The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification

• URL: http://arxiv.org/abs/2306.05204v2
• Date: Thu, 31 Aug 2023 07:01:09 GMT
• Title: The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification
• Authors: Boldizs\'ar Po\'or, Robert I. Booth, Titouan Carette, John van de Wetering, Lia Yeh
• Abstract summary: We derive a simplified rule set that closely resembles the original rules of qubit ZX-calculus. We also demonstrate analogues of the spider-removing local complementation and pivoting rules. We have implemented many of these findings in DiZX, a new open-source Python library for qudit ZX-diagrammatic reasoning.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present a smorgasbord of results on the stabiliser ZX-calculus for odd prime-dimensional qudits (i.e. qupits). We derive a simplified rule set that closely resembles the original rules of qubit ZX-calculus. Using these rules, we demonstrate analogues of the spider-removing local complementation and pivoting rules. This allows for efficient reduction of diagrams to the affine with phases normal form. We also demonstrate a reduction to a unique form, providing an alternative and simpler proof of completeness. Furthermore, we introduce a different reduction to the graph state with local Cliffords normal form, which leads to a novel layered decomposition for qupit Clifford unitaries. Additionally, we propose a new approach to handle scalars formally, closely reflecting their practical usage. Finally, we have implemented many of these findings in DiZX, a new open-source Python library for qudit ZX-diagrammatic reasoning.

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