On the classical complexity of sampling from quantum interference of indistinguishable bosons

Experimental demonstration of the quantum advantage over classical simulations with Boson Sampling is currently under intensive investigation. There seems to be a scalability issue to the necessary number of bosons on the linear optical platforms and the experiments, such as the recent Boson Sampling with $20$ photons on $60$-port interferometer by H.~Wang~\textit{et al}, \textit{Phys. Rev. Lett.} \textbf{123,} 250503 (2019), are usually carried out on a small interferometer, much smaller than the size necessary for the no-collision regime. Before demonstration of quantum advantage, it is urgent to estimate exactly how the classical computations necessary for sampling from the output distribution of Boson Sampling are reduced when a smaller-size interferometer is used. The present work supplies such a result, valid with arbitrarily close to $1$ probability, which reduces in the no-collision regime to the previous estimate by P.~Clifford and R.~Clifford. One of the results with immediate application to current experiments with Boson Sampling is that classically sampling from the interference of $N$ single bosons on an $M$-port interferometer is at least as hard as that with $\mathcal{N}= \frac{N}{1+N/M}$ single bosons in the no-collision regime, i.e., on a much larger interferometer with at least $\mathcal{M}\gg N^2$ ports.