Jeandel, Emmanuel and Perdrix, Simon and Vilmart, Renaud - Completeness of the ZX-Calculus

lmcs:5288 - Logical Methods in Computer Science, June 4, 2020, Volume 16, Issue 2 - https://doi.org/10.23638/LMCS-16(2:11)2020
Completeness of the ZX-Calculus

Authors: Jeandel, Emmanuel and Perdrix, Simon and Vilmart, Renaud

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 for 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.


Volume: Volume 16, Issue 2
Published on: June 4, 2020
Submitted on: March 15, 2019
Keywords: Quantum Physics,Computer Science - Logic in Computer Science


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