Generators and Relations for Real Stabilizers

Abstract

Real stabilizer operators, which are also known as real Clifford operators, are generated, through composition and tensor product, by the Hadamard gate, the Pauli Z gate, and the controlled-Z gate. We introduce a normal form for real stabilizer circuits and show that every real stabilizer operator admits a unique normal form. Moreover, we give a finite set of relations that suffice to rewrite any Clifford circuit to its normal form. This yields a presentation by generators and relations of the strict spatial monoidal category of real stabilizer operators.