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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 density of bosons $\rho =\frac{N}{M}$ for $M\ge N$ affects the number of computations required to produce a single sample. First, Glynn's formula is modified for computation of probabilities of output configurations $\mathbf{m}=(m_1,\ldots,m_n,0,\ldots,0)\;$ with $n 0$ with probability $1-\epsilon$ the number of computations $\mathcal{C}_\mathbf{m}$ scales at least as $O\bigl( N 2^{\frac{1-\delta}{1+\rho}N}\bigr)$ and at most as $O\Bigl(N\left(1+r\right)^{\frac{N}{r}}\Bigr)$, where $\delta = \sqrt{\frac{4(1+\rho)}{N}\ln\left(\frac{2}{\epsilon}\right)}\;$ and $r = \mathrm{max}\left(1,\frac{1+\rho}{1+\delta}\right)$. Third, the upper and lower bounds (essentially the same as above) are found on the number of classical computations in the sampling algorithm of P. Clifford and R. Clifford, which is based on using Glynn's formula and applies uniformly over the output configurations.

On the classical complexity of sampling from quantum interference of indistinguishable bosons