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International Conference on Reversible Computation

RC 2019: Reversible Computation pp 3–19Cite as

Concurrent Quantum Strategies

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Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11497))

Abstract

A game-semantics foundation for quantum computation is presented. It draws on two lines of work: for its temporal dynamics, on concurrent games and strategies, based on event structures; for its quantum interactions, on the mathematical foundations of positive operators and completely positive maps. The two lines are married in the definition of quantum concurrent strategy, obtained via an operator generalisation of the conditions on a probabilistic concurrent strategy. The result is a compact-closed (bi)category of quantum games, whose finite configurations carry finite dimensional Hilbert spaces, and quantum strategies, whose finite configurations carry operators.

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Notes

  1. 1.

    The use of subdensity rather than density operators, where \(\mathrm{tr}(\rho )=1\), is natural in quantum systems which may stick with a non-trivial probability.

  2. 2.

    A Scott-open subset of configurations is upwards-closed w.r.t. inclusion and such that if it contains the union of a directed subset S of configurations then it contains an element of S. A continuous valuation is a function w from the Scott-open subsets of \(\,\!{\mathcal C}^\infty (E)\) to [0, 1] which is ((normalised)\(w(\,\!{\mathcal C}^\infty (E)) = 1\); (strict) \(w(\emptyset ) = 0 \); (monotone)\(U \subseteq V \implies w(U)\le w(V)\); (modular)\(w(U \cup V) + w(U\cap V) = w(U) + w(V)\); and (continuous) \(w(\bigcup _{i\in I} U_i) = \mathrm{sup}_{i\in I} w(U_i)\), for directed unions.

  3. 3.

    We eschew the other obvious possibility in which the game also determines the operators because we want strategies to be quantum, not just probabilistic, in line with the quantum lambda-calculus [5] and earlier definition [6].

References

  1. Winskel, G.: Events, causality and symmetry. Comput. J. 54, 42–57 (2011)

    Article  Google Scholar 

  2. Winskel, G.: Distributed games and strategies. Festschrift for pierre-louis curien on the occasion of his 60th birthday (2014, as yet unpublished)

    Google Scholar 

  3. Winskel, G.: Distributed probabilistic and quantum strategies. Electr. Notes Theor. Comput. Sci. 298, 403–425 (2013)

    Article  MathSciNet  Google Scholar 

  4. Grabbe, J.: An introduction to quantum game theory. arXiv preprint quant-ph/0506219 (2005)

    Google Scholar 

  5. Pagani, M., Selinger, P., Valiron, B.: Applying quantitative semantics to higher-order quantum computing. In: POPL 2014 (2014)

    Google Scholar 

  6. Delbecque, Y., Panagaden, P.: Game semantics for quantum stores. MFPS XXIV, Electr. Notes Theor. Comput. Sci. 218, 153–170 (2008)

    Google Scholar 

  7. Dal Lago, U., Faggian, C., Valiron, B., Yoshimizu, A.: The geometry of parallelism, classical, probabilistic, and quantum effects. In: POPL 2017 (2017)

    Google Scholar 

  8. Brukner, C.: Quantum causality. Nat. Phys. 10, 259 (2014)

    Article  Google Scholar 

  9. Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: LICS 2004 (2004)

    Google Scholar 

  10. Castellan, S., Clairambault, P., Winskel, G.: The parallel intensionally fully abstract games model of PCF. In: LICS 2015 (2015)

    Google Scholar 

  11. Castellan, S., Clairambault, P., Paquet, H., Winskel, G.: The concurrent game semantics of probabilistic PCF. In: LICS 2018 (2018)

    Google Scholar 

  12. Selinger, P., Valiron, B.: On a fully abstract model for a quantum linear functional language. Electr. Notes Theor. Comput. Sci. 210, 123–137 (2008)

    Article  MathSciNet  Google Scholar 

  13. Clairambault, P., de Visme, M., Winskel, G.: Game semantics for quantum programming. In: POPL 2019 (2019)

    Google Scholar 

  14. Joyal, A., Street, R., Verity, D.: Traced monoidal categories. In: Mathematical Proceedings of the Cambridge Philosophical Society (1996)

    Google Scholar 

  15. Selinger, P.: Dagger compact closed categories and completely positive maps: (extended abstract). Electr. Notes Theor. Comput. Sci. 170, 139–163 (2007)

    Article  Google Scholar 

  16. Winskel, G.: Probabilistic and quantum event structures. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds.) Horizons of the Mind. A Tribute to Prakash Panangaden. LNCS, vol. 8464, pp. 476–497. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06880-0_25

    Chapter  Google Scholar 

  17. Alvarez-Manilla, M., Edalat, A., Saheb-Djahromi, N.: An extension result for continuous valuations. J. LMS 61, 629–640 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Leifer, M.S., Spekkens, R.W.: Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference. Phys. Rev. 88, 052130 (2013)

    Article  Google Scholar 

  19. Saunders-Evans, L., Winskel, G.: Event structure spans for nondeterministic dataflow. Electr. Notes Theor. Comput. Sci. 175(3), 109–129 (2007)

    Article  Google Scholar 

  20. Selinger, P.: Towards a quantum programming language. Math. Struct. Comput. Sci. 14, 527–586 (2004)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We are grateful to the ERC for Advanced Grant ECSYM. Discussions with Frank Roumen and Benoît Valiron have been very helpful.

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Authors and Affiliations

  1. Univ Lyon, ENS de Lyon, CNRS, UCB Lyon 1, LIP, Lyon, France

    Pierre Clairambault & Marc de Visme

  2. Computer Laboratory, University of Cambridge, Cambridge, UK

    Glynn Winskel

Authors
  1. Pierre Clairambault
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  2. Marc de Visme
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  3. Glynn Winskel
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Corresponding author

Correspondence to Glynn Winskel .

Editor information

Editors and Affiliations

  1. University of Copenhagen, Copenhagen, Denmark

    Michael Kirkedal Thomsen

  2. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Mathias Soeken

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Clairambault, P., de Visme, M., Winskel, G. (2019). Concurrent Quantum Strategies. In: Thomsen, M., Soeken, M. (eds) Reversible Computation. RC 2019. Lecture Notes in Computer Science(), vol 11497. Springer, Cham. https://doi.org/10.1007/978-3-030-21500-2_1

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  • DOI: https://doi.org/10.1007/978-3-030-21500-2_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21499-9

  • Online ISBN: 978-3-030-21500-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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