skip to main content
research-article
Open Access

Full abstraction for the quantum lambda-calculus

Authors Info & Claims
Published:20 December 2019Publication History
Skip Abstract Section

Abstract

Quantum programming languages permit a hardware independent, high-level description of quantum algorithms. In particular, the quantum λ-calculus is a higher-order language with quantum primitives, mixing quantum data and classical control. Giving satisfactory denotational semantics to the quantum λ-calculus is a challenging problem that has attracted significant interest. In the past few years, both static (the quantum relational model) and dynamic (quantum game semantics) denotational models were given, with matching computational adequacy results. However, no model was known to be fully abstract.

Our first contribution is a full abstraction result for the games model of the quantum λ-calculus. Full abstraction holds with respect to an observational quotient of strategies, obtained by summing valuations of all states matching a given observable. Our proof method for full abstraction extends a technique recently introduced to prove full abstraction for probabilistic coherence spaces with respect to probabilistic PCF.

Our second contribution is an interpretation-preserving functor from quantum games to the quantum relational model, extending a long line of work on connecting static and dynamic denotational models. From this, it follows that the quantum relational model is fully abstract as well.

Altogether, this gives a complete denotational landscape for the semantics of the quantum λ-calculus, with static and dynamic models related by a clean functorial correspondence, and both fully abstract.

