  Determining the vibrational structure of a molecule is central to fundamental
applications in several areas, from atmospheric science to catalysis, fuel
combustion modeling, biochemical imaging, and astrochemistry. However, when
significant anharmonicity and mode coupling are present, the problem is
classically intractable for a molecule of just a few atoms. Here, we outline a
set of quantum algorithms for solving the molecular vibrational structure
problem for both near- and long-term quantum computers. There are previously
unaddressed characteristics of this problem which require approaches distinct
from most instances of the commonly studied quantum simulation of electronic
structure: many eigenstates are often desired, states of interest are often far
from the ground state (requiring methods for "zooming in" to some energy
window), and transition amplitudes with respect to a non-unitary Hermitian
operator must be calculated. We address these hurdles and consider problem
instances of four molecular vibrational Hamiltonians. Finally and most
importantly, we give analytical and numerical results which suggest that, to a
given energy precision, a vibrational problem instance will be simulatable on a
quantum computer before an electronic structure problem instance. These results
imply that more focus in the quantum information community ought to shift
toward scientifically and industrially important quantum vibrational problems.
