  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 recognised
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 initialisation techniques, which improve convergence
behavior relative to conventional initialisation. These techniques are applied
to H$_2$ and malonaldehyde for which results agree with Nuclear-Electronic
Orbital Full Configuration Interaction and Nuclear-Electronic Orbital Complete
Active Space Configuration Interaction benchmarks for ground state energy to
within $10^{-6}$ Ha 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.
