  Using a photonic quantum computer for boson sampling has been demonstrated a
tremendous advantage over classical supercomputers. It is highly desirable to
develop boson sampling algorithms for realistic scientific problems. In this
work, we propose a hybrid quantum-classical sampling (HQCS) algorithm to
calculate the optical spectrum for complex molecules considering anharmonicity
and Duschinsky rotation (DR) effects. The classical sum-over-state method for
this problem has a computational complexity that exponentially increases with
system size. In the HQCS algorithm, an intermediate harmonic potential energy
surface (PES) is created, bridging the initial and final PESs. The magnitude
and sign (-1 or +1) of the overlap between the initial state and the
intermediate state are estimated by quantum boson sampling and by classical
algorithms respectively, achieving an exponential speed-up. Additionally, the
overlap between the intermediate state and the final state is efficiently
evaluated by classical algorithms. The feasibility of HQCS is demonstrated in
calculations of the emission spectrum of a Morse model as well as pyridine
molecule by comparison with the nearly exact time-dependent density matrix
renormalization group solutions. A near-term quantum advantage for realistic
molecular spectroscopy simulation is proposed.
