A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

@article{Jeandel2017ACA,
  title={A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics},
  author={Emmanuel Jeandel and Simon Perdrix and Renaud Vilmart},
  journal={Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science},
  year={2017}
}
We introduce the first complete and approximately universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics -- also called the π/4-fragment of the ZX-Calculus -- was one of the main open questions in categorical quantum mechanics. We prove the… 
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102 Citations

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32 References

Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

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.

Completeness of the ZX-Calculus

This work improves on the known-to-be-complete presentation for the so-called Clifford fragment of the ZX-Calculus, and provides a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.

Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics

Recently, we gave a complete axiomatisation of the ZX-calculus [1] for the overall pure qubit quantum mechanics [4]. In this paper, we first simplify the rule of addition (AD) and show that some…

A Simplified Stabilizer ZX-calculus

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.

Towards a Minimal Stabilizer ZX-calculus

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.

ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics

It is shown 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, implying the incompleteness of the language for the so-called Clifford+T quantum mechanics.

Generalised Supplementarity and new rule for Empty Diagrams to Make the ZX-Calculus More Expressive

To prove the incompleteness of the π 4-fragment, an equality over scalars that is not derivable from the set of axioms is shown, and that will lead to a substitute for the inverse rule, and the obsolescence of the zero rule.

Supplementarity is Necessary for Quantum Diagram Reasoning

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".

The ZX-calculus is complete for stabilizer quantum mechanics

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary…

Interacting quantum observables: categorical algebra and diagrammatics

The ZX-calculus is introduced, an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information and axiomatize phase shifts within this framework.