Concrete Categorical Model of a Quantum Circuit Description Language with Measurement

Authors Dongho Lee, Valentin Perrelle, Benoît Valiron, Zhaowei Xu

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Dongho Lee
  • Université Paris-Saclay, CentraleSupélec, LMF, France & CEA, List, France
Valentin Perrelle
  • Université Paris-Saclay, CEA, List, France
Benoît Valiron
  • Université Paris-Saclay, CentraleSupélec, LMF, France
Zhaowei Xu
  • Université Paris-Saclay, LMF, France


The authors want to thank Christophe Chareton and Sébastien Bardin for enlightening discussions.

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Dongho Lee, Valentin Perrelle, Benoît Valiron, and Zhaowei Xu. Concrete Categorical Model of a Quantum Circuit Description Language with Measurement. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 51:1-51:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In this paper, we introduce dynamic lifting to a quantum circuit-description language, following the Proto-Quipper language approach. Dynamic lifting allows programs to transfer the result of measuring quantum data - qubits - into classical data - booleans -. We propose a type system and an operational semantics for the language and we state safety properties. Next, we introduce a concrete categorical semantics for the proposed language, basing our approach on a recent model from Rios&Selinger for Proto-Quipper-M. Our approach is to construct on top of a concrete category of circuits with measurements a Kleisli category, capturing as a side effect the action of retrieving classical content out of a quantum memory. We then show a soundness result for this semantics.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Categorical semantics
  • Operational semantics
  • Quantum circuit description language


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