DROPS

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DOI: 10.4230/LIPIcs.LICS.2026.56

One Rig to Control Them All

Authors: Chris Heunen, Robin Kaarsgaard, and Louis Lemonnier

Published in: LIPIcs, Volume 380, 41st Annual Symposium on Logic in Computer Science (LICS 2026)

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)

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@InProceedings{heunen_et_al:LIPIcs.LICS.2026.56,
  author =	{Heunen, Chris and Kaarsgaard, Robin and Lemonnier, Louis},
  title =	{{One Rig to Control Them All}},
  booktitle =	{41st Annual Symposium on Logic in Computer Science (LICS 2026)},
  pages =	{56:1--56:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-434-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{380},
  editor =	{Faggian, Claudia and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.LICS.2026.56},
  URN =		{urn:nbn:de:0030-drops-268435},
  doi =		{10.4230/LIPIcs.LICS.2026.56},
  annote =	{Keywords: Quantum control, rig categories, complete equational theories}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.CONCUR.2025.2

Towards Categorical Quantum Concurrency Theory (Invited Talk)

Authors: Chris Heunen

Published in: LIPIcs, Volume 348, 36th International Conference on Concurrency Theory (CONCUR 2025)

Abstract

Quantum computing inherently has concurrent aspects. Even with only local operations, qubits can influence each other. This ability leads to genuinely new quantum communication protocols, but also raises even thornier questions of causality than in classical concurrent computing. Monoidal categories and their string diagrams form a convenient and popular language for quantum computing. After an introduction to quantum concurrency, I will discuss the framework of tensor topology, which aims to analyse the interaction of several agents in monoidal categories, using notions from sheaf theory and ordered locales.

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Chris Heunen. Towards Categorical Quantum Concurrency Theory (Invited Talk). In 36th International Conference on Concurrency Theory (CONCUR 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 348, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{heunen:LIPIcs.CONCUR.2025.2,
  author =	{Heunen, Chris},
  title =	{{Towards Categorical Quantum Concurrency Theory}},
  booktitle =	{36th International Conference on Concurrency Theory (CONCUR 2025)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-389-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{348},
  editor =	{Bouyer, Patricia and van de Pol, Jaco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2025.2},
  URN =		{urn:nbn:de:0030-drops-239527},
  doi =		{10.4230/LIPIcs.CONCUR.2025.2},
  annote =	{Keywords: Quantum computing, causality, monoidal categories, tensor topology}
}

Document

DOI: 10.4230/LIPIcs.CSL.2022.15

Localisable Monads

Authors: Carmen Constantin, Nuiok Dicaire, and Chris Heunen

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract

Monads govern computational side-effects in programming semantics. A collection of monads can be combined together in a local-to-global way to handle several instances of such effects. Indexed monads and graded monads do this in a modular way. Here, instead, we start with a single monad and equip it with a fine-grained structure by using techniques from tensor topology. This provides an intrinsic theory of local computational effects without needing to know how constituent effects interact beforehand. Specifically, any monoidal category decomposes as a sheaf of local categories over a base space. We identify a notion of localisable monads which characterises when a monad decomposes as a sheaf of monads. Equivalently, localisable monads are formal monads in an appropriate presheaf 2-category, whose algebras we characterise. Three extended examples demonstrate how localisable monads can interpret the base space as locations in a computer memory, as sites in a network of interacting agents acting concurrently, and as time in stochastic processes.

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Carmen Constantin, Nuiok Dicaire, and Chris Heunen. Localisable Monads. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{constantin_et_al:LIPIcs.CSL.2022.15,
  author =	{Constantin, Carmen and Dicaire, Nuiok and Heunen, Chris},
  title =	{{Localisable Monads}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.15},
  URN =		{urn:nbn:de:0030-drops-157353},
  doi =		{10.4230/LIPIcs.CSL.2022.15},
  annote =	{Keywords: Monad, Monoidal category, Presheaf, Central idempotent, Graded monad, Indexed monad, Formal monad, Strong monad, Commutative monad}
}

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Documents authored by Heunen, Chris

One Rig to Control Them All

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Towards Categorical Quantum Concurrency Theory (Invited Talk)

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Localisable Monads

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