Strong approximation of stochastic semiclassical Schrödinger equation with multiplicative noise

We consider the stochastic nonlinear Schrödinger equation driven by a multiplicative noise in a semiclassical regime, where the Plank constant $ \varepsilon $ is small. In this regime, the solution of the equation exhibits high-frequency oscillations. We design an efficient numerical approximation by combining the spectral Galerkin method and the temporal midpoint scheme. This accurately approximates the solution, or at least of the associated physical observables. Furthermore, the strong convergence rates for the proposed numerical approximation are derived, which explicitly depend on the scaled Planck constant. This conclusion implies the semiclassical regime's admissible meshing strategies for obtaining 'correct' physical observables.