This work initiates the study of formal guarantees for QNEs of measured (Rényi) relative entropies in the form of non-asymptotic error risk bounds. We further establish exponential tail bounds showing that the error is sub-Gaussian and thus sharply concentrates about the ground truth value. For an appropriate sub-class of density operator pairs on a space of dimension $d$ with bounded Thompson metric, our theory establishes a copy complexity of $O(|\Theta(\mathcal{U})|d/\epsilon^2)$ for QNE with a quantum circuit parameter set $\Theta(\mathcal{U})$, which has minimax optimal dependence on the accuracy $\epsilon$. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to $O(|\Theta(\mathcal{U})|\mathrm{polylog}(d)/\epsilon^2)$. Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.Popular summaryThe paper studies how to reliably use a quantum neural estimator (QNE) to estimate a fundamental entropic quantity in quantum physics and information theory called measured Rényi relative entropy. This quantity subsumes entropy and measured relative entropy as special cases, which respectively measure how much uncertainty (or information) exists in a quantum system and how different two quantum states are. Such quantities are central objects in quantum computing, quantum cryptography, machine learning, and thermodynamics. However, calculating them exactly for large quantum systems is extremely hard. QNE learns to estimate the entropic quantity directly by training a hybrid model composed of a classical neural net and a parametrized quantum circuit, based on measurements of quantum states. Thus, it works differently from trying to reconstruct entire unknown quantum states, which can often be prohibitively expensive. The paper provides the first performance guarantees for QNE via non-asymptotic error bounds and the number of copies of quantum states required for accurate estimation. The bounds are characterized in terms of properties of the neural network, quantum circuit, and the target quantum state class. These results provide practitioners with guidance for choosing network size, circuit depth, and deciding how much quantum resources are needed for reliable estimation before running expensive quantum experiments. Additionally, we discuss how extra symmetries such as permutation invariance of quantum states can be leveraged to improve performance at a reduced resource budget. Overall, the work provides a sound theoretical grounding for QNE while also providing guidance for its implementation.► BibTeX data@article{Sreekumar2026performance, doi = {10.22331/q-2026-05-21-2113}, url = {https://doi.org/10.22331/q-2026-05-21-2113}, title = {Performance {G}uarantees for {Q}uantum {N}eural {E}stimation of {E}ntropies}, author = {Sreekumar, Sreejith and Goldfeld, Ziv and Wilde, Mark M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2113}, month = may, year = {2026} }► References [1] John von Neumann. ``Thermodynamik quantenmechanischer gesamtheiten''. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1927, 273–291 (1927). url: http://eudml.org/doc/59231. http://eudml.org/doc/59231 [2] Claude E. Shannon. ``A mathematical theory of communication''.

In Contemporary Mathematics. Volume 529 of Entropy and the Quantum.

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Performance Guarantees for Quantum Neural Estimation of Entropies

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