Sufficient condition for the mode mismatch of single photons for scalability of the boson-sampling computer

The boson sampler proposed by Aaronson and Arkhipov is a non-universal quantum computer, which can serve as evidence against the extended Church-Turing thesis. It samples the probability distribution at the output of linear unitary optical network, with indistinguishable single photons at the input. Four experimental groups have already tested their small-scale prototypes with up to four photons. The boson sampler with few dozens of single photons is believed to be hard to simulate on a classical computer. For scalability of a realistic boson sampler with current technology it is necessary to know the effect of the photon mode mismatch on its operation. Here a nondeterministic model of the boson sampler is analyzed, which employs partially indistinguishable single photons emitted by identical sources. A sufficient condition on the average mutual fidelity $ \langle \mathcal{F}\rangle$ of the single photons is found, which guarantees that the realistic boson sampler outperforms the classical computer. Moreover, the boson sampler computer with partially indistinguishable single photons is scalable while being beyond the power of classical computers when the single photon mode mismatch $1-\langle \mathcal{F}\rangle$ scales as $ \mathcal{O}(N^{-3/2})$ with the total number of photons $N$.