Algorithm 1: Transition function

$h:=\{i:l_i>0\}$               ▹h is a set that contains indices of non-empty outputs $K:=\left|h\right|$                         ▹K is the amount of non-empty outputs $r:=random\left(\right\{1,\dots ,K\left\}\right)$              ▹ We generate a random number $1\le r\le K$ $l_{h_r}:=l_{h_r}-1$                  ▹ Decrease the number of photons in $h_r$-th output $s:=random(\{1,\dots ,N\}\setminus h_r)$    ▹ We generate a random number $1\le s\le N$ such that $s\ne r$ $l_s:=l_s+1$                    ▹ Increase the number of photons in s-th output return $|l_1,l_2,\dots ,l_N\rangle$