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We start by studying the subgroup structures underlying stabilizer circuitsand 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 theClifford group. It has shape CX-CZ-P-H-CZ-P-H, where CX (resp. CZ) denotes alayer of $\cnot$ (resp. $\cz$) gates, P a layer of phase gates and H a layer ofHadamard gates. Then we consider a normal form for stabilizer states and weshow how to reduce the two-qubit gate count in circuits implementing graphstates. Finally we carry out a few numerical tests on classical and quantumcomputers in order to show the practical utility of our methods. All thealgorithms described in the paper are implemented in the C language as a Linuxcommand available on GitHub.

Reduced quantum circuits for stabilizer states and graph states