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Completeness of the ZX-Calculus

Abstract : The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. It comes equipped with an equational presentation. We focus here on a very important property of the language: completeness, which roughly ensures the equational theory captures all of quantum mechanics. We first improve on the known-to-be-complete presentation or the so-called Clifford fragment of the language - a restriction that is not universal - by adding some axioms. Thanks to a system of back-and-forth translation between the ZX-Calculus and a third-party complete graphical language, we prove that the provided axiomatisation is complete for the first approximately universal fragment of the language, namely Clifford+T. We then prove that the expressive power of this presentation, though aimed at achieving completeness for the aforementioned restriction, extends beyond Clifford+T, to a class of diagrams that we call linear with Clifford+T constants. We use another version of the third-party language - and an adapted system of back-and-forth translation - to complete the language for the ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the added axioms, and finally, we provide a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.
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Contributor : Simon Perdrix <>
Submitted on : Monday, December 9, 2019 - 1:31:58 PM
Last modification on : Tuesday, December 8, 2020 - 9:51:14 AM

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Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart. Completeness of the ZX-Calculus. Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2020, 16 (2), pp.11:1 -- 11:72. ⟨10.23638/LMCS-16(2:11)2020⟩. ⟨hal-02400081⟩



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