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Simpler Presentations for Many Fragments of Quantum Circuits

Author Colin Blake

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LIPIcs.FSCD.2026.6.pdf
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  • 20 pages

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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum circuits
  • Clifford group
  • equational theories
  • minimality
  • qutrit

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Abstract

Equational reasoning is central to quantum circuit optimisation and verification: one replaces subcircuits by provably equivalent ones using a fixed set of rewrite rules viewed as equations. A finite rule set is most informative when it separates the genuine algebra of a circuit fragment from the structural treatment of wires. This paper gives six near-Clifford fragments a common PROP treatment, where wire permutations are structural: qubit Clifford, real Clifford, Clifford+T (up to two qubits), Clifford+CS (up to three qubits), CNOT-dihedral, and qutrit Clifford. Starting from prior completeness theorems, we transfer completeness into this setting and remove redundant non-structural rules, then check minimality by separating interpretations tailored to individual axioms; the resulting presentations are minimal in all arities for qubit Clifford, real Clifford, and CNOT-dihedral, minimal in bounded ranges for the remaining fragments, and comparable by one transfer-and-separation pattern.

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Colin Blake. Simpler Presentations for Many Fragments of Quantum Circuits. In 11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 378, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FSCD.2026.6

Author Details

Colin Blake
  • Inria Mocqua and Université de Lorraine, CNRS, LORIA, F-54000 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 reviews of early versions of the article. I also thank Noé Delorme for many discussions, early presentations of the Clifford and Real Clifford fragments, and joint attempts to derive candidate rules.

References

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