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We study the quantum to classical transition in Boson Sampling by analysinghow $N$-boson interference is affected by inevitable noise in an experimentalsetup. We adopt the Gaussian noise model of Kalai and Kindler for BosonSampling and show that it appears from some realistic experimentalimperfections. We reveal a connection between noise in Boson Sampling andpartial distinguishability of bosons, which allows us to prove efficientclassical simulatability of noisy no-collision Boson Sampling with finite noiseamplitude $\epsilon$, i.e., $\epsilon = \Omega(1)$ as $N\to \infty$. On theother hand, using an equivalent representation of network noise as losses ofbosons compensated by random (dark) counts of detectors, it is proven that fornoise amplitude inversely proportional to total number of bosons, i.e.,$\epsilon=O(1/N)$, noisy no-collision Boson Sampling is as hard to simulateclassically 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 noisyBoson Sampling is also hard to simulate classically. The results significantlyrelax sufficient condition on noise in a network components, such as two-modebeam splitters, for classical hardness of experimental Boson Sampling.

Noise in BosonSampling and the threshold of efficient classical simulatability