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Quantum Subspace Expansion for Molecular Excited States: Representability, Stability, and Practical Considerations

Naythan Z. X. Yeo, Shi Jun Ang

Published Sep 24, 2026• 0 citations
Publisher

Abstract

Accurate treatment of excited states remains challenging for classical electronic-structure methods, motivating quantum algorithms that may provide more favorable resource scaling. We systematically assess the practical performance of Quantum Subspace Expansion (QSE) for low-lying electronic states across 28 organic molecules from the Thiel benchmark set. Statevector simulations reveal that QSE representability depends strongly on the target state, active space, and molecular electronic structure, with multiconfigurational character emerging as an important factor. In several cases, the one-body QSE subspace cannot adequately represent the target states, while explicit double-excitation operators improve representability at the cost of a larger operator space. Analysis of the QSE overlap matrix shows that although eigenmodes with small eigenvalues drive numerical instability, they cannot be discarded indiscriminately without compromising spectral accuracy. Under finite-shot sampling, energy errors exhibit an empirical scaling of approximately n2 orb/ √ Nshots, which, combined with the growth in measurement groups, yields a total measurement cost of approximately O(n11 orb) for constant accuracy under our direct-estimation protocol. These results expose a central tension between representability, numerical stability, and measurement cost. Encouragingly, we find that the projected matrices exhibit recurring structural patterns, while the finite-shot perturbations in the Hamiltonian and overlap matrices show nontrivial covariance, suggesting opportunities for more efficient structure-aware measurement and reconstruction strategies. Overall, our results provide an assessment of QSE across chemically-relevant organic molecules, identifying present limitations while highlighting potential routes toward more practical future implementations.

DOI
10.26434/chemrxiv.15009412/v1

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Quantum Subspace Expansion for Molecular Excited States: Representability, Stability, and Practical Considerations

Naythan Z. X. Yeo, Shi Jun Ang

Accurate treatment of excited states remains challenging for classical electronic-structure methods, motivating quantum algorithms that may provide more favorable resource scaling. We systematically assess the practical performance of Quantum Subspace Expansion (QSE) for low-lying electronic states across 28 organic molecules from the Thiel benchmark set. Statevector simulations reveal that QSE representability depends strongly on the target state, active space, and molecular electronic structure, with multiconfigurational character emerging as an important factor. In several cases, the one-body QSE subspace cannot adequately represent the target states, while explicit double-excitation operators improve representability at the cost of a larger operator space. Analysis of the QSE overlap matrix shows that although eigenmodes with small eigenvalues drive numerical instability, they cannot be discarded indiscriminately without compromising spectral accuracy. Under finite-shot sampling, energy errors exhibit an empirical scaling of approximately n_{\mathrm{orb}}^2/\sqrt{N_{\mathrm{shots}}} , which, combined with the growth in measurement groups, yields a total measurement cost of approximately \mathcal{O}(n_{\mathrm{orb}}^{11}) for constant accuracy under our direct-estimation protocol. These results expose a central tension between representability, numerical stability, and measurement cost. Encouragingly, we find that the projected matrices exhibit recurring structural patterns, while the finite-shot perturbations in the Hamiltonian and overlap matrices show nontrivial covariance, suggesting opportunities for more efficient structure-aware measurement and reconstruction strategies. Overall, our results provide an assessment of QSE across chemically-relevant organic molecules, identifying present limitations while highlighting potential routes toward more practical future implementations.

