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.