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ZX-calculus: cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics. (English) Zbl 1447.81077
Larsen, Kim G. (ed.) et al., 42nd international symposium on mathematical foundations of computer science, MFCS 2017, August 21–25, 2017, Aalborg, Denmark. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik. LIPIcs – Leibniz Int. Proc. Inform. 83, Article 11, 13 p. (2017).
Summary: 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 [the second and last authors, LIPIcs – Leibniz Int. Proc. Inform. 58, Article 76, 14 p. (2016; Zbl 1398.81015)]. 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.
For the entire collection see [Zbl 1376.68011].

##### MSC:
 81P68 Quantum computation 18M30 String diagrams and graphical calculi 81P65 Quantum gates 15A67 Applications of Clifford algebras to physics, etc.
##### Keywords:
quantum computing; ZX-calculus; incompleteness; Clifford+T
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##### References:
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