Improving the Accuracy of the Variational Quantum Eigensolver for Molecular Systems by the Explicitly-Correlated Perturbative [2]-R12-Correction

TL;DR

This paper integrates an explicitly correlated perturbative correction into the VQE method, significantly improving molecular energy accuracy without extra quantum resources by using RDMs and molecular integrals, especially with PNOs.

Contribution

It introduces a VQE extension with [2]-R12 correction using RDMs, enhancing accuracy efficiently with PNOs and minimal basis sets.

Findings

VQE+[2]R12 improves energy accuracy significantly.

PNOs reduce computational cost of the correction.

Combination of PNOs with [2]R12 is highly effective.

Abstract

We provide an integration of the universal, perturbative explicitly correlated [2]$_\text{R12}$-correction in the context of the Variational Quantum Eigensolver (VQE). This approach is able to increase the accuracy of the underlying reference method significantly while requiring no additional quantum resources. Our proposed approach only requires knowledge of the one- and two-particle reduced density matrices (RDMs) of the reference wavefunction; these can be measured after having reached convergence in VQE. The RDMs are then combined with a set of molecular integrals. This computation comes at a cost that scales as the sixth power of the number of electrons. We explore the performance of the VQE+[2]$_\text{R12}$ approach using both conventional Gaussian basis sets and our recently proposed directly determined pair-natural orbitals obtained by multiresolution analysis (MRA-PNOs). Both Gaussian orbital and PNOs are investigated as a potential set of complementary basis functions in the computation of [2]$_\text{R12}$. In particular the combination of MRA-PNOs with [2]$_\text{R12}$ has turned out to be very promising -- persistently throughout our data, this allowed very accurate simulations at a quantum cost of a minimal basis set. Additionally, we found that the deployment of PNOs as complementary basis can greatly reduce the number of complementary basis functions that enter the computation of the correction at a cubic complexity.