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Nuclear quantum effects such as zero-point energy and hydrogen tunnellingplay a central role in many biological and chemical processes. Thenuclear-electronic orbital (NEO) approach captures these effects by treatingselected nuclei quantum mechanically on the same footing as electrons. Onclassical computers, the resources required for an exact solution of NEO-basedmodels grow exponentially with system size. By contrast, quantum computersoffer a means of solving this problem with polynomial scaling. However, due tothe limitations of current quantum devices, NEO simulations are confined to thesmallest systems described by minimal basis sets whereas realistic simulationsbeyond the Born--Oppenheimer approximation require more sophisticated basissets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE methodto the NEO framework in the frozen natural orbital (FNO) basis. We demonstrateon H$_2$ and D$_2$ molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOTcount by a several orders of magnitude relative to the NEO unitary coupledcluster method with singles and doubles (NEO-UCCSD) while maintaining thedesired accuracy. This extreme reduction in the CNOT gate count is sufficientto permit practical computations employing the NEO method -- an important steptoward accurate simulations involving non-classical nuclei andnon--Born--Oppenheimer effects on near-term quantum devices. We further showthat the method can capture isotope effects and we demonstrate that inclusionof correlation energy systematically improves the prediction of $\Delta$ZPE.

Toward Accurate Post-Born-Oppenheimer Molecular Simulations on Quantum  Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen  Natural Orbitals