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Simple Qudit ZX and ZH Calculi, via Integrals

Authors Niel de Beaudrap , Richard D. P. East

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Niel de Beaudrap
  • University of Sussex, Brighton, UK
Richard D. P. East
  • Haiqu

Acknowledgements

Thanks to Patrick Roy, Titouan Carette, John van de Wetering, and Robert Booth for technical discussions, and to the anonymous referees for suggestions on presentation.

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Niel de Beaudrap and Richard D. P. East. Simple Qudit ZX and ZH Calculi, via Integrals. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.20

Abstract

The ZX calculus and ZH calculus use diagrams to denote and compute properties of quantum operations, using "rewrite rules" to transform between diagrams which denote the same operator through a functorial semantic map. Different semantic maps give rise to different rewrite systems, which may prove more convenient for different purposes. Using discrete measures, we describe semantic maps for ZX and ZH diagrams, well-suited to analyse unitary circuits and measurements on qudits of any fixed dimension D > 1 as a single "ZXH-calculus". We demonstrate rewrite rules for the "stabiliser fragment" of the ZX calculus and a "multicharacter fragment" of the ZH calculus.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • ZX-calculus
  • ZH-calculus
  • qudits
  • string diagrams
  • discrete integrals

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