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Quantum Neural Estimation of Entropies

Ziv Goldfeld, Dhrumil Patel, Sreejith Sreekumar, Mark M. Wilde

TL;DR

This work presents a variational quantum estimation framework, termed quantum neural estimation, to estimate quantum entropies and measured relative entropies from copies of unknown states. It combines a classical neural network to parameterize eigenvalues with a parameterized quantum circuit for eigenvectors, enabling a variational objective that can be optimized using sampling on a quantum computer. The approach delivers accurate estimates for von Neumann and Rényi entropies, as well as measured relative entropies and fidelity, demonstrated on two- and six-qubit mixed states, and discusses extensions and practical challenges such as barren-plateau effects. The method offers a scalable alternative to full state tomography for entropy estimation with potential impact on downstream quantum information tasks and experiments.

Abstract

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 Rényi entropies, as well as the measured relative entropy and measured Rényi relative entropy. Our approach first parameterizes 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 as a promising approach for usage in downstream tasks.

Quantum Neural Estimation of Entropies

TL;DR

This work presents a variational quantum estimation framework, termed quantum neural estimation, to estimate quantum entropies and measured relative entropies from copies of unknown states. It combines a classical neural network to parameterize eigenvalues with a parameterized quantum circuit for eigenvectors, enabling a variational objective that can be optimized using sampling on a quantum computer. The approach delivers accurate estimates for von Neumann and Rényi entropies, as well as measured relative entropies and fidelity, demonstrated on two- and six-qubit mixed states, and discusses extensions and practical challenges such as barren-plateau effects. The method offers a scalable alternative to full state tomography for entropy estimation with potential impact on downstream quantum information tasks and experiments.

Abstract

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 Rényi entropies, as well as the measured relative entropy and measured Rényi relative entropy. Our approach first parameterizes 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 as a promising approach for usage in downstream tasks.
Paper Structure (17 sections, 51 equations, 2 figures, 1 algorithm)

This paper contains 17 sections, 51 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Literature Review
  4. Quantum Entropies and Divergences
  5. Measured Relative Entropy and von Neumann Entropy
  6. Measured Rényi Relative Entropy and Rényi Entropy
  7. Quantum Neural Estimation of Entropies
  8. Measured Relative Entropy
  9. Von Neumann Entropy
  10. Measured Rényi Relative Entropy
  11. Rényi Entropy
  12. Root Fidelity
  13. Implementation and Experiments
  14. Methodology
  15. Results
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Two-qubit examples. Convergence of the quantum neural estimation algorithm for the von Neumann entropy, measured relative entropy, Rényi entropy ($\alpha = 2.5$), root fidelity, and measured Rényi relative entropy ($\alpha = 2.5$). Two input instances are generated at random for each of these quantities. For each instance, results are presented both with (blue) and without (purple) the classical neural network approximation of eigenvalues. Each solid or dotted line shows the mean value of the estimate, the shaded region represents the standard deviation, while the ground truth is marked by the dashed line. Python source files for reproducing these plots are available with the arXiv posting of our paper.
  • Figure 2: Six-qubit examples. Convergence of the quantum neural estimation algorithm for the von Neumann entropy, measured relative entropy, Rényi entropy ($\alpha = 1.05$), root fidelity, and measured Rényi relative entropy ($\alpha = 1.05$). Two input instances are generated at random for each of these quantities. For each instance, results are presented both with (blue) and without (purple) the classical neural network approximation of eigenvalues. Each solid or dotted line shows the mean value of the estimate, the shaded region represents the standard deviation, while the ground truth is marked by the dashed line. Python source files for reproducing these plots are available with the arXiv posting of our paper.

Theorems & Definitions (1)

  • Remark 1: Comparison to existing variational estimators