Classical simulation of linear optics subject to nonuniform losses

We present a comprehensive study of the impact of non-uniform, i.e.\ path-dependent, photonic losses on the computational complexity of linear-optical processes. Our main result states that, if each beam splitter in a network induces some loss probability, non-uniform network designs cannot circumvent the efficient classical simulations based on losses. To achieve our result we obtain new intermediate results that can be of independent interest. First, we show that, for any network of lossy beam-splitters, it is possible to extract a layer of non-uniform losses that depends on the network geometry. We prove that, for every input mode of the network it is possible to commute $s_i$ layers of losses to the input, where $s_i$ is the length of the shortest path connecting the $i$th input to any output. We then extend a recent classical simulation algorithm due to P. Clifford and R. Clifford to allow for arbitrary $n$-photon input Fock states (i.e. to include collision states). Consequently, we identify two types of input states where boson sampling becomes classically simulable: (A) when $n$ input photons occupy a constant number of input modes; (B) when all but $O(\log n)$ photons are concentrated on a single input mode, while an additional $O(\log n)$ modes contain one photon each.