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Journal of Molecular Modeling

Assessment of the variational quantum eigensolver (VQE) for bond dissociation of H$$_{2}$$, H$${_3^+}$$, and CH$$_{5}^{+}$$

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

Published Sep 25, 2026• 0 citations
Publisher

Abstract

Abstract Context The variational quantum eigensolver (VQE) is a quantum-classical algorithm for estimating molecular ground-state energies. However, the accuracy depends on the choice of ansatz , active space, and initial variational parameters. Hence, it is relevant to perform assessments of VQE. The dissociation profiles of H $$_2$$ 2 , H $$_3^+$$ 3 + , and CH $$_5^+$$ 5 + were chosen for these assessments. For H $$_2$$ 2 and H $$_3^+$$ 3 + , the UCCSD, GateFabric, and k -UpCCGSD ansätze reproduced the dissociation profiles with high accuracy. For CH $$_5^+$$ 5 + , GateFabric and k -UpCCGSD provided the best compromise between accuracy and scalability. Hardware-efficient ansätze (HEA), particularly HEA-A and HEA-B, produced significant deviations from the reference energies. The results also demonstrate that larger numbers of parameters or logic gates do not necessarily improve VQE performance. Furthermore, some ansätze , especially GRSD and EHA, showed sensitivity to the initialization of the parameters. Methods Reference geometries and energies were obtained at the CISD/STO-3 G level using Gaussian 09. VQE simulations were performed with the PennyLane library employing the Jordan-Wigner mapping. The ansätze evaluated were UCCSD, k -UpCCGSD, GateFabric, GRSD, HEA-A, HEA-B, HEA-C, and Entanglement-variational Hardware-efficient Ansatz (EHA). Dissociation profiles were generated by varying interatomic distances and angular coordinates for H $$_2$$ 2 and H $$_3^+$$ 3 + , as well as the C-H $$_2$$ 2 dissociation coordinate for CH $$_5^+$$ 5 + . Active-space ranging from 2/2 to 8/8 electrons/orbitals was employed for CH $$_5^+$$ 5 + . Variational parameters were optimized using stochastic gradient descent with a convergence threshold of $$1 \times 10^{-6}$$ 1 × 10

