  The combined quantum electron-nuclear dynamics is often associated with the
Born-Huang expansion of the molecular wave function and the appearance of
nonadiabatic effects as a perturbation. On the other hand, native
multicomponent representations of electrons and nuclei also exist, which do not
rely on any a priori approximation. However, their implementation is hampered
by prohibitive scaling costs and therefore quantum computers offer a unique
opportunity for extending their use to larger systems. Here, we propose a
quantum algorithm for the simulation of the time-evolution of molecular systems
in the second quantization framework, which is applied to the simulation of the
proton transfer dynamics in malonaldehyde. After partitioning the dynamics into
slow and fast components, we show how the entanglement between the electronic
and nuclear degrees of freedom can persist over long times if electrons are not
adiabatically following the nuclear displacement. The proposed quantum
algorithm may become a valid candidate for the study of electron-nuclear
quantum phenomena when sufficiently powerful quantum computers become
available.
