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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.
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