DOI
10.1007/s00894-026-06951-w

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The Variational Quantum Eigensolver (VQE) is a hybrid algorithm that combines quantum and classical computing to determine the ground-state energy of molecular systems. In this context, this study applies VQE to investigate the ground state of protocatechuic acid, analyzing its performance with various Ansatzes and active spaces. Subsequently, all VQE results were compared to those obtained with the CISD and FCI methods. The results demonstrate that Ansatzes, like Unitary Coupled Cluster Singles and Doubles (UCCSD) and variations of Hardware-Efficient Ansatzes, generally achieve accuracy close to that of FCI. In conclusion, this study highlights the effectiveness of VQE as a robust method for investigating molecular ground-state energies. Additionally, the findings emphasize the pivotal role of Ansatz design and active space selection in optimizing VQE performance, offering meaningful insights into its capabilities and constraints.
In recent years, the Variational Quantum Eigensolver (VQE) has emerged as one of the most popular algorithms for solving the electronic structure problem on near-term quantum computers. The utility of VQE is often hindered by the limitations of current quantum hardware, including, but not limited to, short qubit coherence times and low gate fidelities. These limitations become particularly pronounced when VQE is used along with deep quantum circuits, such as those required by the ”Unitary Coupled Cluster Singles and Doubles” (UCCSD) ansatz, often resulting in significant errors. To address these issues, we propose a low-depth ansatz based on parallelized Givens rotations, which can recover substantial correlation energy while drastically reducing circuit depth and two-qubit gate counts for an arbitrary active space (AS). Also, considering the current hardware architectures with low qubit counts, we introduce a systematic way to select molecular orbitals to define active spaces (ASs) that retain significant electron correlation. We validate our approach by computing bond dissociation profiles of water and strongly correlated systems, such as molecular nitrogen and oxygen, across various ASs. Noiseless simulations using the new ansatz yield ground-state energies comparable to those from the UCCSD ansatz while reducing circuit depth by 50–70%. Moreover, in noisy simulations, our approach achieves energy error rates an order of magnitude lower than that of UCCSD. Considering the efficiency and practical usage of our ansatz, we hope that it becomes a potential choice for performing quantum chemistry calculations on near-term quantum devices.
High Resolution Image Download MS PowerPoint Slide The variational quantum eigensolver (VQE) algorithm has positioned quantum chemistry calculation methods as promising applications for noisy intermediate-scale quantum (NISQ) devices. In NISQ devices, where the quantum noise is considerable, it is desirable to have an ansatz for the quantum circuit that is practical to implement and capable of achieving high chemical accuracy. Hardware-efficient Ansätze (HEA), such as the RyRz linear ansatz (RLA) and its modified form that preserves physical quantities─the symmetry preserving ansatz (SPA)─are constructed using quantum gates that can be implemented easily on a quantum computer. However, the extent to which HEA can accurately yield the electronic state energies of actual molecules and the number of quantum gates required to achieve chemical accuracy─which, in practical cases, should be within 1 kcal/mol of the exact energy value─are not well understood. In this study, we aimed to gain a detailed understanding of these aspects by performing noiseless simulations to obtain the ground and low-lying excited state using high-depth HEA quantum circuits for several molecules, including LiH, H 2 O, BeH 2, CH 4, and N 2 . From the results, we demonstrated that symmetry preserving HEA, such as SPA, can achieve accurate computational results that maintain CCSD-level chemical accuracy by increasing the number of layers. We quantitatively analyze how the expressibility and entangling capability of RLA and SPA quantum circuits evolve with circuit depth. Our results reveal that while increased layers generally expand accessible Hilbert space and entanglement, differences in their limitations highlight the importance of considering physically allowed entanglement for performance of a parametric quantum circuit in VQE. We also studied the potential energy surface of the dissociation of these molecules and found that quantum circuits using SPA can capture static electron correlation, which is challenging to address with classical single-reference quantum chemistry methods such as CCSD. These results from SPA demonstrate its potential to represent electronic states, as it achieves highly accurate results with fewer gate operations compared to physically inspired quantum circuits such as the Unitary Coupled Cluster (UCC) method. This study establishes a possible path for the application of quantum devices in solving quantum chemistry problems.
Abstract We present a comprehensive quantum chemical study of nine molecules H₂, HeH⁺, LiH, BeH₂, H₂O, NH₃, CH₄, CO₂, and N₂ using the Variational Quantum Eigensolver (VQE) algorithm implemented in Qiskit 2.4.1. Molecular Hamiltonians are constructed in the STO-3G minimal basis set, transformed to qubit operators via the Jordan-Wigner mapping, and optimized using the n-local hardware-efficient ansatz with the SLSQP optimizer. Ground state energies are benchmarked against Full Configuration Interaction (FCI)/STO-3G reference values from authoritative literature sources. VQE achieves accuracies from 96.46% (HeH⁺) to 99.99% (H₂), average 98.68%. Potential energy surfaces are generated for all nine molecules and compared with high level ab initio literature PES references including the Kolos-Wolniewicz H₂ curve, the Polyansky water surface, and the Bytautas-Ruedenberg N₂ FCI surface. Equilibrium bond lengths agree with experiment to within 0.000–0.006 Å. Quantum ansatz circuits are analyzed in detail for all molecules. These results confirm the viability of near-term VQE algorithms for molecular quantum chemistry across diverse bond types and geometries.
This study investigates the Variational Quantum Eigensolver (VQE), a hybrid quantum-classical algorithm that partitions the computational workload between a quantum processing unit and a classical optimiser to determine the ground-state energies of molecular systems. The Qiskit SDK v1.x framework is employed, utilising the PySCFDriver module within the qiskit-nature package to construct second-quantised molecular electronic Hamiltonians for two representative systems: Hydrogen ( ) and Lithium Hydride ( ). For the complete active space is retained without spatial reduction, mapping the full Hamiltonian onto a 4-qubit register. However for the full STO-3G representation demands 12 qubits, rendering direct classical parameter optimisation computationally intractable. This bottleneck is resolved by applying an ActiveSpaceTransformer (num_electrons=2, num_spatial_orbitals=3) to freeze the chemically inactive lithium core electrons, reducing the active register to 6 qubits. Fermionic operators are subsequently converted to Pauli strings via the Jordan-Wigner transformation ( ) enabling the quantum processor to evaluate the molecular Hamiltonian as a circuit while preserving the fermionic anti-commutation relations. Under ideal, noise-free simulation using the StatevectorEstimator primitive paired with the COBYLA optimiser (maxiter=200), all VQE energy points converge precisely onto the Full Configuration Interaction (FCI) classical baseline across the full range of interatomic separations. This confirms that the hardware-efficient TwoLocal ansatz ( rotation layers and entanglement layers with reps=3) possesses sufficient expressive power to capture the molecular ground-state wavefunction. Introducing gate-level noise through AerSimulator backends calibrated from the FakeManilaV2 and FakeNairobiV2 hardware profiles—with optimisation driven by the SPSA algorithm (maxiter=150)—reveals two critical failure modes. A systematic upward energy bias is observed, attributable to depolarising gate errors and measurement readout faults that prevent the optimiser from reaching the true minimum energy eigenvalue. Additionally, severe stochastic oscillations between 1.0 Å and 1.5 Å on the curve induce premature convergence of the SPSA optimiser to spurious local minima. These results demonstrate that unmitigated hardware noise renders VQE incapable of reliably predicting equilibrium bond lengths on current NISQ devices, motivating the future integration of Zero-Noise Extrapolation (ZNE), readout error mitigation, and Virtual Distillation into the computational pipeline. Keywords—Variational Quantum Eigensolver (VQE); Qiskit 1.x; Potential Energy Curve; Molecular Ground-State Energy; Hardware-Efficient Ansatz; Gate-Level Noise Simulation; ActiveSpaceTransformer; NISQ Devices.
High Resolution Image Download MS PowerPoint Slide Quantum computers have the potential to efficiently solve the electronic structure problem but are currently limited by noise and shallow circuits. We present an enhanced Variational Quantum Eigensolver (VQE) ansatz based on the Qubit Coupled Cluster (QCC) approach that requires optimization of only n parameters, where n is the number of Pauli string generators, rather than the typical n + 2 m parameters, where m is the number of qubits. We evaluate the ground state energies and molecular dissociation curves of strongly correlated molecules, namely O 3 and Li 4, using active spaces of varying sizes in conjunction with our enhanced QCC ansatz, Unitary Coupled Cluster Single–Double (UCCSD) ansatz, and the classical Coupled Cluster Singles and Doubles (CCSD) method. Compared to UCCSD, our approach significantly reduces the number of parameters while maintaining high accuracy. Numerical simulations demonstrate the effectiveness of our approach, and experiments on superconducting and trapped-ion quantum computers showcase its practicality on real hardware. By eliminating the need for symmetry-restoring gates and reducing the number of parameters, our enhanced QCC ansatz enables accurate quantum chemistry calculations on near-term quantum devices for strongly correlated systems.