Constrained Nuclear-Electronic Orbital Theory for Quantum Computation

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 H₂, HD, and D₂ 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.