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arXiv:2502.17932 · New Journal of Physics

Efficient excited-state calculations for molecules based on contextual subspace method and symmetry optimizations

Qianjun Yao, He Li

Published Sep 1, 2025• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/ae0360

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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.
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.
Problems in quantum chemical simulations, especially achieving accurate excited-state potential energy surfaces, are among the primary applications to achieve quantum utility. On near-term quantum hardware, variants of the variational quantum eigensolver (VQE) algorithms are the primary choice for chemistry simulation. In this study, a combination of leading ground and excited state quantum algorithms for general excited states, namely, ADAPT-VQE/LUCJ and q-sc-EOM, are utilized to calculate accurate excited state potential energy surfaces in challenging bond-breaking scenarios and compared with the classical scalable EOM-CCSD method. This work investigates avenues toward quantum utility in excited-state quantum chemistry using the q-sc-EOM approach. We assess its accuracy while mitigating major scaling bottlenecks through the Davidson algorithm and basis rotation grouping, reducing the measurement scaling from O(N12^{12}12) to O(N5^{5}5), and implementing the method on quantum hardware with various error mitigation strategies to reduce gate and measurement errors in excited states. The hardware implementation of the q-sc-EOM algorithm, augmented by mitigation of M3 readout error and symmetry projection, produces reasonably accurate excited-state energies with gate noise identified as the predominant source of error. This paves the way for accurate and scalable, generally applicable quantum excited-state methods with potential for quantum utility while identifying critical problems that require advancements.
We introduce a spin zzz-component (SzS_{z}Sz​) conserving symmetry-preserving ansatz and a shallow quantum phase estimation (QPE) routine of spin xxx (SxS_xSx​), and combine them into a spin-filtering variational quantum deflation (sfVQD) scheme for noisy intermediate-scale quantum (NISQ) computing era excited state calculations. The scheme encodes the spin information into a small ancilla register through controlled rotations under exp(iθS^x)\mathrm{exp} (iθ\hat{S}_{x})exp(iθS^x​) with only modest circuit overhead. The encoded information is then utilized to suppress spin contamination by screening, avoiding costly explicit evaluation on the total spin ⟨S^2⟩\langle\hat{S}^{2}\rangle⟨S^2⟩. Because the screening module operates independently of the variational ansatz, it can also be employed with other excited-state calculation schemes based on variational quantum eigensolvers. As a demonstration, we apply sfVQD to LiH and BeH2_22​ with varying geometries to show markedly improved separation of singlet and triplet manifolds over conventional VQD without QPE-derived screening. These results suggest that ancilla-assisted symmetry screening provides a modular and NISQ-compatible route to securing excited state calculations of physically meaningful properties. We discuss how our scheme may naturally be extended to computing other conserved quantities.
Variational quantum algorithms on bosonic quantum processors are an emerging paradigm for quantum chemistry calculations, exploiting the natural alignment between molecular structure and harmonic oscillator-based hardware. We introduce the qumode-based variational quantum deflation framework (QumVQD) for finding both electronic and vibrational excited state energies on qumode-based architectures. We validate the approach through electronic structure calculations on H2_{\text{2}}2​ and linear H4_{\text{4}}4​, where we introduce Hamming-weight filtering of the Fock basis to enforce particle number conservation and eliminate spurious eigenstates by reducing the required Hilbert space, which reduces the required number of qumodes in turn. We achieve agreement with full configuration interaction (FCI) using the STO-3G basis set within the chemical accuracy threshold at most points along the potential energy surfaces. Extending to the vibrational structure, we combine QumVQD with an existing Hamiltonian fragmentation approach based on Cartan subalgebra, allowing us to compute the vibrational eigenenergies of CO2_{\text{2}}2​ and H2_{\text{2}}2​S to spectroscopic accuracy with per-fragment circuits that scale as O(N)O(N)O(N) in single-qumode gates and O(N2)O(N^2)O(N2) in beam-splitter gates for NNN qumodes. For the case of CO2_{\text{2}}2​, we get total gate counts more than an order of magnitude smaller than those reported for qubit-based vibrational algorithms at this system size. These results demonstrate that bosonic quantum devices are a viable platform for excited-state quantum chemistry, particularly for vibrational problems where qubit-based methods incur substantial boson-to-qubit mapping overhead.
Simulating electronic structure on a quantum computer requires encoding of fermionic systems onto qubits. Common encoding methods transform a fermionic system of NNN spin-orbitals into an NNN-qubit system, but many of the fermionic configurations do not respect the required conditions and symmetries of the system so the qubit Hilbert space in this case may have unphysical states and thus cannot be fully utilized. We propose a generalized qubit-efficient encoding (QEE) scheme that requires the qubit number to be only logarithmic in the number of configurations that satisfy the required conditions and symmetries. For the case of considering only the particle-conserving and singlet configurations, we reduce the qubit count to an upper bound of O(mlog2N)\mathcal{O}(m{log}_{2}N)O(mlog2​N), where mmm is the number of particles. This QEE scheme is demonstrated on an H2{\mathrm{H}}_{2}H2​ molecule in the 6-31G basis set and a LiH\mathrm{LiH}LiH molecule in the STO-3G basis set using fewer qubits than the common encoding methods. We calculate the ground-state energy surfaces using a variational quantum eigensolver algorithm with a hardware-efficient ansatz circuit. We choose to use a hardware-efficient ansatz since most of the Hilbert space in our scheme is spanned by desired configurations so a heuristic search for an eigenstate is sensible. The simulations are performed on IBM Quantum machines and the Qiskit simulator with a noise model implemented from a IBM Quantum machine. Using the methods of measurement error mitigation and error-free linear extrapolation, we demonstrate that most of the distributions of the extrapolated energies using our QEE scheme agree with the exact results obtained by Hamiltonian diagonalization in the given basis sets within chemical accuracy. Our proposed scheme and results show the feasibility of quantum simulations for larger molecular systems in the noisy intermediate-scale quantum (NISQ) era. The number of terms in the Hamiltonian has an upper bound of $\mathcal{O}(\frac{{N}^{2m+1}}{(m\ensuremath{-}1)!\phantom{\rule{0.16em}{0ex}}m!})$ for the QEE scheme while it scales as O(N4)\mathcal{O}({N}^{4})O(N4) for the Jordan-Wigner encoding scheme. Nevertheless, we present several cases where QEE is useful.
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.