# Improving Energy and Molecular Properties by Convergence of the One‐Particle Reduced Density Matrix in Variational Quantum Eigensolvers (VQE)

## TL;DR

This paper introduces new strategies to improve the accuracy of quantum simulations of molecular systems by ensuring convergence of the one-particle reduced density matrix.

## Contribution

The novel contribution is the introduction of two new VQE algorithms, VQE* and VQE-LD, that incorporate 1-RDM convergence for better energy and molecular property predictions.

## Key findings

- VQE* and VQE-LD improved accuracy of energy and density-based properties like dipole moments and electron density.
- For GateFabric ansatz, both approaches significantly enhanced energy accuracy and 1-RDM quality.
- The convergence of 1-RDM provides a simple yet effective strategy for reliable molecular predictions on quantum computers.

## Abstract

The variational quantum eigensolver (VQE) is a relevant method for simulating molecular systems on near‐term quantum computers. While its primary application is the estimation of ground‐state energies, VQE also produces the one‐particle reduced density matrix (1‐RDM), from which other relevant molecular properties can be obtained. The accuracy of these properties depends on the reliability and convergence of the 1‐RDM, which is not guaranteed by energy‐only optimization. Thus, two new algorithms were introduced: VQE* that incorporates the RMSD of consecutive 1‐RDM as a convergence criterion and VQE‐LD that modifies the cost function by adding to the energy a term involving the RMSD of 1‐RDM weighted by a proper factor. These algorithms were tested for protonated methane, CH, at equilibrium and four dissociation geometries, with the k‐UpCCGSD (4,4)‐ and GateFabric (2,2)‐active space ansätze. For k‐UpCCGSD, whose energies are already close to CASCI(4,4), improvements were mainly observed in density‐dependent properties such as electron density, dipole moments, and Mulliken charges. For GateFabric, which initially displayed larger energy deviations, both approaches significantly improved the energy accuracy and the quality of the 1‐RDM. Overall, our findings show that the convergence of the energy and of the 1‐RDM provides a simple yet effective strategy to improve the accuracy of energies and molecular properties in variational quantum algorithms.

Two new strategies that include the convergence of the one‐particle reduced density matrix (1‐RDM) were introduced in Variational Quantum Eigensolvers (VQE). Improved energy and, most importantly, density‐based properties (dipole moments, electron density) were obtained. So, simple strategies for more reliable molecular predictions with near‐term quantum computers were devised.