  We propose a Fourier pseudo-spectral scheme for the space-fractional
nonlinear Schr\"odinger equation. The proposed scheme has the following
features: it is linearly implicit, it preserves two invariants of the equation,
its unique solvability is guaranteed without any restrictions on space and time
step sizes. The scheme requires solving a complex symmetric linear system per
time step. To solve the system efficiently, we also present a certain variable
transformation and preconditioner.
