  It is known that, by accounting for the multiboson interferences up to a
finite order, the output distribution of noisy Boson Sampling, with
distinguishability of bosons serving as noise, can be approximately sampled
from in a time polynomial in the total number of bosons. The drawback of this
approach is that the joint probabilities of completely distinguishable bosons,
i.e., those that do not interfere at all, have to be computed also. In trying
to restore the ability to sample from the distinguishable bosons with
computation of only the single-boson probabilities, one faces the following
issue: the quantum probability factors in a convex-sum expression, if truncated
to a finite order of multiboson interference, have, on average, a finite amount
of negativity in a random interferometer. The truncated distribution does
become a proper one, while allowing for sampling from it in a polynomial time,
only in a vanishing domain close to the completely distinguishable bosons.
Nevertheless, the conclusion that the negativity issue is inherent to all
efficient classical approximations to noisy Boson Sampling may be premature. I
outline the direction for a whole new program, which seem to point to a
solution. However its success depends on the asymptotic behavior of the
symmetric group characters, which is not known.
