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oai:drops-oai.dagstuhl.de:8117 2024-03-12T11:58:21Z ddc:004 open_access

ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics Jeandel, Emmanuel Perdrix, Simon Vilmart, Renaud Wang, Quanlong Categorical Quantum Mechanincs ZX-Calculus Completeness Cyclotomic Supplmentarity Clifford+T The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language - i.e. the ability to derive any true equation - is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity - called cyclotomic supplementarity - which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning. We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Emmanuel Jeandel and Simon Perdrix and Renaud Vilmart and Quanlong Wang 2017 Is Part Of LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2017.11 urn:nbn:de:0030-drops-81173 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.11 eng https://creativecommons.org/licenses/by/3.0/legalcode