A Recipe for Quantum Graphical Languages

Authors Titouan Carette, Emmanuel Jeandel

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Titouan Carette
  • Université de Lorraine, CNRS, Inria, LORIA, Nancy, France
Emmanuel Jeandel
  • Université de Lorraine, CNRS, Inria, LORIA, Nancy, France

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Titouan Carette and Emmanuel Jeandel. A Recipe for Quantum Graphical Languages. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 118:1-118:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Different graphical calculi have been proposed to represent quantum computation. First the ZX-calculus [Coecke and Duncan, 2011], followed by the ZW-calculus [Hadzihasanovic, 2015] and then the ZH-calculus [Backens and Kissinger, 2018]. We can wonder if new ZX-like calculi will continue to be proposed forever. This article answers negatively. All those language share a common core structure we call Z^*-algebras. We classify Z^*-algebras up to isomorphism in two dimensional Hilbert spaces and show that they are all variations of the aforementioned calculi. We do the same for linear relations and show that the calculus of [Bonchi et al., 2017] is essentially the unique one.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Semantics and reasoning
  • Mathematics of computing
  • Theory of computation → Equational logic and rewriting
  • Categorical Quantum Mechanics
  • Quantum Computing
  • Category Theory


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