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A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs

Author Colin Blake

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ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Categorical semantics
  • Theory of computation → Equational logic and rewriting
Keywords
  • Qudit circuits
  • Quantum circuits
  • Completeness
  • Control
  • Categorical quantum mechanics

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Abstract

High-dimensional quantum computation needs a native circuit-level equational theory for qudits. We give the first finite schematic equational theory that is sound and complete for exact unitary qudit circuits in every finite dimension at least two. Circuits are built from local gates, sequential and parallel composition, and value-controls; equality is derivable exactly when the standard unitary denotations agree. For each dimension, a finite list of local bounded-arity axiom schemata presents the theory, and the diagrammatic shapes do not depend on d. Primitive value-control makes control on a chosen basis value part of the language, so local rules generate the internal algebra of controlled operations within the circuit PROP. This gives a finite, dimension-uniform basis for exact equational reasoning about qudit circuits.

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Colin Blake. A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.6

Author Details

Colin Blake
  • Université de Lorraine, CNRS, Inria, LORIA, Nancy, France

Funding

This work is supported by the Plan France 2030 through the PEPR integrated project EPiQ (ANR-22-PETQ-0007) and the HQI platform (ANR-22-PNCQ-0002); by the European Union through the MSCA Staff Exchanges project QCOMICAL (Grant Agreement ID: 101182520); and by the Maison du Quantique MaQuEst.

Acknowledgements

I thank my PhD advisors, Simon Perdrix and Miriam Backens, for their helpful feedback and for reviewing earlier versions of this article.

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