  It has been recently proposed that late time behavior of holographic
complexity in a uncharged black brane solution of Einstein-Hilbert theory with
boundary cut off is consistent with Lloyd's bound if we have a cut off behind
the horizon. Interestingly, the value of this new cut off is fixed by the
boundary cut off. In this paper, we extend this analysis to the charged black
holes. Concretely, we find the value of this new cut off for charged small
black hole solutions of Einstein-Hilbert-Maxwell theory, in which the proposed
bound on the complexification is saturated. We also explore this new cut off in
Gauss-Bonnet-Maxwell theory.
