  We propose a $\theta$-scheme to discretize the $d$-dimensional stochastic
cubic Schr\"odinger equation in Stratono\-vich sense. A uniform bound for the
Hamiltonian of the discrete problem is obtained, which is a crucial property to
verify the convergence in probability towards a mild solution.
  Furthermore, based on the uniform bounds of iterates in ${\mathbb
H}^2(\mathcal{O})$ for $\mathcal{O}\subset\mathbb{R}^{1}$, the optimal
convergence order 1 in strong local sense is obtained.
