The probabilities of vibronic transitions in molecules are referred to as Franck-Condon factors (FCFs). Although several approaches for calculating FCFs have been developed, such calculations are still challenging. Recently it was shown that there exists a correspondence between the problem of calculating FCFs and boson sampling. However, if the output photon number distribution of boson sampling is sparse, then it can be classically simulated. Exploiting these results, we develop a method to approximately reconstruct the distribution of FCFs of certain molecules. We demonstrate its proof of concept by applying it to formic acid and thymine at 0 K. In our method, we first obtain the marginal photon number distributions for pairs of modes of a Gaussian state associated with the molecular transition. We then apply a compressive sensing method called polynomial-time matching pursuit to recover FCFs.
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