A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness

TL;DR

This paper introduces a generic normal form for ZX-diagrams applicable to Clifford+T quantum mechanics and rational angle fragments, providing a constructive approach to completeness proofs and an algorithm for normalization.

Contribution

It presents a generic normal form for ZX-diagrams, sufficient conditions for axiomatisation completeness, and an algorithm to normalize diagrams across various quantum fragments.

Findings

Normal form for ZX-diagrams in Clifford+T and rational angle fragments.

A complete axiomatisation for these fragments using the normal form.

Introduction of a simple cancellation rule for non-dyadic rational angles.

Abstract

Recent completeness results on the ZX-Calculus used a third-party language, namely the ZW-Calculus. As a consequence, these proofs are elegant, but sadly non-constructive. We address this issue in the following. To do so, we first describe a generic normal form for ZX-diagrams in any fragment that contains Clifford+T quantum mechanics. We give sufficient conditions for an axiomatisation to be complete, and an algorithm to reach the normal form. Finally, we apply these results to the Clifford+T fragment and the general ZX-Calculus -- for which we already know the completeness--, but also for any fragment of rational angles: we show that the axiomatisation for Clifford+T is also complete for any fragment of dyadic angles, and that a simple new rule (called cancellation) is necessary and sufficient otherwise.