  The main objective of this paper is to present an efficient
structure-preserving scheme, which is based on the idea of the scalar auxiliary
variable approach, for solving the space fractional nonlinear Schr\"{o}dinger
equation. First, we reformulate the equation as a Hamiltonian system, and
obtain a new equivalent system via introducing a scalar variable. Then, we
construct a semi-discrete energy-preserving scheme by using the Fourier
pseudo-spectral method to discretize the equivalent system in space direction.
After that, applying the Crank-Nicolson method on the temporal direction gives
a linear implicit scheme in the fully-discrete version. As expected, the
proposed scheme can preserve the energy exactly and more efficient in the sense
that only decoupled equations with constant coefficients need to be solved at
each time step. Finally, numerical experiments are provided to demonstrate the
effectiveness and conservation of the scheme.
