A finite presentation of cnot-dihedral perators

Abstract

We give a finite presentation by generators and relations of the unitary operators expressible over the fCNOT;T;Xg gate set, also known as CNOT-dihedral operators. To this end, we introduce a notion of normal form for CNOT-dihedral circuits and prove that every CNOT-dihedral operator admits a unique normal form. Moreover, we show that in the presence of certain structural rules only finitely many circuit identities are required to reduce an arbitrary CNOT-dihedral circuit to its normal form. By appropriately restricting our relations, we obtain a finite presentation of unitary operators expressible over the fCNOT;Tg gate set as a corollary.