A Recipe for Quantum Graphical Languages

Authors Titouan Carette, Emmanuel Jeandel



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2020.118.pdf
  • Filesize: 0.58 MB
  • 17 pages

Document Identifiers

Author Details

Titouan Carette
  • Université de Lorraine, CNRS, Inria, LORIA, Nancy, France
Emmanuel Jeandel
  • Université de Lorraine, CNRS, Inria, LORIA, Nancy, France

Cite AsGet BibTex

Titouan Carette and Emmanuel Jeandel. A Recipe for Quantum Graphical Languages. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 118:1-118:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.118

Abstract

Different graphical calculi have been proposed to represent quantum computation. First the ZX-calculus [Coecke and Duncan, 2011], followed by the ZW-calculus [Hadzihasanovic, 2015] and then the ZH-calculus [Backens and Kissinger, 2018]. We can wonder if new ZX-like calculi will continue to be proposed forever. This article answers negatively. All those language share a common core structure we call Z^*-algebras. We classify Z^*-algebras up to isomorphism in two dimensional Hilbert spaces and show that they are all variations of the aforementioned calculi. We do the same for linear relations and show that the calculus of [Bonchi et al., 2017] is essentially the unique one.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Semantics and reasoning
  • Mathematics of computing
  • Theory of computation → Equational logic and rewriting
Keywords
  • Categorical Quantum Mechanics
  • Quantum Computing
  • Category Theory

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Miriam Backens and Aleks Kissinger. ZH: A complete graphical calculus for quantum computations involving classical non-linearity. arXiv preprint, 2018. URL: http://arxiv.org/abs/1805.02175.
  2. Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi. Interacting Hopf algebras. Journal of Pure and Applied Algebra, 221(1):144-184, 2017. Google Scholar
  3. Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi. Deconstructing Lawvere with distributive laws. Journal of logical and algebraic methods in programming, 95:128-146, 2018. Google Scholar
  4. Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13(4):043016, 2011. Google Scholar
  5. Bob Coecke and Aleks Kissinger. The compositional structure of multipartite quantum entanglement. In International Colloquium on Automata, Languages, and Programming, pages 297-308. Springer, 2010. Google Scholar
  6. Joseph Collins and Ross Duncan. Hopf-Frobenius algebras and a simpler Drinfeld double. Electronic Proceedings in Theoretical Computer Science, 2019. Google Scholar
  7. Niel de Beaudrap and Dominic Horsman. The ZX calculus is a language for surface code lattice surgery. Quantum, 4:218, 2020. Google Scholar
  8. Khadra Dekkar and Abdenacer Makhlouf. Bialgebra structures of 2-associative algebras. arXiv preprint, 2008. URL: http://arxiv.org/abs/0809.1144.
  9. Yukio Doi and Mitsuhiro Takeuchi. BiFrobenius algebras. Contemporary Mathematics, 267:67-98, 2000. Google Scholar
  10. Ross Duncan and Kevin Dunne. Interacting Frobenius Algebras are Hopf. In 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-10. IEEE, 2016. Google Scholar
  11. Ross Duncan, Aleks Kissinger, Simon Perdrix, and John Van De Wetering. Graph-theoretic simplification of quantum circuits with the ZX-calculus. arXiv preprint, 2019. URL: http://arxiv.org/abs/1902.03178.
  12. Amar Hadzihasanovic. A diagrammatic axiomatisation for qubit entanglement. In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 573-584. IEEE, 2015. Google Scholar
  13. Amar Hadzihasanovic. The algebra of entanglement and the geometry of composition. PhD thesis, University of Oxford, 2017. URL: http://arxiv.org/abs/1709.08086.
  14. Amar Hadzihasanovic, Kang Feng Ng, and Quanlong Wang. Two complete axiomatisations of pure-state qubit quantum computing. In 2018 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 502-511. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209128.
  15. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A complete axiomatisation of the ZX-calculus for Clifford+ T quantum mechanics. In 2018 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 559-568, 2018. Google Scholar
  16. André Joyal and Ross Street. The geometry of tensor calculus, I. Advances in mathematics, 88(1):55-112, 1991. Google Scholar
  17. M Koppinen. On algebras with two multiplications, including Hopf algebras and Bose-Mesner algebras. Journal of Algebra, 182(1):256-273, 1996. Google Scholar
  18. Saunders Mac Lane. Categories for the Working Mathematician. Springer, 1971. Google Scholar
  19. Kang Feng Ng and Quanlong Wang. Completeness of the zx-calculus for pure qubit clifford+ t quantum mechanics. arXiv preprint, 2018. URL: http://arxiv.org/abs/1801.07993.
  20. Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information, 2002. Google Scholar
  21. J.S. Ponizovskii. Semigroup rings. Semigroup Forum, 36:1-46, 1987. Google Scholar
  22. Bjorn Poonen. Isomorphism types of commutative algebras of finite rank. Computational arithmetic geometry, 463:111-120, 2008. Google Scholar
  23. Peter Selinger. A survey of graphical languages for monoidal categories. In New structures for physics, pages 289-355. Springer, 2010. Google Scholar
  24. E Study. Über systeme complexer zahlen und ihre anwendung in der theorie der transformationsgruppen. Monatshefte für Mathematik und Physik, 1:283-354, 1890. Google Scholar
  25. Moss E. Sweedler. Hopf Algebras. W.A. Benjamin, Inc., 1969. Google Scholar
  26. Fabio Zanasi. Interacting hopf algebras: the theory of linear systems. arXiv preprint, 2018. URL: http://arxiv.org/abs/1805.03032.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail