Quantum Neural Estimation Achieves sub-Gaussian Error Risk Bounds for Rényi Relative Entropies in Machine Learning

Estimating entropies and divergences presents a fundamental challenge across physics, information theory, and machine learning, and researchers are increasingly turning to quantum neural estimators (QNEs) as a promising computational approach. Sreejith Sreekumar from L2S, CNRS, CentraleSupélec, University of Paris-Saclay, alongside Ziv Goldfeld and Mark M. Wilde from Cornell University, now establish formal performance guarantees for these hybrid quantum-classical estimators when applied to measured Rényi relative entropies. Their work delivers non-asymptotic error risk bounds and demonstrates that the estimation error concentrates sharply around the true value, offering a level of reliability previously lacking in this field. Crucially, the team establishes a copy complexity of for QNEs under certain conditions, achieving optimal accuracy and paving the way for principled implementation and more efficient hyperparameter tuning in practical applications.

These estimators combine classical neural networks with parametrized quantum circuits, and their effective deployment often requires careful tuning of hyperparameters controlling sample size, network architecture, and circuit topology. This research initiates a formal study of guarantees for quantum noise estimation (QNE) of measured Rényi relative entropies, establishing non-asymptotic error risk bounds. The researchers further demonstrate exponential tail bounds, proving that the error concentrates sharply around the true value, and providing a rigorous foundation for understanding the accuracy and reliability of these estimators in practical applications.

Quantum Information, Estimation, and Channel Theory

This compilation represents a comprehensive bibliography related to quantum information theory, machine learning, statistical estimation, and related mathematical fields. It encompasses key themes and a categorized breakdown to facilitate navigation, focusing on quantum entropy and divergences, such as Von Neumann and Rényi entropies, and their applications, alongside work on semi-definite programming and limit distributions. It also covers quantum channels and operations, detailing their characterization and efficient manipulation using techniques like Schur and Clebsch-Gordan transforms. A significant portion addresses quantum state tomography and estimation, employing statistical estimation techniques to determine quantum states and parameters, including entanglement measures like the relative entropy of entanglement.

A substantial section focuses on quantum machine learning (QML), particularly variational quantum algorithms (VQAs) utilizing parametrized quantum circuits (PQCs) as neural network analogs, addressing challenges like barren plateaus, exploring initialization strategies, and investigating expressibility. References also cover the broader theory and practice of quantum neural networks, including their limitations and potential advantages, and recent work exploiting symmetry and equivariance in QNNs to improve performance and overcome barren plateaus. The bibliography extends to statistical estimation and approximation theory, encompassing entropy and capacity estimation, both in classical and quantum settings, with references to minimax rates and best polynomial approximation. It includes work on universal approximation theorems, concentration inequalities, limit theorems, and theoretical foundations for statistical inference.

Mathematical tools and techniques are also well represented, including functional analysis concepts like ε-entropy and ε-capacity, classical results on interpolation and approximation theory, group-theoretic approaches to quantum information, and algorithms for approximating quantum gates. Key research areas highlighted within the bibliography include overcoming barren plateaus in VQAs, determining the statistical limits of quantum estimation, balancing expressibility and trainability in PQCs, leveraging symmetry in QML, and applying statistical learning theory to quantum data. Overall, this bibliography represents a comprehensive resource for researchers working at the intersection of quantum information theory, machine learning, and statistical inference, reflecting the rapid growth and increasing complexity of this interdisciplinary field.

Quantum Estimation Achieves Polynomial Scaling with Accuracy

Scientists have established formal guarantees for the performance of quantum neural estimators (QNEs) when measuring Rényi relative entropies, delivering non-asymptotic error risk bounds and demonstrating sub-Gaussian error concentration. The research establishes that for appropriate density operator pairs in a space of dimension d, QNE achieves a copy complexity of O(d) with accuracy dependent on the error, representing a significant advancement in efficient quantum estimation. Furthermore, when applied to permutation invariant density operators, the dimension dependence improves to a polynomial scaling, leveraging the principles of Schur-Weyl duality to reduce the degrees of freedom required for accurate estimation. Experiments reveal that QNE achieves an upper bound on the expected absolute error for estimating measured relative entropy and measured Rényi relative entropy over a defined class of density operators.

Specifically, the team demonstrated sub-Gaussian tails for the absolute error, indicating a predictable and well-behaved concentration of results around the true value. Analysis of shallow neural networks within QNE shows that the achieved error bounds align closely with the theoretical lower bounds for any estimator, confirming the efficiency of the approach. The team quantified the copy complexity of QNE, showing that while it scales exponentially with the number of qudits, N, it achieves a much more favorable polynomial dependence for permutation invariant states across N qudits. Measurements confirm that QNE’s performance is not susceptible to classical simulability arguments, as it relies on direct access to quantum samples, exploiting intrinsically quantum input data. This work delivers a crucial step towards principled implementation of QNEs and guides hyperparameter tuning for measured relative entropies, paving the way for more efficient quantum information processing.

Rényi Entropy Estimation, Scalable Quantum Neural Networks

This work establishes formal guarantees for quantum neural estimators (QNEs) used to estimate measured Rényi relative entropies, a crucial task in fields ranging from physics to machine learning. Researchers successfully derived non-asymptotic error risk bounds and demonstrated that the estimation error concentrates sharply around the true value, exhibiting sub-Gaussian behaviour. Importantly, the team established a copy complexity of for QNEs, meaning the number of samples required for accurate estimation scales favourably with the desired accuracy and system dimension. This result represents a significant advance in understanding the efficiency of these estimators.

Furthermore, for specific cases involving permutation-invariant density operators, the researchers improved the dimension dependence of the copy complexity, demonstrating enhanced performance in these scenarios. These findings facilitate principled implementation of QNEs and provide guidance for tuning hyperparameters in practical applications. The authors acknowledge that the derived bounds rely on certain assumptions regarding the smoothness of the underlying functions and the size of the quantum circuit parameters. Future research could focus on relaxing these assumptions and extending the analysis to more complex systems and estimators, potentially broadening the applicability of these techniques.