  Calculating the observable properties of chemical systems is often
classically intractable and is widely viewed as a promising application of
quantum information processing. Yet one of the most common and important
chemical systems in nature - the hydrogen bond - has remained a challenge to
study using quantum hardware on account of its anharmonic potential energy
landscape. Here, we introduce a framework for solving hydrogen-bond systems and
more generic chemical dynamics problems using quantum logic. We experimentally
demonstrate a proof-of-principle instance of our method using the QSCOUT
ion-trap quantum computer, in which we experimentally drive the ion-trap system
to emulate the quantum wavepacket of the shared-proton within a hydrogen bond.
Following the experimental creation of the shared-proton wavepacket, we then
extract measurement observables such as its time-dependent spatial projection
and its characteristic vibrational frequencies to spectroscopic accuracy (3.3
cm$^{-1}$ wavenumbers, corresponding to > 99.9% fidelity). Our approach
introduces a new paradigm for studying the quantum chemical dynamics and
vibrational spectra of molecules, and when combined with existing algorithms
for electronic structure, opens the possibility to describe the complete
behavior of complex molecular systems with unprecedented accuracy.
