Noise in BosonSampling and the threshold of efficient classical simulatability

We study the quantum to classical transition in Boson Sampling by analysing how $N$-boson interference is affected by inevitable noise in an experimental setup. We adopt the Gaussian noise model of Kalai and Kindler for Boson Sampling and show that it appears from some realistic experimental imperfections. We reveal a connection between noise in Boson Sampling and partial distinguishability of bosons, which allows us to prove efficient classical simulatability of noisy no-collision Boson Sampling with finite noise amplitude $\epsilon$, i.e., $\epsilon = \Omega(1)$ as $N\to \infty$. On the other hand, using an equivalent representation of network noise as losses of bosons compensated by random (dark) counts of detectors, it is proven that for noise amplitude inversely proportional to total number of bosons, i.e., $\epsilon=O(1/N)$, noisy no-collision Boson Sampling is as hard to simulate classically as in the noiseless case. Moreover, the ratio of ``noise clicks" (lost bosons compensated by dark counts) to the total number of bosons $N$ vanishes as $N\to \infty$ for arbitrarily small noise amplitude, i.e., $\epsilon = o(1)$ as $N\to \infty$, hence, we conjecture that such a noisy Boson Sampling is also hard to simulate classically. The results significantly relax sufficient condition on noise in a network components, such as two-mode beam splitters, for classical hardness of experimental Boson Sampling.