  We explore the possibility of efficient classical simulation of linear optics
experiments under the effect of particle losses. Specifically, we investigate
the canonical boson sampling scenario in which an $n$-particle Fock input state
propagates through a linear-optical network and is subsequently measured by
particle-number detectors in the $m$ output modes. We examine two models of
losses. In the first model a fixed number of particles is lost. We prove that
in this scenario the output statistics can be well approximated by an efficient
classical simulation, provided that the number of photons that is left grows
slower than $\sqrt{n}$. In the second loss model, every time a photon passes
through a beamsplitter in the network, it has some probability of being lost.
For this model the relevant parameter is $s$, the smallest number of
beamsplitters that any photon traverses as it propagates through the network.
We prove that it is possible to approximately simulate the output statistics
already if $s$ grows logarithmically with $m$, regardless of the geometry of
the network. The latter result is obtained by proving that it is always
possible to commute $s$ layers of uniform losses to the input of the network
regardless of its geometry, which could be a result of independent interest. We
believe that our findings put strong limitations on future experimental
realizations of quantum computational supremacy proposals based on boson
sampling.
