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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msubsup><mml:mrow></mml:mrow> <mml:mrow><mml:mn>5</mml:mn></mml:mrow> <mml:mrow><mml:mo>+</mml:mo></mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation>$$ {}_5^{+} $$</mml:annotation></mml:semantics> </mml:math> , 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.