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Journal of Computational Chemistry

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

Amanda Marques de Lima, Erico S. Teixeira, Eivson Darlivam Rodrigues de Aguiar Silva, Ricardo Luiz Longo

Published Jan 5, 2026• 4 citations
Publisher

Abstract

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.

DOI
10.1002/jcc.70289

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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.
The variational quantum eigensolver (VQE) is a promising method for simulating molecular systems on near-term quantum computers. This approach employs energy estimation; however, other relevant molecular properties can be extracted from the one-particle reduced density matrix (1-RDM) generated by VQE. The accuracy of these properties strongly depends on the reliability and convergence of the 1-RDM, which is not guaranteed by energy-only optimization. Thus, we investigate the effect of optimizing the 1-RDM within VQE to improve the accuracy of both the energy and molecular properties. A two-step algorithm was implemented that optimizes the energy and 1-RDM by incorporating a penalty term in the cost function to enforce the convergence of the 1-RDM. The first step focuses on energy minimization, while in the second step, a weighted penalty is added to the cost function to promote simultaneous improvement of the energy and 1-RDM. This approach was tested and validated for the k-UpCCGSD and GateFabric ansätzes with active spaces (4,4) and (2,2), respectively. k-UpCCGSD produces energies close to CISD, so optimizing 1-RDM has little effect on the energy but significantly improves electronic properties such as electron density, dipole moments, and atomic charges. GateFabric initially shows higher energy deviations from CISD, but optimizing 1-RDM substantially improves both the energy accuracy and the quality of 1-RDM. These results demonstrate that simultaneous optimization of energy and 1-RDM is an effective strategy to improve the accuracy of energies and molecular properties in variational quantum algorithms.
ABSTRACT The variational quantum eigensolver (VQE) has emerged as one of the leading quantum algorithms for solving electronic structure problems on near‐term noisy intermediate‐scale quantum devices. However, its practical application to quantum chemistry remains challenging due to the limited coherence time, imperfect quantum gate fidelity, and the large number of measurements required, which together confine current electronic structure simulations to relatively small active spaces. In this work, we present a measurement‐reduced orbital‐optimized (OO) VQE framework based on orbital compression, designed to improve the accuracy of electronic structure calculations while maintaining relatively small active spaces. Frozen natural orbitals (FNO) and split virtual orbitals (SVO) are first employed to construct compact active spaces for VQE simulations, leading to the FNO/SVO‐VQE approach. Orbital optimization is then incorporated to further recover electron correlation effects, resulting in the FNO/SVO‐OO‐VQE methods. We apply the proposed method to simulate potential energy surfaces for molecular dissociation and the activation energy of formaldehyde decomposition. Numerical results demonstrate that both FNO‐OO‐VQE and SVO‐OO‐VQE improve the variational accuracy while substantially reducing measurement cost.
Exponential scaling presents a significant challenge in electronic structure calculations performed on classical computers. This paper explores how quantum computer algorithms can accurately represent quantum systems. Variational Quantum Eigensolver (VQE) algorithm is used to compute the ground state energy of hydrogen and helium sequences by implementing variational principle and quantum gates as trial wavefunction. This technique combines classical optimization with quantum computing calculations to simulate quantum systems on noisy and resource-limited computers. The resulting calculated energy is highly consistent to the corresponding exact values and Hartree-Fock calculations with a trend of when the number of atoms increases the calculated energy becomes more negative, leading to a decrease in the percentage error. Moreover, the convergence of the ground state energy of hydrogen and helium atoms was effectively optimized. The desired energy was reached, proven by adjusting the expectation value, and gradually achieving unity in state overlap. These findings demonstrate the VQE method's accuracy in calculating simple quantum systems and its scalability for larger atomic and molecular system, such as those in quantum chemistry and material science. However, challenges in quantum computer simulations, such as limited in qubit numbers and the presence of noise, require further advancements. Therefore, implementing a larger basis sets, advanced qubit mapping, specific chemistry ansatz, and flexible optimization techniques is one way to improve overall calculation.
We propose a modification of the Variational Quantum Eigensolver algorithm for electronic structure optimization using quantum computers, named nonunitary Variational Quantum Eigensolver (nu-VQE), in which a nonunitary operator is combined with the original system Hamiltonian leading to a new variational problem with a simplified wave function ansatz. In the present work, as nonunitary operator, we use the Jastrow factor, inspired from classical Quantum Monte Carlo techniques for simulation of strongly correlated electrons. The method is applied to prototypical molecular Hamiltonians for which we obtain accurate ground-state energies with shallower circuits, at the cost of an increased number of measurements. Finally, we also show that this method achieves an important error mitigation effect that drastically improves the quality of the results for VQE optimizations on today's noisy quantum computers. The absolute error in the calculated energy within our scheme is 1 order of magnitude smaller than the corresponding result using traditional VQE methods, with the same circuit depth.
Abstract The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the circuit depth of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.
Abstract Materials simulations involving strongly correlated electrons pose fundamental challenges to state‐of‐the‐art electronic structure methods but are hypothesized to be the ideal use case for quantum computing algorithms. To date, no quantum computer has simulated a molecule of a size and complexity relevant to real‐world applications, despite the fact that the variational quantum eigensolver (VQE) algorithm can predict chemically accurate total energies. Nevertheless, because of the many applications of moderately sized, strongly correlated systems, such as molecular catalysts, the successful use of the VQE stands as an important waypoint in the advancement toward useful chemical modeling on near‐term quantum processors. In this paper, we take a significant step in this direction. We lay out the steps, write, and run parallel code for an (emulated) quantum computer to compute the bond dissociation curves of the TiH, LiH, NaH, and KH diatomic hydride molecules using the VQE. TiH was chosen as a relatively simple chemical system that incorporates d orbitals and strong electron correlation. Because current VQE implementations on existing quantum hardware are limited by qubit error rates, the number of qubits available, and the allowable gate depth, recent studies using it have focused on chemical systems involving s and p block elements. Through VQE + UCCSD calculations of TiH, we evaluate the near‐term feasibility of modeling a molecule with d‐orbitals on real quantum hardware. We demonstrate that the inclusion of d‐orbitals and the use of the UCCSD ansatz, which are both necessary to capture the correct TiH physics, dramatically increase the cost of this problem. We estimate the approximate error rates necessary to model TiH on current quantum computing hardware using VQE + UCCSD and show them to likely be prohibitive until significant improvements in hardware and error correction algorithms are available.