  A block lower triangular Toeplitz system arising from time-space fractional
diffusion equation is discussed. For efficient solutions of such the linear
system, the preconditioned biconjugate gradient stabilized method and flexible
general minimal residual method are exploited. The main contribution of this
paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is
developed for the block lower triangular Toeplitz system, whose storage is of
$\mathcal{O}(N)$ with $N$ being the spatial grid number; (ii) A new
skew-circulant preconditioner is designed to fast calculate the inverse of the
block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical
experiments are given to demonstrate the efficiency of our preconditioners.
