Quantum algorithms for the generalized eigenvalue problem

TL;DR

This paper introduces a variational quantum algorithm for solving generalized eigenvalue problems, enabling the calculation of eigenvalues using shallow quantum circuits and demonstrating robustness in noisy simulations.

Contribution

The paper presents a novel full quantum eigensolver (FQGE) for generalized eigenvalues, utilizing quantum gradient descent and suitable loss functions for near-term quantum devices.

Findings

Successfully simulated a 2-qubit generalized eigenvalue problem

Demonstrated robustness of FQGE under Gaussian noise

Proposed a method to compute eigenvalues and derivatives on shallow quantum circuits

Abstract

The generalized eigenvalue (GE) problems are of particular importance in various areas of science engineering and machine learning. We present a variational quantum algorithm for finding the desired generalized eigenvalue of the GE problem, $\mathcal{A}|\psi\rangle=\lambda\mathcal{B}|\psi\rangle$, by choosing suitable loss functions. Our approach imposes the superposition of the trial state and the obtained eigenvectors with respect to the weighting matrix $\mathcal{B}$ on the Rayleigh-quotient. Furthermore, both the values and derivatives of the loss functions can be calculated on near-term quantum devices with shallow quantum circuit. Finally, we propose a full quantum generalized eigensolver (FQGE) to calculate the minimal generalized eigenvalue with quantum gradient descent algorithm. As a demonstration of the principle, we numerically implement our algorithms to conduct a 2-qubit simulation and successfully find the generalized eigenvalues of the matrix pencil $(\mathcal{A},\,\mathcal{B})$. The numerically experimental result indicates that FQGE is robust under Gaussian noise.