  Giving a convincing experimental evidence of the quantum supremacy over
classical simulations is a challenging goal. Noise is considered to be the main
problem in such a demonstration, hence it is urgent to understand the effect of
noise. Recently found classical algorithms can efficiently approximate, to any
small error, the output of boson sampling with finite-amplitude noise. In this
work it is shown analytically and confirmed by numerical simulations that one
can efficiently distinguish the output distribution of such a noisy boson
sampling from the approximations accounting for low-order quantum multiboson
interferences, what includes the mentioned classical algorithms. The number of
samples required to tell apart the quantum and classical output distributions
is strongly affected by the previously unexplored parameter: density of bosons,
i.e., the ratio of total number of interfering bosons to number of input ports
of interferometer. Such critical dependence is strikingly reminiscent of the
quantum-to-classical transition in systems of identical particles, which sets
in when the system size scales up while density of particles vanishes.
