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One Rig to Control Them All

Authors Chris Heunen , Robin Kaarsgaard , Louis Lemonnier

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LIPIcs.LICS.2026.56.pdf
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ACM Subject Classification
  • Theory of computation → Control primitives
  • Theory of computation → Categorical semantics
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum control
  • rig categories
  • complete equational theories

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Abstract

Controlled commands - computations whose execution depends on a separate input - play a central role in reversible Boolean circuits and quantum circuits. However, existing formalisms typically treat control only implicitly, entangled with other aspects of computation. From a semantic perspective, control is most naturally expressed in semisimple rig categories, which - unlike standard circuit models such as props - support both parallel and conditional composition.
We present a construction that freely adjoins an explicit syntactic notion of control to a circuit theory specified as a suitable prop, subject to eight universally quantified equations. Our main result is that these equations are sound and complete for the intended semantics of control: the resulting theory satisfies a universal property, identifying it exactly as the circuit subtheory of the free semisimple rig completion. The proof combines coherence for rig categories with a new method based on induction over Gray codes.
We illustrate the usefulness of the framework by showing that it simplifies several existing sound and complete axiomatisations of quantum circuits, isolating a small and conceptually clean set of generators and equations. In addition, the same equations yield a sound and complete axiomatisation of the multiply controlled Toffoli gate set, that is universal for reversible Boolean circuits.

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Chris Heunen, Robin Kaarsgaard, and Louis Lemonnier. One Rig to Control Them All. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 56:1-56:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.LICS.2026.56

Author Details

Chris Heunen
  • University of Edinburgh, UK
Robin Kaarsgaard
  • University of Southern Denmark, Odense, Denmark
Louis Lemonnier
  • University of Edinburgh, UK

Funding

This research was funded by the Engineering and Physical Sciences Research Council (EPSRC) under project EP/X025551/1 "Rubber DUQ: Flexible Dynamic Universal Quantum programming". Robin Kaarsgaard was supported by Sapere Aude: DFF-Research Leader grant 5251-00024B.

Acknowledgements

The authors would like to thank the members of the quantum programming group, specifically Wang Fang, and of the category theory group at Edinburgh for their support and feedback. We extend our thanks to Robert Booth, Kostia Chardonnet, Noé Delorme, Nicolas Heurtel, Simon Perdrix, and Peter Selinger for comments and fruitful discussions that helped us improve the paper. We also thank the anonymous reviewers for their comments and suggestions.

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