Abstract The variational quantum Eigensolver (VQE) provides a scalable route to molecular ground-state energies on near-term quantum hardware, but its accuracy degrades rapidly as system size grows because hardware-efficient ansätze cannot span the full correlation subspace. We introduce a physics-informed ridge regression correction that is trained on the difference between VQE and Hartree–Fock (HF) energies and applied as a postprocessing step requiring no additional quantum resources. The feature set─the VQE–HF energy gap ΔE, its square ΔE2, the active-space qubit count, the number of active electrons, and two cross-term interactions─captures the dominant drivers of ansatz error in a low-dimensional, interpretable form. The model is benchmarked on H2, H2O, NH3, CH4, C2H6, and CH3OH using the STO-3G basis set and a state vector VQE simulator using a two-layer Ry-CNOT hardware-efficient ansatz. To assess cross-molecule generalizability, the corrected framework is further tested on CO2, N2, and HCN under both untrained and trained conditions. Absolute energy errors fall from a range of 0.02–0.79 Ha (raw VQE) to 0.0007–0.094 Ha after correction, corresponding to improvement factors of 3× to 640× relative to CASCI reference values. Zero-shot generalization tests yield factors of 9–58×, and including molecules in training improves results by a further order of magnitude in several cases. The three extended molecules exhibit improvement factors of 4–6× in untrained settings and 5–6× after training, providing preliminary evidence of transferability across chemically distinct systems. These findings suggest that the VQE–HF energy gap encodes physically transferable information about correlation-energy saturation, and that even the smallest training set can produce practically useful corrections for near-term quantum chemistry.
Abstract The variational quantum Eigensolver (VQE) provides a scalable route to molecular ground-state energies on near-term quantum hardware, but its accuracy degrades rapidly as system size grows because hardware-efficient ansätze cannot span the full correlation subspace. We introduce a physics-informed ridge regression correction that is trained on the difference between VQE and Hartree–Fock (HF) energies and applied as a postprocessing step requiring no additional quantum resources. The feature set─the VQE–HF energy gap ΔE, its square ΔE2, the active-space qubit count, the number of active electrons, and two cross-term interactions─captures the dominant drivers of ansatz error in a low-dimensional, interpretable form. The model is benchmarked on H2, H2O, NH3, CH4, C2H6, and CH3OH using the STO-3G basis set and a state vector VQE simulator using a two-layer Ry-CNOT hardware-efficient ansatz. To assess cross-molecule generalizability, the corrected framework is further tested on CO2, N2, and HCN under both untrained and trained conditions. Absolute energy errors fall from a range of 0.02–0.79 Ha (raw VQE) to 0.0007–0.094 Ha after correction, corresponding to improvement factors of 3× to 640× relative to CASCI reference values. Zero-shot generalization tests yield factors of 9–58×, and including molecules in training improves results by a further order of magnitude in several cases. The three extended molecules exhibit improvement factors of 4–6× in untrained settings and 5–6× after training, providing preliminary evidence of transferability across chemically distinct systems. These findings suggest that the VQE–HF energy gap encodes physically transferable information about correlation-energy saturation, and that even the smallest training set can produce practically useful corrections for near-term quantum chemistry.
Classical simulation of molecular electronic structure is often limited by the rapid increase in computational cost associated with describing electron correlation. The Variational Quantum Eigensolver (VQE) is one hybrid quantum-classical approach ideally adapted to the Noisy Intermediate-Scale Quantum (NISQ) regime. This work applies the VQE algorithm to compute ground state energies of a set of molecules:H2,H3+,LiH,HF,OH−,H2O,BH3,CH4, andNH3. The approach utilizes the Unitary Coupled Cluster for Single and Double excitations (UCCSD) ansatz along with the Bravyi-Kitaev transformation for fermionic-to-qubit encoding. Quantum simulation outputs are systematically benchmarked against Full Configuration Interaction (FCI), a numerically exact traditional approach, and Unrestricted Hartree-Fock (UHF). The results establish a robust consistency of the VQE-obtained molecular ground state energies against standard FCI and UHF benchmarks. This research provides a critical benchmark, confirming that VQE has the capability for high-precision molecular simulations and playing the role of a pivotal stepping stone toward revolutionizing applications in drug discovery and materials science.
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.
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.
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.
Assessment of the variational quantum eigensolver (VQE) for bond dissociation of H2, H3+, and CH5+
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 1×10−6 1 × 10
Variational quantum eigensolver ansätze hold considerable promise for ground-state energy calculations on near-term quantum hardware, yet most promising ansatz designs currently strongly depend on how well the molecular orbital basis captures the electronic correlation of the system. Computing optimized orbital coefficients via classical routines is computationally expensive and must be performed independently for each molecular geometry -- a bottleneck that limits scalability across chemical space. We present a graph neural network framework that predicts optimized orbital coefficients directly from molecular geometry and pair-wise bonding structure. Trained on hydrogenic systems of modest size (H4 and H6) across tens of thousands of geometries, our model transfers to larger, unseen systems (H8, H10 and H12) without retraining -- demonstrating strong out-of-distribution generalization with respect to system size. When evaluating on structured and random configurations, and comparing against energies obtained with full classical optimization, our model reaches mean absolute energy errors O(102) and O(10) milli-Hartrees, respectively. Beyond energy estimation, the predicted orbitals serve as high-quality warm-start initializations that substantially reduce optimizer iterations to ground-state energy convergence. These results establish graph neural networks as an effective and scalable strategy for accelerating orbital optimization in hybrid quantum-classical workflows, directly reducing the classical pre-processing overhead that currently limits the practical deployment of variational quantum eigensolver on near-term quantum hardware.