  In this paper, we first investigate the global existence of a solution for
the stochastic fractional nonlinear Schr\"odinger equation with radially
symmetric initial data in a suitable energy space $H^{\alpha}$. We then show
that the stochastic fractional nonlinear Schr\"odinger equation in the
Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system,
with its phase flow preserving symplecticity. Finally, we develop a stochastic
midpoint scheme for the stochastic fractional nonlinear Schr\"odinger equation
from the perspective of symplectic geometry. It is proved that the stochastic
midpoint scheme satisfies the corresponding symplectic law in the discrete
sense. A numerical example is conducted to validate the efficiency of the
theory.
