A Non-Orthogonal Variational Quantum Eigensolver

TL;DR

This paper introduces a non-orthogonal variational quantum eigensolver that improves ground state approximations by solving a generalized eigenvalue problem within a parametrized quantum state subspace, reducing circuit complexity.

Contribution

It presents a novel extension to the variational quantum eigensolver that allows systematic wavefunction improvements without significantly increasing circuit complexity.

Findings

Effective measurement strategies for Hamiltonian and overlap matrices.

Monte Carlo scheme for energy uncertainty estimation.

Application to strongly correlated molecular systems.

Abstract

Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H$_4$ and the $\pi$-system of Hexatriene (C$_6$H$_8$).