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We start by studying the subgroup structures underlying stabilizer circuits.Then we apply our results to provide two normal forms for stabilizer circuits.These forms are computed by induction using simple conjugation rules in theClifford group and our algorithms do not rely on a special decomposition in thesymplectic group. The first normal form has shape CX-CZ-P-Z-X-H-CZ-P-H, whereCX (resp. CZ) denotes a layer of CNOT (resp. controlled-Z) gates, P a layer ofphase gates, X (resp. Z) a layer of Pauli-X (resp. Pauli-Z) gates. Then wereplace most of the controlled-Z gates by CNOT gates to obtain a second normalform of type P-CX-CZ-CX-Z-X-H-CZ-CX-P-H. In this second form, both controlled-Zlayers have depth 1 and together contain therefore at most n controlled-Zgates. We also consider normal forms for stabilizer states and graph states.Finally we carry out a few tests on classical and quantum computers in order toshow experimentally the utility of these normal forms to reduce the gate countof a stabilizer circuit.

Reducing stabilizer circuits without the symplectic group