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.
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.
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.
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.
High Resolution Image Download MS PowerPoint Slide Quantum chemistry is a promising application for noisy intermediate-scale quantum (NISQ) devices. However, quantum computers have thus far not succeeded in providing solutions to problems of real scientific significance, with algorithmic advances being necessary to fully utilize even the modest NISQ machines available today. We discuss a method of ground state energy estimation predicated on a partitioning of the molecular Hamiltonian into two parts: one that is noncontextual and can be solved classically, supplemented by a contextual component that yields quantum corrections obtained via a Variational Quantum Eigensolver (VQE) routine. This approach has been termed Contextual Subspace VQE (CS-VQE); however, there are obstacles to overcome before it can be deployed on NISQ devices. The problem we address here is that of the ansatz, a parametrized quantum state over which we optimize during VQE; it is not initially clear how a splitting of the Hamiltonian should be reflected in the CS-VQE ansätze. We propose a “noncontextual projection” approach that is illuminated by a reformulation of CS-VQE in the stabilizer formalism. This defines an ansatz restriction from the full electronic structure problem to the contextual subspace and facilitates an implementation of CS-VQE that may be deployed on NISQ devices. We validate the noncontextual projection ansatz using a quantum simulator and demonstrate chemically precise ground state energy calculations for a suite of small molecules at a significant reduction in the required qubit count and circuit depth.
Due to numerous limitations including restrictive qubit topologies, short coherence times, and prohibitively high noise floors, few quantum chemistry experiments performed on existing noisy intermediate-scale quantum hardware have achieved the high bar of chemical precision, namely energy errors to within 1.6 mHa of full configuration interaction. To have any hope of doing so, we must layer contemporary resource reduction techniques with best-in-class error mitigation methods; in particular, we combine the techniques of qubit tapering and the contextual subspace variational quantum eigensolver with several error mitigation strategies comprised of measurement-error mitigation, symmetry verification, zero-noise extrapolation, and dual-state purification. We benchmark these strategies across a suite of eight 27-qubit IBM Falcon series quantum processors, taking preparation of the HCl molecule's ground state as our testbed.
Abstract Quantum computing methods for excited-state calculations remain underexplored in noisy intermediate-scale quantum hardware, despite their critical role in photochemistry and material science. Herein, we propose a resource-efficient framework that integrates the contextual subspace method with the variational quantum deflation algorithm to enable systematic excited-state calculations for molecules while reducing qubit requirements. On the basis of the numerical results, we find that it is unproblematic to utilize this combination in calculating the excited state to reduce qubits. Furthermore, we demonstrate that the implementation of a spin-conserving hardware-efficient ansatz, namely the N ( θ x , θ y , θ z ) block ansatz, allows exploitation of spin symmetry within the projected subspace, thereby achieving further reductions in computational resource demands. Compared to the commonly used R Y R Z ansatz, using the N ( θ x , θ y , θ z ) ansatz can reduce the number of optimization iterations by up to three times at a similar circuit depth.
The quantum-selected configuration interaction identifies important determinantal basis functions through real-time evolution of a reference wavefunction and diagonalizing the Hamiltonian matrix in the resulting selected subspace. However, implementing the full electronic Hamiltonian on noisy quantum devices leads to rapidly increasing circuit complexity, limiting its scalability. To address this issue, we identify the dominant fermionic excitation operators and perform reference-state fidelity loss analysis to construct a compact Hamiltonian, reducing computational overhead while retaining high precision. Applied to Group IIIA monofluorides (BF, AlF, GaF, InF, and TlF), the proposed framework achieves a near-quadratic improvement in Hamiltonian-term scaling, enabling resource-efficient simulations. We employ this framework to compute the relativistic ground-state energies and permanent electric dipole moments (PDMs) of the systems under consideration. After validating the framework via simulations, we demonstrate hardware execution for AlF and TlF on the IBM Marrakesh processor using active spaces of up to 20 qubits. For a 20-qubit TlF system, the reduced Hamiltonian yields a reduction of higher than $ 98\%$ in both circuit depth and two-qubit gate counts, with the resulting PDMs from quantum hardware matching complete active space configuration interaction values within 99.99%. These results demonstrate the scalability of this approach on noisy intermediate-scale quantum devices.