A Finite Presentation of CNOT-Dihedral Operators

TL;DR

This paper provides a finite set of generators and relations for CNOT-dihedral operators, introducing a unique normal form for circuits and simplifying their analysis and synthesis.

Contribution

It introduces a normal form for CNOT-dihedral circuits and proves the existence of a finite presentation for these operators, including a subset with only CNOT and T gates.

Findings

Unique normal form for CNOT-dihedral circuits

Finite presentation with generators and relations

Simplified circuit reduction to normal form

Abstract

We give a finite presentation by generators and relations of the unitary operators expressible over the {CNOT, T, X} 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 {CNOT, T} gate set as a corollary.