  We consider a finite dimensional approximation of the stochastic nonlinear
Schr\"odinger equation driven by multiplicative noise, which is derived by
applying a symplectic method to the original equation in spatial direction.
Both the unique ergodicity and the charge conservation law for this finite
dimensional approximation are obtained on the unit sphere. To simulate the
ergodic limit over long time for the finite dimensional approximation, we
discretize it further in temporal direction to obtain a fully discrete scheme,
which inherits not only the stochastic multi-symplecticity and charge
conservation law of the original equation but also the unique ergodicity of the
finite dimensional approximation. The temporal average of the fully discrete
numerical solution is proved to converge to the ergodic limit with order one
with respect to the time step for a fixed spatial step. Numerical experiments
verify our theoretical results on charge conservation, ergodicity and weak
convergence.
