Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions

Authors Robert I. Booth , Titouan Carette

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Robert I. Booth
  • University of Edinburgh, Edinburgh, United Kingdom
  • Sorbonne Université, CNRS, LIP6, 4 place Jussieu, F-75005 Paris, France
  • LORIA CNRS, Inria Mocqua, Université de Lorraine, F-54000 Nancy, France
Titouan Carette
  • Université Paris-Saclay, Inria, CNRS, LMF, 91190, Gif-sur-Yvette, France


We thank Cole Comfort and Simon Perdrix for enlightening discussions.

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Robert I. Booth and Titouan Carette. Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We introduce a family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions. These calculi recover many of the nice features of the qubit ZX-calculus which were lost in previous proposals for higher-dimensional systems. We then prove that these calculi are complete, i.e. provide a set of rewrite rules which can be used to prove any equality of stabiliser quantum operations. Adding a discard construction, we obtain a calculus complete for mixed state stabiliser quantum mechanics in odd prime dimensions, and this furthermore gives a complete axiomatisation for the related diagrammatic language for affine co-isotropic relations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • ZX-calculus
  • completeness
  • quantum
  • stabiliser
  • qudits


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