  This paper discusses a numerical method for computing the evolution of large
interacting system of quantum particles. The idea of the random batch method is
to replace the total interaction of each particle with the $N-1$ other
particles by the interaction with $p<N$ particles chosen at random at each time
step, multiplied by $(N-1)/p$. This reduces the computational cost of computing
the interaction partial per time step from $O(N^2)$ to $O(N)$. For simplicity,
we consider only in this work the case $p=1$. In other words, we assume that
$N$ is even, and that at each time step, the $N$ particles are organized in
$N/2$ pairs, with a random reshuffling of the pairs at the beginning of each
time step. We obtain a convergence estimate for the Wigner transform of the
single-particle reduced density matrix of the particle system at time $t$ that
is uniform in $N>1$ and independent of the Planck constant $\hbar$.
