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

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arxiv 1705.11151 v2 pith:OMTMEANT submitted 2017-05-31 quant-ph

classification quant-ph
keywords cliffordmechanicsquantumzx-calculuscompletelanguagecompletenessdiagrammatic
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We introduce the first complete and approximatively 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 four new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open questions in categorical quantum mechanics. We prove the completeness of the Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that the Clifford+T fragment of the ZX-Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.

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    quant-ph 2026-08 conditional novelty 6.0 of 10

    The authors define Coecke flow lines as branch-independent paths through quantum protocol diagrams that survive every semantics-preserving rewrite down to a bare wire.

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