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

Renaud Vilmart 1
1 MOCQUA - Designing the Future of Computational Models
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : 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 π 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.
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Renaud Vilmart. A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond. QPL 2018, Jun 2018, Halifax, Canada. pp.313-344, ⟨10.4204/EPTCS.287.18⟩. ⟨hal-01762264⟩

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