Scaling of boson sampling experiments

Boson sampling is the problem of generating a multiphoton state whose counting probability is the permanent of an $n\times n$ matrix. This is created as the output $n$-photon coincidence rate of a prototype quantum computing device with $n$ input photons. It is a fundamental challenge to verify boson sampling, and therefore the question of how output count rates scale with matrix size $n$ is crucial. Here we apply results from random matrix theory as well as the characteristic function approach from quantum optics to establish analytical scaling laws for average count rates. We treat boson sampling experiments with arbitrary inputs, outputs, and losses. Using the scaling laws we analyze grouping of channel outputs and the count rates for this case.

Figure 1

Average subunitary permanent squared $P_{n|m}$ with $t=1$ for $k=1,2,4,6$, with $k=1$ at the top and $k=6$ at the bottom.

Figure 2

Relative sampling errors in the subunitary permanent squared $P_{n|m}$ with $t=1$ for $k=2$. Numerical results for 40000 random unitaries (solid lines with error bars) are compared to exact results (dashed lines) and the asymptotic power law form (dotted line) for up to $n=25$.

Figure 3

Upper bound on count rates $R_n^m$ for $n$ photons occurring, in any $n$ output channels, for $k=2,...6$, with $k=6$ at the top and $k=2$ at the bottom.