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.
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) 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) 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 an orbital-optimized 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.
Abstract Variational quantum eigensolvers (VQE) are among the most promising approaches for solving electronic structure problems on near-term quantum computers. A critical challenge for VQE in practice is that one needs to strike a balance between the expressivity of the VQE ansatz versus the number of quantum gates required to implement the ansatz, given the reality of noisy quantum operations on near-term quantum computers. In this work, we consider an orbital-optimized pair-correlated approximation to the unitary coupled cluster with singles and doubles (uCCSD) ansatz and report a highly efficient quantum circuit implementation for trapped-ion architectures. We show that orbital optimization can recover significant additional electron correlation energy without sacrificing efficiency through measurements of low-order reduced density matrices (RDMs). In the dissociation of small molecules, the method gives qualitatively accurate predictions in the strongly-correlated regime when running on noise-free quantum simulators. On IonQ’s Harmony and Aria trapped-ion quantum computers, we run end-to-end VQE algorithms with up to 12 qubits and 72 variational parameters—the largest full VQE simulation with a correlated wave function on quantum hardware. We find that even without error mitigation techniques, the predicted relative energies across different molecular geometries are in excellent agreement with noise-free simulators.
Variational quantum eigensolvers (VQEs) are among the most promising quantum algorithms for solving electronic structure problems in quantum chemistry, particularly during the noisy intermediate-scale quantum (NISQ) era. In this study, we investigate the capabilities and limitations of VQE algorithms implemented on current quantum hardware for determining molecular ground-state energies, focusing on the adaptive derivative-assembled pseudo-Trotter ansatz VQE (ADAPT-VQE). To address the significant computational challenges posed by molecular Hamiltonians, we explore various well known strategies to simplify the Hamiltonian, optimize the ansatz, and improve classical parameter optimization through modifications of the COBYLA optimizer. These enhancements are integrated into a tailored quantum computing implementation designed to minimize the circuit depth and computational cost. Using benzene as a benchmark system, we demonstrate the application of these optimizations on an IBM quantum computer. Despite these improvements, our results highlight the limitations imposed by current quantum hardware, particularly the impact of quantum noise on state preparation and energy measurement. The noise levels in today's devices prevent meaningful evaluations of molecular Hamiltonians with sufficient accuracy to produce reliable quantum chemical insights. Finally, we extrapolate the requirements for future quantum hardware to enable practical and scalable quantum chemistry calculations using VQE algorithms. This work provides an assessment of current quantum algorithms for molecular modeling on real quantum hardware, highlighting the impact of noise and hardware limitations on the achievable accuracy.
Variational quantum eigensolver (VQE) is demonstrated to be the promising methodology for quantum chemistry based on near-term quantum devices. However, many problems are yet to be investigated for this methodology, such as the influences of optimization algorithm and basis size on the accuracy and efficiency for quantum computing. To address these issues, five molecules (H2, LiH, HF, N2 and F2) are studied in this work based on the VQE method using unitary coupled cluster (UCC) ansatz. The performance of the gradient optimization L-BFGS-B is compared with that of the direct search method COBYLA. The former converges more quickly, but the accuracy of energy surface is a little lower. The basis set shows a vital influence on the accuracy and efficiency. A large basis set generally provides an accurate energy surface, but induces a significant increase in computing time. The 631g basis is generally required from the energy surface of the simplest H2 molecule. For practical applications of VQE, complete active space (CAS) is suggested based on limited quantum resources. With the same number of qubits, more occupied orbitals included in CAS gives a better accuracy for the energy surface and a smaller evaluation number in the VQE optimization. Additionally, the electronic structure, such as filling fraction of orbitals, the bond strength of a molecule and the maximum nuclear charge also influences the performance of optimization, where half occupation of orbitals generally requires a large computation cost.
Variational quantum eigensolver (VQE) is demonstrated to be the promising methodology for quantum chemistry based on near-term quantum devices. However, many problems are yet to be investigated for this methodology, such as the influences of optimization algorithm and basis size on the accuracy and efficiency for quantum computing. To address these issues, five molecules (H2, LiH, HF, N2 and F2) are studied in this work based on the VQE method using unitary coupled cluster (UCC) ansatz. The performance of the gradient optimization L-BFGS-B is compared with that of the direct search method COBYLA. The former converges more quickly, but the accuracy of energy surface is a little lower. The basis set shows a vital influence on the accuracy and efficiency. A large basis set generally provides an accurate energy surface, but induces a significant increase in computing time. The 631g basis is generally required from the energy surface of the simplest H2 molecule. For practical applications of VQE, complete active space (CAS) is suggested based on limited quantum resources. With the same number of qubits, more occupied orbitals included in CAS gives a better accuracy for the energy surface and a smaller evaluation number in the VQE optimization. Additionally, the electronic structure, such as filling fraction of orbitals, the bond strength of a molecule and the maximum nuclear charge also influences the performance of optimization, where half occupation of orbitals generally requires a large computation cost.