  In this paper we improve the layered implementation of arbitrary stabilizer
circuits introduced by Aaronson and Gottesman in {\it Phys. Rev. A 70(052328)},
2004. In particular, we reduce their 11-stage computation
-H-C-P-C-P-C-H-P-C-P-C- into an 8-stage computation of the form
-H-C-CZ-P-H-P-CZ-C-. We show arguments in support of using -CZ- stages over the
-C- stages: not only the use of -CZ- stages allows a shorter layered
expression, but -CZ- stages are simpler and appear to be easier to implement
compared to the -C- stages. Relying on the 8-stage decomposition we develop a
two-qubit depth-$(14n-4)$ implementation of stabilizer circuits over the gate
library {P,H,CNOT}, executable in the LNN architecture, improving best
previously known depth-$25n$ circuit, also executable in the LNN architecture.
Our constructions rely on folding arbitrarily long sequences $($-P-C-$)^m$ into
a 3-stage computation -P-CZ-C-, as well as efficient implementation of the -CZ-
stage circuits.
