Document Open Access Logo

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

Authors Alexandre Clément , Simon Perdrix

Thumbnail PDF


  • Filesize: 0.5 MB
  • 14 pages

Document Identifiers

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


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)


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
  • Quantum Computing
  • Diagrammatic Language
  • Completeness
  • Quantum Control
  • Polarising Beam Splitter
  • Categorical Quantum Mechanics
  • Quantum Switch


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Alastair A. Abbott, Julian Wechs, Dominic Horsman, Mehdi Mhalla, and Cyril Branciard. Communication through coherent control of quantum channels. arXiv preprint, October 2018.
  2. Thorsten Altenkirch and Jonathan Grattage. A functional quantum programming language. In 20th Annual IEEE Symposium on Logic in Computer Science (LICS'05), pages 249-258. IEEE, 2005. URL:
  3. Mateus Araújo, Fabio Costa, and Časlav Brukner. Computational advantage from quantum-controlled ordering of gates. Physical review letters, 113(25):250402, 2014. URL:
  4. Miriam Backens and Aleks Kissinger. ZH: A complete graphical calculus for quantum computations involving classical non-linearity. Electronic Proceedings in Theoretical Computer Science, 287:23–42, January 2019. URL:
  5. Filippo Bonchi, Joshua Holland, Robin Piedeleu, Paweł Sobociński, and Fabio Zanasi. Diagrammatic algebra: from linear to concurrent systems. Proceedings of the ACM on Programming Languages, 3(POPL):1-28, 2019. URL:
  6. Filippo Bonchi, Robin Piedeleu, Pawel Sobociński, and Fabio Zanasi. Graphical affine algebra. In 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-12. IEEE, 2019. URL:
  7. Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi. Interacting hopf algebras. Journal of Pure and Applied Algebra, 221(1):144-184, 2017. preprint available from URL:
  8. Giulio Chiribella. Perfect discrimination of no-signalling channels via quantum superposition of causal structures. Physical Review A, 86(4):040301, 2012. URL:
  9. Alexandre Clément and Simon Perdrix. PBS-calculus: A graphical language for coherent control of quantum computations. arXiv preprint, 2020.
  10. Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13(4):043016, 2011. URL:
  11. Timoteo Colnaghi, Giacomo Mauro D'Ariano, Stefano Facchini, and Paolo Perinotti. Quantum computation with programmable connections between gates. Physics Letters A, 376(45):2940-2943, 2012. URL:
  12. Giovanni de Felice, Amar Hadzihasanovic, and Kang Feng Ng. A diagrammatic calculus of fermionic quantum circuits. Logical Methods in Computer Science, 15(3), 2019. URL:
  13. Gilles Dowek and Pablo Arrighi. Lineal: A linear-algebraic lambda-calculus. Logical Methods in Computer Science, 13(1), 2017. URL:
  14. Daniel Ebler, Sina Salek, and Giulio Chiribella. Enhanced communication with the assistance of indefinite causal order. Physical review letters, 120(12):120502, March 2018. URL:
  15. Stefano Facchini and Simon Perdrix. Quantum circuits for the unitary permutation problem. In International Conference on Theory and Applications of Models of Computation, pages 324-331. Springer, 2015. URL:
  16. Adrien Feix, Mateus Araújo, and Časlav Brukner. Quantum superposition of the order of parties as a communication resource. Physical Review A, 92(5):052326, 2015. URL:
  17. Philippe Allard Guérin, Adrien Feix, Mateus Araújo, and Časlav Brukner. Exponential communication complexity advantage from quantum superposition of the direction of communication. Physical review letters, 117(10):100502, 2016. URL:
  18. Amar Hadzihasanovic. The algebra of entanglement and the geometry of composition. PhD thesis, University of Oxford, 2017. URL:
  19. Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103(15), October 2009. URL:
  20. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A complete axiomatisation of the ZX-calculus for Clifford+T quantum mechanics. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, pages 559-568, 2018. URL:
  21. André Joyal, Ross Street, and Dominic Verity. Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society, 119(3):447–468, 1996. URL:
  22. Gregory M Kelly and Miguel L Laplaza. Coherence for compact closed categories. Journal of pure and applied algebra, 19:193-213, 1980. URL:
  23. Saunders Mac Lane. Categorical algebra. Bull. Amer. Math. Soc., 71:40-106, 1965. URL:
  24. Lorenzo M Procopio, Amir Moqanaki, Mateus Araújo, Fabio Costa, Irati Alonso Calafell, Emma G Dowd, Deny R Hamel, Lee A Rozema, Časlav Brukner, and Philip Walther. Experimental superposition of orders of quantum gates. Nature communications, 6:7913, 2015. URL:
  25. Giulia Rubino, Lee A Rozema, Adrien Feix, Mateus Araújo, Jonas M Zeuner, Lorenzo M Procopio, Časlav Brukner, and Philip Walther. Experimental verification of an indefinite causal order. Science advances, 3(3):e1602589, 2017. URL:
  26. Amr Sabry, Benoît Valiron, and Juliana Kaizer Vizzotto. From symmetric pattern-matching to quantum control. In International Conference on Foundations of Software Science and Computation Structures, pages 348-364. Springer, 2018. URL:
  27. Peter Selinger. A survey of graphical languages for monoidal categories. In Bob Coecke, editor, New Structures for Physics, volume 813 of Lecture Notes in Physics, pages 289-355. Springer, 2011. Also available from URL:
  28. Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484–1509, October 1997. URL:
  29. Mingsheng Ying, Nengkun Yu, and Yuan Feng. Alternation in quantum programming: from superposition of data to superposition of programs. arXiv preprint, 2014.
  30. Fabio Zanasi. Interacting Hopf Algebras- the Theory of Linear Systems. PhD thesis, Ecole normale supérieure de lyon - ENS LYON, October 2015. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail