Fig. 3: Simulation in the finite setting.
Analysis of total variation distance (TVD) between the non-linear process and its linear-optical simulation, and bunching probability at the non-linear phase shift site for n = 4 photons. The TVD is calculated between the exact probability distribution [Eq. (4)], and the post-selected linear-optical simulation using k < n auxiliary photons. In these plots, the exact distribution is calculated using the linear-optical simulation scheme with n auxiliary photons. \({P}_{{{{\rm{bunch}}}}}^{x}\) is the probability of having more than k photons at the nonlinearity site x. a Parametric plot showing a correlation between the mean values \(\overline{{{{\rm{TVD}}}}}\) and \(\overline{{P}_{{{{\rm{bunch}}}}}^{x}}\), averaged over 100 different evolutions W and V for fixed ϕ = π/2. Each point corresponds to a different value of m, in the range [5, 7, …, 45]. b–e Correlation between TVD and \({P}_{{{{\rm{bunch}}}}}^{x}\) for fixed number of modes m. Here, each point corresponds to different evolutions W and V for fixed ϕ = π/2. In all plots: blue points correspond to k = 1, orange points to k = 2, and green points to k = 3.
