What is the most efficient way to generate a random permutation from probabilistic pairwise swaps?
A working algorithm that I described in a comment above is the following:


First start by bringing a random element with probability $1/n$ in position $n$: swap positions 1 and 2 with proba $1/2$, then 2 and 3 with proba $2/3$, ... then $n-1$ and $n$ with proba $(n-1)/n$.
Apply the same procedure to bring a random element in position $n-1$: swap positions 1 and 2 with prob $1/2$ ... then positions $n-2$ and $n-1$ with proba $(n-2)/(n-1)$.
Etc


The number of gates required by this algorithm is $(n-1)+(n-2)+ \cdots + 2+1 = n(n-1)/2 = O(n^2)$.