  We consider the nonlinear Schr\"odinger equation (NLS) on a torus of
arbitrary dimension. The equation is studied in presence of an external
potential field whose time-dependent amplitude is taken as control. Assuming
that the potential satisfies a saturation property, we show that the NLS
equation is approximately controllable between any pair of eigenstates in
arbitrarily small time. The proof is obtained by developing a multiplicative
version of a geometric control approach introduced by Agrachev and Sarychev. We
give an application of this result to the study of the large time behavior of
the NLS equation with random potential. More precisely, we assume that the
amplitude of the potential is a random process whose law is $1$-periodic in
time and non-degenerate. Combining the controllability with a stopping time
argument and the Markov property, we show that the trajectories of the random
equation are almost surely unbounded in regular Sobolev spaces.
