We consider a nonlinear Schr\"odinger equation (NLS) with random
coefficients, in a regime of separation of scales corresponding to
diffusion approximation. The primary goal is to propose and
study an efficient numerical scheme in this framework. We use a
pseudo-spectral splitting scheme and we establish the order of the
global error. In particular we show that we can take an integration step
larger than the smallest scale of the problem, here the correlation
length of the random medium. We study
the asymptotic behavior of the numerical solution in the diffusion
approximation regime.