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Accurate treatment of excited states remains challenging for classical electronic-structure methods, motivating quantum algorithms that may provide more favorable resource scaling. We systematically assess the practical performance of Quantum Subspace Expansion (QSE) for low-lying electronic states across 28 organic molecules from the Thiel benchmark set. Statevector simulations reveal that QSE representability depends strongly on the target state, active space, and molecular electronic structure, with multiconfigurational character emerging as an important factor. In several cases, the one-body QSE subspace cannot adequately represent the target states, while explicit double-excitation operators improve representability at the cost of a larger operator space. Analysis of the QSE overlap matrix shows that although eigenmodes with small eigenvalues drive numerical instability, they cannot be discarded indiscriminately without compromising spectral accuracy. Under finite-shot sampling, energy errors exhibit an empirical scaling of approximately n2 orb/ √ Nshots, which, combined with the growth in measurement groups, yields a total measurement cost of approximately O(n11 orb) for constant accuracy under our direct-estimation protocol. These results expose a central tension between representability, numerical stability, and measurement cost. Encouragingly, we find that the projected matrices exhibit recurring structural patterns, while the finite-shot perturbations in the Hamiltonian and overlap matrices show nontrivial covariance, suggesting opportunities for more efficient structure-aware measurement and reconstruction strategies. Overall, our results provide an assessment of QSE across chemically-relevant organic molecules, identifying present limitations while highlighting potential routes toward more practical future implementations.
Accurate treatment of excited states remains challenging for classical electronic-structure methods, motivating quantum algorithms that may provide more favorable resource scaling. We systematically assess the practical performance of Quantum Subspace Expansion (QSE) for low-lying electronic states across 28 organic molecules from the Thiel benchmark set. Statevector simulations reveal that QSE representability depends strongly on the target state, active space, and molecular electronic structure, with multiconfigurational character emerging as an important factor. In several cases, the one-body QSE subspace cannot adequately represent the target states, while explicit double-excitation operators improve representability at the cost of a larger operator space. Analysis of the QSE overlap matrix shows that although eigenmodes with small eigenvalues drive numerical instability, they cannot be discarded indiscriminately without compromising spectral accuracy. Under finite-shot sampling, energy errors exhibit an empirical scaling of approximately n_{\mathrm{orb}}^2/\sqrt{N_{\mathrm{shots}}} , which, combined with the growth in measurement groups, yields a total measurement cost of approximately \mathcal{O}(n_{\mathrm{orb}}^{11}) for constant accuracy under our direct-estimation protocol. These results expose a central tension between representability, numerical stability, and measurement cost. Encouragingly, we find that the projected matrices exhibit recurring structural patterns, while the finite-shot perturbations in the Hamiltonian and overlap matrices show nontrivial covariance, suggesting opportunities for more efficient structure-aware measurement and reconstruction strategies. Overall, our results provide an assessment of QSE across chemically-relevant organic molecules, identifying present limitations while highlighting potential routes toward more practical future implementations.
Predicting spectra and photochemical pathways relies on the computation of molecular excited states. However, the practicality of near-term variational quantum algorithms is threatened by the prohibitive growth of measurement overheads. Integrating Quantum Subspace Expansion with the Contextual Subspace (CS) method, we propose a CS-QSE framework to resolve the scaling bottlenecks in excited-state energy estimation. By confining excitation operators to a compact CS, the framework removes redundant degrees of freedom and reduces Pauli string counts, thereby easing the measurement burden. The primary strength of CS-QSE lies in its substantial reduction of the operator-pool size. Benchmarking on LiH, HF, H2O, and HCl shows that CS-QSE achieves errors within the target tolerance of 1.6 × 10-3 Ha relative to full configuration interaction benchmarks in the same basis set while mitigating the prohibitive scaling inherent in the full-space method. Numerical simulations reveal that the operator pool size is consistently pruned by over 90% across all systems, with the reduction reaching as high as 99.7% for molecules such as HCl. This CS-QSE framework establishes a resource-efficient route for molecular simulations tailored to near-term noisy intermediate-scale quantum devices.
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.
Abstract The simulation of molecular electronic structure is an important application of quantum devices. Recently, it has been shown that quantum devices can be effectively combined with classical supercomputing centers in the context of the sample-based quantum diagonalization (SQD) algorithm. This allowed the largest electronic structure quantum simulation to date (77 qubits) and opened near-term devices to practical use cases in chemistry toward the hundred-qubit mark. However, the description of many important physical and chemical properties of those systems, such as photo-absorption/-emission, requires a treatment that goes beyond the ground state alone. In this work, we extend the SQD algorithm to determine low-lying molecular excited states. The extended-SQD method improves over the original SQD method in accuracy, at the cost of an additional computational step. It also improves over quantum subspace expansion based on single and double electronic excitations, a widespread approach to excited states on pre-fault-tolerant quantum devices, in both accuracy and efficiency. We employ the extended SQD method to compute the first singlet (S 1) and triplet (T 1) excited states of the nitrogen molecule with a correlation-consistent basis set, and the ground- and excited-state properties of the [2Fe-2S] cluster.
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 H2_22​O and N2_22​ 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 H2_22​O. 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).
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_22​ 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.