A Complete Graphical Language for Linear Optical Circuits with Finite-Photon-Number Sources and Detectors

Author Nicolas Heurtel



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Nicolas Heurtel
  • Quandela, 7 Rue Léonard de Vinci, 91300 Massy, France
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France

Acknowledgements

We would like to thank Marc de Visme and Vladimir Zamdzhiev for helpful discussions, Alexandre Clément for the insight into the angles of ({E3}) and the derivation of ({oE3}), and particularly Shane Mansfield, Benoît Valiron and Renaud Vilmart for helpful discussions, support and reviews of some parts of the paper. We would also like to thank all the anonymous reviewers for their insightful comments and suggestions, which greatly helped to improve the quality of this manuscript.

Cite As Get BibTex

Nicolas Heurtel. A Complete Graphical Language for Linear Optical Circuits with Finite-Photon-Number Sources and Detectors. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 38:1-38:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.38

Abstract

Graphical languages are powerful and useful to represent, rewrite and simplify different kinds of processes. In particular, they have been widely used for quantum processes, improving the state of the art for compilation, simulation and verification. In this work, we focus on one of the main carrier of quantum information and computation: linear optical circuits. We introduce the LO_fi-calculus, the first graphical language to reason on the infinite-dimensional photonic space with circuits only composed of the four core elements of linear optics: the phase shifter, the beam splitter, and auxiliary sources and detectors with bounded photon number. First, we study the subfragment of circuits composed of phase shifters and beam splitters, for which we provide the first minimal equational theory. Next, we introduce a rewriting procedure on those LO_fi-circuits that converge to normal forms. We prove those forms to be unique, establishing both a novel and unique representation of linear optical processes. Finally, we complement the language with an equational theory that we prove to be complete: two LO_fi-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LO_fi-calculus.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Axiomatic semantics
  • Hardware → Quantum computation
Keywords
  • Quantum Computing
  • Graphical Language
  • Linear Optical Circuits
  • Linear Optical Quantum Computing
  • Completeness
  • Fock Space

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