  In order to inherit numerically the ergodicity of the damped stochastic
nonlinear Schr\"odinger equation with additive noise, we propose a fully
discrete scheme, whose spatial direction is based on spectral Galerkin method
and temporal direction is based on a modification of the implicit Euler scheme.
We not only prove the unique ergodicity of the numerical solutions of both
spatial semi-discretization and full discretization, but also present error
estimations on invariant measures, which gives order $2$ in spatial direction
and order ${\frac12}$ in temporal direction.
