Analytical formulation of the second-order derivative of energy for orbital-optimized variational quantum eigensolver: application to polarizability

TL;DR

This paper introduces a quantum-classical hybrid algorithm for efficiently calculating the second-order energy derivatives, such as polarizability, in molecular systems using near-term quantum computers, validated through simulations.

Contribution

It provides a novel analytical derivative formula for OO-VQE that can be computed without ancillary qubits and demonstrates its advantages over numerical derivatives.

Findings

Analytical derivatives require fewer quantum measurements than numerical ones.

The method successfully computes polarizabilities of water, thiophene, and furan molecules.

Numerical simulations confirm the efficiency and accuracy of the proposed approach.

Abstract

We develop a quantum-classical hybrid algorithm to calculate the analytical second-order derivative of the energy for the orbital-optimized variational quantum eigensolver (OO-VQE), which is a method to calculate eigenenergies of a given molecular Hamiltonian by utilizing near-term quantum computers and classical computers. We show that all quantities required in the algorithm to calculate the derivative can be evaluated on quantum computers as standard quantum expectation values without using any ancillary qubits. We validate our formula by numerical simulations of quantum circuits for computing the polarizability of the water molecule, which is the second-order derivative of the energy with respect to the electric field. Moreover, the polarizabilities and refractive indices of thiophene and furan molecules are calculated as a testbed for possible industrial applications. We finally analyze the error-scaling of the estimated polarizabilities obtained by the proposed analytical derivative versus the numerical one obtained by the finite difference. Numerical calculations suggest that our analytical derivative requires fewer measurements (runs) on quantum computers than the numerical derivative to achieve the same fixed accuracy.