  The four-wave interaction in quantum nonlinear Schr\"odinger lattices with
disorder is shown to destroy the Anderson localization of waves, giving rise to
unlimited spreading of the nonlinear field to large distances. Moreover, the
process is not thresholded in the quantum domain, contrary to its "classical"
counterpart, and leads to an accelerated spreading of the subdiffusive type,
with the dispersion $\langle(\Delta n)^2\rangle \sim t^{1/2}$ for
$t\rightarrow+\infty$. The results, presented here, shed new light on the
origin of subdiffusion in systems with a broad distribution of relaxation
times.
