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Performance Guarantees for Quantum Neural Estimation of Entropies

Sreejith Sreekumar, Ziv Goldfeld, Mark M. Wilde

TL;DR

The paper develops a theoretical foundation for quantum neural estimators (QNE) to estimate measured relative entropies and their Rényi variants from quantum samples. It presents non-asymptotic risk bounds and sub-Gaussian concentration results, with copy complexity bounds that depend on the classical-quantum circuit parameters and system dimension, and improves dimension dependence under permutation invariance via Schur-Weyl duality. The framework integrates neural network approximation of eigenvalues with PQC-driven eigenvectors, under assumptions on covering entropies and embedding schemes, and explicit results for shallow and deep networks. This work provides principled guidance for hyperparameter tuning and delineates regimes where QNE is minimax-optimal in $n$ and where symmetry can dramatically reduce resource demands, thereby informing practical deployment in quantum entropy estimation tasks.

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.

Performance Guarantees for Quantum Neural Estimation of Entropies

TL;DR

The paper develops a theoretical foundation for quantum neural estimators (QNE) to estimate measured relative entropies and their Rényi variants from quantum samples. It presents non-asymptotic risk bounds and sub-Gaussian concentration results, with copy complexity bounds that depend on the classical-quantum circuit parameters and system dimension, and improves dimension dependence under permutation invariance via Schur-Weyl duality. The framework integrates neural network approximation of eigenvalues with PQC-driven eigenvectors, under assumptions on covering entropies and embedding schemes, and explicit results for shallow and deep networks. This work provides principled guidance for hyperparameter tuning and delineates regimes where QNE is minimax-optimal in and where symmetry can dramatically reduce resource demands, thereby informing practical deployment in quantum entropy estimation tasks.

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 with bounded Thompson metric, our theory establishes a copy complexity of for QNE with a quantum circuit parameter set , which has minimax optimal dependence on the accuracy . Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to . Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.

Paper Structure

This paper contains 30 sections, 14 theorems, 154 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Summary of Contributions
  4. Literature Review and Related Work
  5. Paper Organization
  6. Preliminaries
  7. Measured Relative Entropy and Measured Rényi Relative Entropy
  8. Minimax Estimation of Measured Relative Entropies
  9. Quantum Neural Estimator for Estimating Measured Relative Entropies
  10. Approximating Functions and Operators in Quantum Neural Estimation
  11. Classical Neural Networks
  12. Parametrized Quantum Circuits
  13. Performance Guarantees for Quantum Neural Estimation
  14. Measured Relative Entropy
  15. Measured Rényi Relative entropy
  16. ...and 15 more sections

Key Result

Theorem 1

Consider a QNE for estimating $\mathsf{D}_{\mathsf{M}}(\rho\Vert\sigma)$ with classical neural net $\mathcal{F} \subseteq \mathcal{F}_b$ satisfying Assumption Assump1 and a PQC $\mathcal{U}(\delta,\mathcal{S})$ parametrized by $\Theta(\delta,\mathcal{S})$ for $\mathcal{S} \subseteq \mathcal{S}_d(b)$ where $\kappa(\mathcal{F})$ and $\varepsilon(\mathcal{F},\mathcal{S})$ are as given in eq:finentrin

Theorems & Definitions (20)

  • Definition 1: Covering entropy
  • Theorem 1: Performance guarantees for measured relative entropy
  • Remark 1: Effective copy complexity
  • Definition 2: Neural network class
  • Corollary 2: Performance guarantees with shallow nets
  • Remark 2: Lower bound on minimax risk
  • Proposition 3: Relation between classes of density operators
  • Proposition 4: Performance guarantees with deep nets
  • Theorem 5: Performance guarantees for measured Rényi relative entropy
  • Corollary 6: Performance guarantees with shallow neural net
  • ...and 10 more