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Journal of Chemical Theory and Computation

Constrained Nuclear-Electronic Orbital Theory for Quantum Computation

Tanner Culpitt, Zehua Chen, Fabijan Pavošević, Yang Yang

Published Aug 12, 2025• 8 citations
Publisher

Abstract

Quantum computing offers promising advantages for computational chemistry, particularly through algorithms that efficiently model correlated methods. In this work, we apply quantum computing techniques to the constrained nuclear-electronic orbital (CNEO) framework, which enables the inclusion of nuclear quantum effects in chemical simulations while preserving a well-defined molecular structure. We present the development and implementation of two correlated wave function methods within this framework: CNEO full configuration interaction (CNEO-FCI) and CNEO unitary coupled-cluster with singles and doubles (CNEO-UCCSD), with the latter solved using the variational quantum eigensolver algorithm. These methods were applied to the hydrogen isotopologues H 2, HD, and D 2 and used to calculate potential energy surfaces, equilibrium geometries, harmonic vibrational frequencies, and von Neumann entropies. The CNEO-UCCSD results are in excellent agreement with CNEO-FCI, recovering over 99% of the correlation energy and accurately capturing geometries and vibrational frequencies. Additionally, we observe a strong connection between CNEO-FCI and CNEO-UCCSD energy and entropy differences as a function of bond length, highlighting the role of quantum entanglement in molecular dissociation. These results demonstrate the viability of CNEO-based quantum algorithms for capturing nuclear quantum effects and lay the groundwork for future quantum simulations and entropy analysis of multicomponent systems within the CNEO paradigm.

