Completeness of the ZX-Calculus

@article{Jeandel2020CompletenessOT,
  title={Completeness of the ZX-Calculus},
  author={Emmanuel Jeandel and Simon Perdrix and Renaud Vilmart},
  journal={Log. Methods Comput. Sci.},
  year={2020},
  volume={16}
}
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… Expand
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A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics
TLDR
The ZX-Calculus is made complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language, and it is proved that the π/4-fragment of the ZX -Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals. Expand
Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics
TLDR
It is shown that the axiomatisation for Clifford+T is not complete in general but can be completed by adding a single (non linear) axiom, providing a simpler axiom atisation of the ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang. Expand
Towards a Minimal Stabilizer ZX-calculus
TLDR
It is shown that most of the remaining rules of the language are necessary, however leaving as an open question the necessity of two rules, including the bialgebra rule, which is an axiomatisation of complementarity, the cornerstone of the ZX-calculus. Expand
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TLDR
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TLDR
It is proved that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the stabilizer ZX-calculus, can in fact be derived. Expand
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TLDR
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TLDR
A ZX-calculus augmented with triangle nodes is considered, and the form of the matrices it represents is precisely shown, and an axiomatisation is provided which makes the language complete for the Toffoli-Hadamard quantum mechanics. Expand
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TLDR
It is proved that its π 4-fragment is not complete, in other words the ZX-calculus is notcomplete for the so called "Clifford+T quantum mechanics". Expand
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TLDR
This paper presents a full graphical axiomatisation of the relations between GHZ and W: the ZW calculus, which refines a version of the preexisting ZX calculus, while keeping its most desirable characteristics: undirected ness, a large degree of symmetry, and an algebraic underpinning. Expand
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TLDR
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