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PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations

Authors Alexandre Clément , Simon Perdrix

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Author Details

Alexandre Clément
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Simon Perdrix
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France

Funding

This work is funded by ANR-17-CE25-0009 SoftQPro, ANR-17-CE24-0035 VanQuTe, PIA-GDN/Quantex, and LUE / UOQ.

Acknowledgements

The authors want to thank Mehdi Mhalla, Emmanuel Jeandel and Titouan Carette for fruitful discussions. All diagrams were written with the help of TikZit.

Cite AsGet BibTex

Alexandre Clément and Simon Perdrix. PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.MFCS.2020.24

Abstract

We introduce the PBS-calculus to represent and reason on quantum computations involving coherent control of quantum operations. Coherent control, and in particular indefinite causal order, is known to enable multiple computational and communication advantages over classically ordered models like quantum circuits. The PBS-calculus is inspired by quantum optics, in particular the polarising beam splitter (PBS for short). We formalise the syntax and the semantics of the PBS-diagrams, and we equip the language with an equational theory, which is proved to be sound and complete: two diagrams are representing the same quantum evolution if and only if one can be transformed into the other using the rules of the PBS-calculus. Moreover, we show that the equational theory is minimal. Finally, we consider applications like the implementation of controlled permutations and the unrolling of loops.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Axiomatic semantics
  • Theory of computation → Categorical semantics
  • Hardware → Quantum computation
  • Hardware → Quantum communication and cryptography
Keywords
  • Quantum Computing
  • Diagrammatic Language
  • Completeness
  • Quantum Control
  • Polarising Beam Splitter
  • Categorical Quantum Mechanics
  • Quantum Switch

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