  We start by studying the subgroup structures underlying stabilizer circuits
and we use our results to propose a new normal form for stabilizer circuits.
This normal form is computed by induction using simple conjugation rules in the
Clifford group. It has shape CX-CZ-P-H-CZ-P-H, where CX (resp. CZ) denotes a
layer of $\cnot$ (resp. $\cz$) gates, P a layer of phase gates and H a layer of
Hadamard gates. Then we consider a normal form for stabilizer states and we
show how to reduce the two-qubit gate count in circuits implementing graph
states. Finally we carry out a few numerical tests on classical and quantum
computers in order to show the practical utility of our methods. All the
algorithms described in the paper are implemented in the C language as a Linux
command available on GitHub.
