Abstract: Boson sampling has sparked the imagination of theorists and experimentalists since it was introduced by Aaronson and Arkhipov. Here, single photon Fock states are launched into a multimode interferometer where, due to boson statistics, the probability of any output distribution of photons is related to the permanent of a matrix (derived from the interferometer transformation). This makes the output distribution of events difficult to sample from unless certain computational complexity classes are equivalent.
The use of input states different from Fock states, namely Gaussian states, is interesting both theoretically and experimentally. In this work we show that the output distribution of photon numbers from a Gaussian state is given by a matrix function which can be conjectured to be as difficult as the matrix permanent. We then relate the problem of the typical model of Fock boson sampling to the Gaussian model, showing the former can be seen as a special case of the latter.