  This paper proposes and analyzes an efficient difference scheme for the
nonlinear complex Ginzburg-Landau equation involving fractional Laplacian. The
scheme is based on the implicit midpoint rule for the temporal discretization
and a weighted and shifted Gr\"unwald difference operator for the spatial
fractional Laplacian. By virtue of a careful analysis of the difference
operator, some useful inequalities with respect to suitable fractional Sobolev
norms are established. Then the numerical solution is shown to be bounded, and
convergent in the $l^2_h$ norm with the optimal order $O(\tau^2+h^2)$ with time
step $\tau$ and mesh size $h$. The a priori bound as well as the convergence
order hold unconditionally, in the sense that no restriction on the time step
$\tau$ in terms of the mesh size $h$ needs to be assumed. Numerical tests are
performed to validate the theoretical results and effectiveness of the scheme.
