Combining the scale auxiliary variable (SAV) approach with leapfrog finite difference methods, an unconditional energy-stable, non-couple and linearly implicit numerical scheme is presented for solving spatial fractional Cahn–Hilliard equations. The fully-discrete scheme gives an indefinite and ill-conditioned system of linear algebraic equations, whose coefficient matrix is a 2 × 2 block with block Toeplitz matrix. The preconditioned minimum residual method with a positive preconditioner is employed for the resulting system. Numerical results are presented to confirm the effectiveness of the proposed numerical scheme and preconditioner.