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

Emmanuel Jeandel 1 Simon Perdrix 1 Renaud Vilmart 1
1 MOCQUA - Designing the Future of Computational Models
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
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.
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Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart. A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness. 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), Jun 2019, Vancouver, Canada. ⟨10.1109/LICS.2019.8785754⟩. ⟨hal-01791791⟩

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