A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

Renaud Vilmart
(Université de Lorraine, CNRS, Inria, LORIA, F 54000 Nancy, France)

We consider a ZX-calculus augmented with triangle nodes which is well-suited to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics. We precisely show the form of the matrices it represents, and we provide an axiomatisation which makes the language complete for the Toffoli-Hadamard quantum mechanics. We extend the language with arbitrary angles and show that any true equation involving linear diagrams which constant angles are multiple of Pi are derivable. We show that a single axiom is then necessary and sufficient to make the language equivalent to the ZX-calculus which is known to be complete for Clifford+T quantum mechanics. As a by-product, it leads to a new and simple complete axiomatisation for Clifford+T quantum mechanics.

In Peter Selinger and Giulio Chiribella: Proceedings of the 15th International Conference on Quantum Physics and Logic (QPL 2018), Halifax, Canada, 3-7th June 2018, Electronic Proceedings in Theoretical Computer Science 287, pp. 313–344.
Contains Appendix.
Published: 31st January 2019.

ArXived at: https://dx.doi.org/10.4204/EPTCS.287.18 bibtex PDF
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