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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.