Euclidean time method in Generalized Eigenvalue Equation

TL;DR

This paper extends the Euclidean time method of the variational quantum eigensolver to solve generalized eigenvalue problems involving hermitian operators, demonstrating its effectiveness through numerical tests and an application to the hydrogen atom.

Contribution

The authors develop a modified Euclidean time formalism tailored for generalized eigenvalue equations, expanding the applicability of the variational quantum eigensolver.

Findings

The formalism performs well in numerical tests.

It successfully computes the hydrogen atom's electric polarizability.

Results are slightly less than perturbation method estimates.

Abstract

We develop the Euclidean time method of the variational quantum eigensolver for solving the generalized eigenvalue equation $A \ket{\phi_n} = \lambda_n B \ket{\phi_n}$, where $A$ and $B$ are hermitian operators, and $\ket{\phi_n}$ and $\lambda_n$ are called the eigenvector and the corresponding eigenvalue of this equation respectively. For the purpose we modify the usual Euclidean time formalism, which was developed for solving the time-independent Schr\"{o}dinger equation. We apply our formalism to three numerical examples for test. It is shown that our formalism works very well in all numerical examples. We also apply our formalism to the hydrogen atom and compute the electric polarizability. It turns out that our result is slightly less than that of the perturbation method.