A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness
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.
Cite
@article{arxiv.1805.05296,
title = {A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness},
author = {Emmanuel Jeandel and Simon Perdrix and Renaud Vilmart},
journal= {arXiv preprint arXiv:1805.05296},
year = {2018}
}