DOI
10.1021/acs.jctc.5c00815

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Quantum computing offers promising advantages for computational chemistry, particularly through algorithms that efficiently model correlated methods. In this work, we apply quantum computing techniques to the constrained nuclear-electronic orbital (CNEO) framework, which enables the inclusion of nuclear quantum effects in chemical simulations while preserving a well-defined molecular structure. We present the development and implementation of two correlated wave function methods within this framework: CNEO full configuration interaction (CNEO-FCI) and CNEO unitary coupled-cluster with singles and doubles (CNEO-UCCSD), with the latter solved using the variational quantum eigensolver algorithm. These methods were applied to the hydrogen isotopologues H 2, HD, and D 2 and used to calculate potential energy surfaces, equilibrium geometries, harmonic vibrational frequencies, and von Neumann entropies. The CNEO-UCCSD results are in excellent agreement with CNEO-FCI, recovering over 99% of the correlation energy and accurately capturing geometries and vibrational frequencies. Additionally, we observe a strong connection between CNEO-FCI and CNEO-UCCSD energy and entropy differences as a function of bond length, highlighting the role of quantum entanglement in molecular dissociation. These results demonstrate the viability of CNEO-based quantum algorithms for capturing nuclear quantum effects and lay the groundwork for future quantum simulations and entropy analysis of multicomponent systems within the CNEO paradigm.
Nuclear quantum effects are critical for describing proton transfer and hydrogen bonding, but their incorporation into quantum chemistry calculations is often computationally prohibitive on classical hardware. A promising alternative are quantum computers due to their linear scaling in the space requirements with system size. We introduce a novel hybrid quantum-classical algorithm for ab-initio molecular dynamics that incorporates nuclear quantum effects via the nuclear-electronic orbital method. The proposed algorithm evaluates ground state energies and forces on the quantum computer using a variational quantum eigensolver, while the molecular geometries are updated classically. We validate our approach through simulations of H2\text{H}_2H2​, H2O\text{H}_2\text{O}H2​O and the Zundel ion H5O2+\text{H}_5\text{O}_2^+H5​O2+​, comparing the simulated vibrational spectra with experimental data. Upon inclusion of nuclear quantum effects, the proton shuttling movement in the Zundel ion becomes effectively barrierless, and errors in the simulated frequencies improve significantly. Employing compact hardware-efficient ansatz circuits we achieve results comparable to the more accurate UCCSD ansatzes, which hints towards the feasibility of executing our algorithm on near-term quantum devices.
Nuclear quantum effects such as zero-point energy and hydrogen tunneling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets, whereas realistic simulations beyond the Born–Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H 2 and D 2 molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method─an important step toward accurate simulations involving nonclassical nuclei and non-Born–Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects, and we demonstrate that inclusion of correlation energy systematically improves the prediction of difference in the zero-point energy (ΔZPE) between isotopes.
The Coupled Cluster (CC) method is used to compute the electronic correlation energy in atoms and molecules and often leads to highly accurate results. However, due to its single-reference nature, standard CC in its projected form fails to describe quantum states characterized by strong electronic correlations and multi-reference projective methods become necessary. On the other hand, quantum algorithms for the solution of many-electron problems have also emerged recently. The quantum unitary variant of CC (UCC) with singles and doubles (q-UCCSD) is a popular wavefunction Ansatz for the variational quantum eigensolver algorithm. The variational nature of this approach can lead to significant advantages compared to its classical equivalent in the projected form, in particular, for the description of strong electronic correlation. However, due to the large number of gate operations required in q-UCCSD, approximations need to be introduced in order to make this approach implementable in a state-of-the-art quantum computer. In this work, we evaluate several variants of the standard q-UCCSD Ansatz in which only a subset of excitations is included. In particular, we investigate the singlet and pair q-UCCD approaches combined with orbital optimization. We show that these approaches can capture the dissociation/distortion profiles of challenging systems, such as H4, H2O, and N2 molecules, as well as the one-dimensional periodic Fermi–Hubbard chain. These results promote the future use of q-UCC methods for the solution of challenging electronic structure problems in quantum chemistry.
Nuclear quantum phenomena beyond the Born-Oppenheimer approximation are known to play an important role in a growing number of chemical and biological processes. While there exists no unique consensus on a rigorous and efficient implementation of coupled electron-nuclear quantum dynamics, it is recognized that these problems scale exponentially with system size on classical processors and, therefore, may benefit from quantum computing implementations. Here, we introduce a methodology for the efficient quantum treatment of the electron-nuclear problem on near-term quantum computers, based upon the Nuclear-Electronic Orbital (NEO) approach. We generalize the electronic two-qubit tapering scheme to include nuclei by exploiting symmetries inherent in the NEO framework, thereby reducing the Hamiltonian dimension, number of qubits, gates, and measurements needed for calculations. We also develop parameter transfer and initialization techniques, which improve convergence behavior relative to conventional initialization. These techniques are applied to H2 and malonaldehyde for which results agree with NEO full configuration interaction and NEO complete active space configuration interaction benchmarks for ground state energy to within 10-6 hartree and entanglement entropy to within 10-4. These implementations therefore significantly reduce resource requirements for full quantum simulations of molecules on near-term quantum devices while maintaining high accuracy.
Applications of quantum simulation algorithms to obtain electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices require careful consideration of resources describing the complex electron correlation effects. In modeling second-quantized problems, the biggest challenge confronted is that the number of qubits scales linearly with the size of the molecular basis. This poses a significant limitation on the size of the basis sets and the number of correlated electrons included in quantum simulations of chemical processes. To address this issue and enable more realistic simulations on NISQ computers, we employ the double unitary coupled-cluster (DUCC) method to effectively downfold correlation effects into the reduced-size orbital space, commonly referred to as the active space. Using downfolding techniques, we demonstrate that properly constructed effective Hamiltonians can capture the effect of the whole orbital space in small-size active spaces. Combining the downfolding preprocessing technique with the variational quantum eigensolver, we solve for the ground-state energy of H 2, Li 2, and BeH 2 in the cc-pVTZ basis using the DUCC-reduced active spaces. We compare these results to full configuration-interaction and high-level coupled-cluster reference calculations.
Abstract Variational quantum eigensolvers (VQE) are among the most promising approaches for solving electronic structure problems on near-term quantum computers. A critical challenge for VQE in practice is that one needs to strike a balance between the expressivity of the VQE ansatz versus the number of quantum gates required to implement the ansatz, given the reality of noisy quantum operations on near-term quantum computers. In this work, we consider an orbital-optimized pair-correlated approximation to the unitary coupled cluster with singles and doubles (uCCSD) ansatz and report a highly efficient quantum circuit implementation for trapped-ion architectures. We show that orbital optimization can recover significant additional electron correlation energy without sacrificing efficiency through measurements of low-order reduced density matrices (RDMs). In the dissociation of small molecules, the method gives qualitatively accurate predictions in the strongly-correlated regime when running on noise-free quantum simulators. On IonQ’s Harmony and Aria trapped-ion quantum computers, we run end-to-end VQE algorithms with up to 12 qubits and 72 variational parameters—the largest full VQE simulation with a correlated wave function on quantum hardware. We find that even without error mitigation techniques, the predicted relative energies across different molecular geometries are in excellent agreement with noise-free simulators.