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DOI: 10.4230/LIPIcs.MFCS.2022.35
LO_v-Calculus: A Graphical Language for Linear Optical Quantum Circuits

Authors: Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, and Benoît Valiron

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
We introduce the LO_v-calculus, a graphical language for reasoning about linear optical quantum circuits with so-called vacuum state auxiliary inputs. We present the axiomatics of the language and prove its soundness and completeness: two LO_v-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LO_v-calculus. We give a confluent and terminating rewrite system to rewrite any polarisation-preserving LO_v-circuit into a unique triangular normal form, inspired by the universal decomposition of Reck et al. (1994) for linear optical quantum circuits.
Cite as

Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, and Benoît Valiron. LO_v-Calculus: A Graphical Language for Linear Optical Quantum Circuits. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{clement_et_al:LIPIcs.MFCS.2022.35,
  author =	{Cl\'{e}ment, Alexandre and Heurtel, Nicolas and Mansfield, Shane and Perdrix, Simon and Valiron, Beno\^{i}t},
  title =	{{LO\underlinev-Calculus: A Graphical Language for Linear Optical Quantum Circuits}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{35:1--35:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.35},
  URN =		{urn:nbn:de:0030-drops-168334},
  doi =		{10.4230/LIPIcs.MFCS.2022.35},
  annote =	{Keywords: Quantum Computing, Graphical Language, Linear Optical Circuits, Linear Optical Quantum Computing, Completeness}
}
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