Quantum Learning of Classical Correlations with continuous-domain Pauli Correlation Encoding

TL;DR

This paper introduces a quantum machine learning framework for estimating classical covariance matrices using parameterized quantum circuits, with two novel estimators and analysis of their properties.

Contribution

It presents two new quantum covariance estimators, analyzes their trade-offs, and demonstrates their effectiveness through numerical simulations.

Findings

The C-Estimator enforces positive semidefiniteness via Cholesky factorization.

The E-Estimator offers computational efficiency by direct expectation value estimation.

Regularization parameters can mitigate barren plateau issues in training.

Abstract

We propose a quantum machine learning framework for estimating classical covariance matrices using parameterized quantum circuits within the Pauli-Correlation-Encoding (PCE) paradigm. We introduce two quantum covariance estimators: the C-Estimator, which constructs the covariance matrix through a Cholesky factorization to enforce positive (semi)definiteness, and a computationally efficient E-Estimator, which directly estimates covariance entries from observable expectation values. We analyze the trade-offs between the two estimators in terms of qubit requirements and learning complexity, and derive sufficient conditions on regularization parameters to ensure positive (semi)definiteness of the estimators. Furthermore, we show that the barren plateau phenomenon in training the variational quantum circuit for E-estimator can be mitigated by appropriately choosing the regularization parameters in the loss function for HEA ansatz. The proposed framework is evaluated through numerical simulations using randomly generated covariance matrices. We examine the convergence behavior of the estimators, their sensitivity to low-rank assumptions, and their performance in covariance completion with partially observed matrices. The results indicate that the proposed estimators provide a robust approach for learning covariance matrices and offer a promising direction for applying quantum machine learning techniques to high-dimensional statistical estimation problems.