On the classical complexity of sampling from quantum interference of indistinguishable bosons (1904.02013v5)

Abstract: 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.