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

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 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.

arXiv commentIn Proceedings QPL 2018, arXiv:1901.09476. Contains Appendix

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• Speedy Contraction of ZX Diagrams with Triangles via Stabiliser Decompositions
• Completeness of the Phase-free ZH-calculus
• Generators and Relations for Real Stabilizer Operators
• Well-tempered ZX and ZH Calculi
• Rewriting and Completeness of Sum-Over-Paths in Dyadic Fragments of Quantum Computing
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• A Sound and Complete Equational Theory for 3-Qubit Toffoli-Hadamard Circuits
• A Graphical #SAT Algorithm for Formulae with Small Clause Density