The Virtual Quantum Subspace Expansion (VQSE) extends the Variational Quantum Eigensolver (VQE) by leveraging additional measurements on the reference state to capture the influence of excluded virtual orbitals. This makes VQSE attractive for chemical applications where accurate energy differences along potential energy surfaces are crucial for modeling reaction rates and kinetics. In this work, we analyze VQSE performance on H$_2$ dissociation including references that use Hartree--Fock molecular orbitals with broken spin symmetry. We identify two mechanisms which affect accuracy: overlap of the reference state with the exact full configuration interaction (FCI) wavefunction and operator pool expressivity. We show these mechanisms are strongly co-dependent. When operators are restricted to act only from the active to the virtual space, results become highly sensitive to the reference, and enlarging the active space does not guarantee improved accuracy. In this case, prioritizing reference overlap over energy minimization is therefore essential. Adding single excitations and number operators within the active space recovers the accuracy of MR-CISD (multi-reference configuration interaction singles and doubles) regardless of the reference. In our noisy hardware experiments, we achieve chemical accuracy by adding additional operators and using strict regularization. These findings motivate careful co-design of reference fidelity, pool expressivity, and hardware constraints for practical VQSE deployment.
The Virtual Quantum Subspace Expansion (VQSE) extends the Variational Quantum Eigensolver (VQE) by leveraging additional measurements on the reference state to capture the influence of excluded virtual orbitals. This makes VQSE attractive for chemical applications where accurate energy differences along potential energy surfaces are crucial for modeling reaction rates and kinetics. In this work, we analyze VQSE performance on H2 dissociation including references that use Hartree--Fock molecular orbitals with broken spin symmetry. We identify two mechanisms which affect accuracy: overlap of the reference state with the exact full configuration interaction (FCI) wavefunction and operator pool expressivity. We show these mechanisms are strongly co-dependent. When operators are restricted to act only from the active to the virtual space, results become highly sensitive to the reference, and enlarging the active space does not guarantee improved accuracy. In this case, prioritizing reference overlap over energy minimization is therefore essential. Adding single excitations and number operators within the active space recovers the accuracy of MR-CISD (multi-reference configuration interaction singles and doubles) regardless of the reference. In our noisy hardware experiments, we achieve chemical accuracy by adding additional operators and using strict regularization. These findings motivate careful co-design of reference fidelity, pool expressivity, and hardware constraints for practical VQSE deployment.
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.
The wave function prepared by a variational quantum eigensolver (VQE) can contain a substantial classically tractable component, yet its energy contribution is sampled on the quantum device. Representing the wave function as a combination of a classical reference and separately prepared quantum states allows the reference energy to be evaluated without quantum sampling. However, near-linear dependence among these components can amplify measurement errors and destabilize the resulting energy estimate. We introduce Classical Active-Space Hybrid Quantum Subspace Expansion (CASH-QSE), which retains a complete-active-space self-consistent field (CASSCF) reference classically and uses occupation structure to construct quantum states exactly orthogonal to the reference and to one another. The energy follows from an ordinary Hermitian eigenvalue problem without overlap measurements, while the same occupation constraints simplify the measured operators. Using full configuration interaction to guide component selection, we test CASH-QSE along H2O and N2 bond-stretching coordinates spanning weakly to strongly correlated regimes. The benchmarks use STO--3G for both molecules and a restricted cc-pVDZ orbital space for H2O. CASH-QSE reaches chemical accuracy while limiting the largest complete measurement circuits to a few hundred all-to-all logical controlled-NOT gates. In favorable cases, it also reduces the idealized final-energy sampling cost by several orders of magnitude relative to VQE with an adaptive derivative-assembled pseudo-Trotter ansatz (ADAPT-VQE).
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
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.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) hardware to estimate eigenvalues of many-body Hamiltonians. Unlike fully quantum approaches such as quantum phase estimation (QPE), VQE trades deep coherent circuits for repeated state preparation, measurement, and classical optimization, making it more compatible with limited qubit counts and finite coherence times. Recent developments have focused on reducing quantum resource requirements while retaining chemically meaningful wavefunction structure. In this paper, we examine recent progress in VQE methods for molecular electronic structure with an emphasis on three themes: (i) strategies for circuit and ansatz complexity reduction, including adaptive and selectively screened approaches, (ii) chemically motivated workflows that combine VQE with orbital optimization, fragmentation, and localized active-space ideas to better address strong correlation, and (iii) extensions of VQE to excited-state calculations. Throughout, we emphasize the tradeoffs among parameter count, gate depth, symmetry preservation, measurement overhead, and classical preprocessing, and discuss where these approaches may become most useful for chemically challenging active spaces. We also highlight benchmarking considerations for assessing both accuracy and resource requirements, and conclude with a perspective on regimes in which VQE may offer the greatest long-term value, particularly multireference active spaces and low-lying excited-state manifolds.
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing.Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace.In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4-8.With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H 4 -square, H 4 -chain, and H 6 -hexagon molecules, while the number of CNOT gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35.Furthermore, we introduced an efficient "score" parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first.Using the H 4 square and H 4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.