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arXiv:2511.19289 · Quantum

Performance Guarantees for Quantum Neural Estimation of Entropies

Sreejith Sreekumar, Ziv Goldfeld, Mark M. Wilde

quant-phcs.ITcs.LGSubmitted Nov 24, 2025• Announced Nov 25, 2025• 0 citations• Revised May 14, 2026 (v2)
v1 · Nov 24, 2025v2 · May 14, 2026
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Abstract

Estimating 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 $O(|Θ(\mathcal{U})|d/ε^2)$ for QNE with a quantum circuit parameter set $Θ(\mathcal{U})$, which has minimax optimal dependence on the accuracy $ε$. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to $O(|Θ(\mathcal{U})|\mathrm{polylog}(d)/ε^2)$. Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.

Journal reference
Quantum 10, 2113 (2026)
DOI
10.22331/q-2026-05-21-2113
Comments
43 pages

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Estimating 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 ddd with bounded Thompson metric, our theory establishes a copy complexity of O(∣Θ(U)∣d/ε2)O(|Θ(\mathcal{U})|d/ε^2)O(∣Θ(U)∣d/ε2) for QNE with a quantum circuit parameter set Θ(U)Θ(\mathcal{U})Θ(U), which has minimax optimal dependence on the accuracy εεε. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to O(∣Θ(U)∣polylog(d)/ε2)O(|Θ(\mathcal{U})|\mathrm{polylog}(d)/ε^2)O(∣Θ(U)∣polylog(d)/ε2). Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.
Entropy measures quantify the amount of information and correlation present in a quantum system. In practice, when the quantum state is unknown and only copies thereof are available, one must resort to the estimation of such entropy measures. Here we propose a variational quantum algorithm for estimating the von Neumann and R\'enyi entropies, as well as the measured relative entropy and measured R\'enyi relative entropy. Our approach first parametrizes a variational formula for the measure of interest by a quantum circuit and a classical neural network and then optimizes the resulting objective over parameter space. Numerical simulations of our quantum algorithm are provided, using a noiseless quantum simulator. The algorithm provides accurate estimates of the various entropy measures for the examples tested, which renders it a promising approach for usage in downstream tasks.
A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.
A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.
A large body of recent work has begun to explore the potential of parametrized quantum circuits PQCs as machine learning models, within the framework of hybrid quantum classical optimization. In particular, theoretical guarantees on the out of sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC based models that depend explicitly on the strategy used for data encoding. These imply bounds on the performance of trained PQC based models on unseen data. Moreover, our results facilitate the selection of optimal data encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC based models via trigonometric functions. Our generalization bounds emphasize the importance of well considered data encoding strategies for PQC based models
Fidelity is a fundamental measure for the closeness of two quantum states, which is important both from a theoretical and a practical point of view. Yet, in general, it is difficult to give good estimates of fidelity, especially when one works with mixed states over Hilbert spaces of very high dimension. Although, there has been some progress on fidelity estimation, all prior work either requires a large number of identical copies of the relevant states, or relies on unproven heuristics. In this work, we improve on both of these aspects by developing new and efficient quantum algorithms for fidelity estimation with provable performance guarantees in case at least one of the states is approximately low-rank. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation, as well as density matrix exponentiation and quantum spectral sampling. As a complementary result, we prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general, by giving a sample complexity lower bound that depends polynomially on the dimension. Moreover, if circuit descriptions for the relevant states are provided, we show that the task is hard for the complexity class called (honest verifier) quantum statistical zero knowledge via a reduction to a closely related result by Watrous.
The remarkable generalization capabilities of overparameterized neural networks remain one of the most profound mysteries in modern machine learning. Despite having sufficient capacity to simply memorize training data, these networks consistently achieve superior performance on unseen test examples. Traditional analyses focusing on weight distributions, loss landscapes, and optimization dynamics have provided partial insights but fail to capture the complex correlational structure that emerges between neurons during learning. This paper introduces a comprehensive quantum information framework for analyzing neural network representations, treating hidden layer activations as quantum states and characterizing them through density matrices, von Neumann entropy, purity, and related quantum information metrics. Through systematic experiments on synthetic classification tasks and algorithmic reasoning problems spanning over 3000 cumulative training epochs and 5000 Wang-Landau sampling iterations, we demonstrate that these quantum metrics reveal structure completely invisible to classical analysis. Our results show that stochastic gradient descent systematically reduces von Neumann entropy while increasing purity as networks learn, transitioning from highly mixed quantum states with entropy approximately 1.42 and purity approximately 0.29 to purer configurations with entropy approximately 1.19 and purity approximately 0.37 that achieve superior test accuracy of 99.5%. Using Wang-Landau sampling to explore the equilibrium entropy landscape, we discover that neural networks naturally favor high-entropy states with entropy approximately 1.42 and moderate accuracy of 62.0%, and that optimization actively purifies the quantum state to achieve task performance. On modular arithmetic tasks exhibiting the grokking phenomenon, we observe that von Neumann entropy increases dramatically from 1.55 to 3.29, representing a 112% increase, during an extended 2000-epoch memorization phase before any improvement in generalization, while purity decreases correspondingly from 0.444 to 0.042, a 90.5% decrease, approaching the maximally mixed limit of 0.01. This entropy accumulation phase appears necessary for eventual generalization, with the network building quantum correlations at a rate of approximately 0.00097 nats per epoch during the plateau. Weight entropy, in contrast, shows minimal change of only 11% increase over the same period. Correlation analysis reveals that von Neumann entropy and purity are almost perfectly anticorrelated with a correlation coefficient of 1 -0.99, while weight entropy shows weak correlation with quantum metrics with absolute correlation coefficients less than 0.3. The effective rank of the density matrix stabilizes near 2.7 to 3.0 for the 6-neuron classifier, indicating that the network effectively uses approximately three independent activation modes. These findings establish quantum information metrics as powerful tools for understanding neural network learning dynamics, provide the first quantitative evidence for the entropy accumulation interpretation of grokking, and suggest that neural networks undergo a quantum-inspired phase transition during learning. We discuss implications for generalization theory, optimization algorithm design, and the statistical physics of deep learning, and outline a comprehensive research agenda for extending these methods to larger architectures and more complex tasks.