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DOI: 10.4230/LIPIcs.MFCS.2026.6

A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs

Authors: Colin Blake

Published in: LIPIcs, Volume 386, 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)

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)

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@InProceedings{blake:LIPIcs.MFCS.2026.6,
  author =	{Blake, Colin},
  title =	{{A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs}},
  booktitle =	{51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-442-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{386},
  editor =	{Kouck\'{y}, Michal and Petrișan, Daniela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.6},
  URN =		{urn:nbn:de:0030-drops-273875},
  doi =		{10.4230/LIPIcs.MFCS.2026.6},
  annote =	{Keywords: Qudit circuits, Quantum circuits, Completeness, Control, Categorical quantum mechanics}
}

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DOI: 10.4230/LIPIcs.FSCD.2026.6

Simpler Presentations for Many Fragments of Quantum Circuits

Authors: Colin Blake

Published in: LIPIcs, Volume 378, 11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026)

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)

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@InProceedings{blake:LIPIcs.FSCD.2026.6,
  author =	{Blake, Colin},
  title =	{{Simpler Presentations for Many Fragments of Quantum Circuits}},
  booktitle =	{11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026)},
  pages =	{6:1--6:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-433-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{378},
  editor =	{Pfenning, Frank},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2026.6},
  URN =		{urn:nbn:de:0030-drops-263562},
  doi =		{10.4230/LIPIcs.FSCD.2026.6},
  annote =	{Keywords: Quantum circuits, Clifford group, equational theories, minimality, qutrit}
}

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A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs

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Simpler Presentations for Many Fragments of Quantum Circuits

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