On the classical complexity of quantum interference of indistinguishable bosons
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The classical complexity of sampling from the probability distribution of quantum interference of N indistinguishable single bosons on unitary network with M input and output ports is studied with the focus on how boson density ρ =N/M for M> N affects the number of computations required to produce a single sample. Glynn's formula is modified for computation of probabilities of output configurations m=(m_1,...,m_n,0,...,0) with n<N output ports occupied by bosons, requiring only C_m≡ O(N∏_l=1^n (m_l+1)/min(m_l+1)) computations. It is found that in a unitary network chosen according to the Haar probability measure the tails of the distribution of the total number n of output ports occupied by bosons are bounded by those of a binomial distribution. This fact allows to prove that for any ϵ>0 with probability 1-ϵ the number of computations in the classical sampling algorithm of P. Clifford and R. Clifford scales as at least O( N 2^1-δ/1+ρN) and at most O(N(1+r)^N/r), where δ = √(4(1+ρ)/N(2/ϵ)) and r = max(1,1+ρ/1+δ).
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