Skip Supplemental Material Section

Supplemental Material

a63-clairambault.webm

References

  1. Samson Abramsky, Radha Jagadeesan, and Pasquale Malacaria. 2000. Full Abstraction for PCF. Inf. Comput. 163, 2 (2000), 409–470. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Samson Abramsky and Guy McCusker. 1996. Linearity, Sharing and State: a fully abstract game semantics for Idealized Algol with active expressions. Electr. Notes Theor. Comput. Sci. 3 (1996), 2–14. Google ScholarGoogle ScholarCross RefCross Ref
  3. Samson Abramsky and Paul-André Melliès. 1999. Concurrent Games and Full Completeness. In 14th Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, July 2-5, 1999. 431–442. Google ScholarGoogle ScholarCross RefCross Ref
  4. Patrick Baillot, Vincent Danos, Thomas Ehrhard, and Laurent Regnier. 1997. Timeless Games. In Computer Science Logic, 11th International Workshop, CSL ’97, Annual Conference of the EACSL, Aarhus, Denmark, August 23-29, 1997, Selected Papers. 56–77. Google ScholarGoogle ScholarCross RefCross Ref
  5. Gavin Bierman. 1993. On intuitionistic linear logic. Ph.D. Dissertation. University of Cambridge Computer Laboratory.Google ScholarGoogle Scholar
  6. Pierre Boudes. 2009. Thick Subtrees, Games and Experiments. In Typed Lambda Calculi and Applications, 9th International Conference, TLCA 2009, Brasilia, Brazil, July 1-3, 2009. Proceedings. 65–79. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Ana C. Calderon and Guy McCusker. 2010. Understanding Game Semantics Through Coherence Spaces. Electr. Notes Theor. Comput. Sci. 265 (2010), 231–244. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Simon Castellan. 2017. Concurrent structures in game semantics. (Structures concurrentes en sémantique des jeux). Ph.D. Dissertation. University of Lyon, France. https://tel.archives- ouvertes.fr/tel- 01587718Google ScholarGoogle Scholar
  9. Simon Castellan and Pierre Clairambault. 2016. Causality vs. Interleavings in Concurrent Game Semantics. In 27th International Conference on Concurrency Theory, CONCUR 2016, August 23-26, 2016, Québec City, Canada. 32:1–32:14. Google ScholarGoogle ScholarCross RefCross Ref
  10. Simon Castellan, Pierre Clairambault, Hugo Paquet, and Glynn Winskel. 2018. The concurrent game semantics of Probabilistic PCF. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018. 215–224. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Simon Castellan, Pierre Clairambault, Silvain Rideau, and Glynn Winskel. 2017. Games and Strategies as Event Structures. LMCS 13, 3 (2017).Google ScholarGoogle Scholar
  12. Simon Castellan, Pierre Clairambault, and Glynn Winskel. 2014. Symmetry in concurrent games. In Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, Vienna, Austria, July 14 - 18, 2014. 28:1–28:10. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Simon Castellan, Pierre Clairambault, and Glynn Winskel. 2015. The Parallel Intensionally Fully Abstract Games Model of PCF. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015. 232–243. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Simon Castellan, Pierre Clairambault, and Glynn Winskel. 2019. Thin Games with Symmetry and Concurrent Hyland-Ong Games. Logical Methods in Computer Science 15, 1 (2019). https://lmcs.episciences.org/5248Google ScholarGoogle Scholar
  15. Simon Castellan and Nobuko Yoshida. 2019. Two sides of the same coin: session types and game semantics: a synchronous side and an asynchronous side. PACMPL 3, POPL (2019), 27:1–27:29. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Pierre Clairambault, Marc de Visme, and Glynn Winskel. 2019. Game semantics for quantum programming. PACMPL 3, POPL (2019), 32:1–32:29. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Pierre Clairambault, Julian Gutierrez, and Glynn Winskel. 2012. The Winning Ways of Concurrent Games. In Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science, LICS 2012, Dubrovnik, Croatia, June 25-28, 2012. 235–244. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. J.R.B. Cockett and R.A.G. Seely. 1997. Weakly distributive categories. Journal of Pure and Applied Algebra 114, 2 (1997), 133–173.Google ScholarGoogle ScholarCross RefCross Ref
  19. Ugo Dal Lago and Olivier Laurent. 2008. Quantitative Game Semantics for Linear Logic. In Computer Science Logic, 22nd International Workshop, CSL 2008, 17th Annual Conference of the EACSL, Bertinoro, Italy, September 16-19, 2008. Proceedings. 230–245. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Yannick Delbecque. 2011. Game Semantics for Quantum Data. Electr. Notes Theor. Comput. Sci. 270, 1 (2011), 41–57. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Peter Dybjer and Andrzej Filinski. 2000. Normalization and Partial Evaluation. In Applied Semantics, International Summer School, APPSEM 2000, Caminha, Portugal, September 9-15, 2000, Advanced Lectures. 137–192.Google ScholarGoogle Scholar
  22. Thomas Ehrhard. 2012. The Scott model of linear logic is the extensional collapse of its relational model. Theor. Comput. Sci. 424 (2012), 20–45. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Thomas Ehrhard, Christine Tasson, and Michele Pagani. 2014. Probabilistic coherence spaces are fully abstract for probabilistic PCF. In The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL ’14, San Diego, CA, USA, January 20-21, 2014. 309–320. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, and Hugo Zbinden. 2002. Quantum cryptography. Reviews of modern physics 74, 1 (2002), 145.Google ScholarGoogle Scholar
  25. Lov K. Grover. 1996. A Fast Quantum Mechanical Algorithm for Database Search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996. 212–219.Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Russell Harmer and Guy McCusker. 1999. A Fully Abstract Game Semantics for Finite Nondeterminism. In 14th Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, July 2-5, 1999. 422–430. Google ScholarGoogle ScholarCross RefCross Ref
  27. Ichiro Hasuo and Naohiko Hoshino. 2017. Semantics of higher-order quantum computation via geometry of interaction. Ann. Pure Appl. Logic 168, 2 (2017), 404–469. Google ScholarGoogle ScholarCross RefCross Ref
  28. J. M. E. Hyland and C.-H. Luke Ong. 2000. On Full Abstraction for PCF: I, II, and III. Inf. Comput. 163, 2 (2000), 285–408. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Martin Hyland and Andrea Schalk. 1999. Abstract Games for Linear Logic. Electr. Notes Theor. Comput. Sci. 29 (1999), 127–150. Google ScholarGoogle ScholarCross RefCross Ref
  30. Martin Hyland and Andrea Schalk. 2003. Glueing and orthogonality for models of linear logic. Theor. Comput. Sci. 294, 1/2 (2003), 183–231. Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. André Joyal, Ross Street, and Dominic Verity. 1996. Traced monoidal categories. In Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 119. Cambridge University Press, 447–468.Google ScholarGoogle ScholarCross RefCross Ref
  32. Jim Laird, Giulio Manzonetto, and Guy McCusker. 2013. Constructing differential categories and deconstructing categories of games. Inf. Comput. 222 (2013), 247–264. Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. Octavio Malherbe. 2013. Categorical models of computation: partially traced categories and presheaf models of quantum computation. Ph.D. Dissertation. University of Ottawa.Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. Octavio Malherbe, Philip Scott, and Peter Selinger. 2013. Presheaf Models of Quantum Computation: An Outline. In Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky - Essays Dedicated to Samson Abramsky on the Occasion of His 60th Birthday. 178–194. Google ScholarGoogle ScholarCross RefCross Ref
  35. Paul-André Melliès. 2005. Asynchronous Games 4: A Fully Complete Model of Propositional Linear Logic. In 20th IEEE Symposium on Logic in Computer Science (LICS 2005), 26-29 June 2005, Chicago, IL, USA, Proceedings. 386–395. Google ScholarGoogle ScholarDigital LibraryDigital Library
  36. Paul-André Melliès. 2006. Asynchronous games 2: The true concurrency of innocence. Theor. Comput. Sci. 358, 2-3 (2006), 200–228. Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. Paul-André Melliès and Nicolas Tabareau. 2010. Resource modalities in tensor logic. Ann. Pure Appl. Logic 161, 5 (2010), 632–653. Google ScholarGoogle ScholarCross RefCross Ref
  38. Paul-André Melliès. 2009. Categorical semantics of linear logic. Panoramas et syntheses 27 (2009), 15–215.Google ScholarGoogle Scholar
  39. Robin Milner. 1977. Fully Abstract Models of Typed lambda-Calculi. Theor. Comput. Sci. 4, 1 (1977), 1–22.Google ScholarGoogle ScholarCross RefCross Ref
  40. Andrzej S. Murawski and C.-H. Luke Ong. 2003. Exhausting strategies, joker games and full completeness for IMLL with Unit. Theor. Comput. Sci. 294, 1/2 (2003), 269–305. Google ScholarGoogle ScholarDigital LibraryDigital Library
  41. Michele Pagani, Peter Selinger, and Benoît Valiron. 2014. Applying quantitative semantics to higher-order quantum computing. In The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL ’14, San Diego, CA, USA, January 20-21, 2014. 647–658. Google ScholarGoogle ScholarDigital LibraryDigital Library
  42. John Power and Edmund Robinson. 1997. Premonoidal Categories and Notions of Computation. Mathematical Structures in Computer Science 7, 5 (1997), 453–468. Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. John Power and Hayo Thielecke. 1999. Closed Freyd- and kappa-categories. In ICALP’99 (LNCS), Vol. 1644. Springer.Google ScholarGoogle ScholarDigital LibraryDigital Library
  44. Silvain Rideau and Glynn Winskel. 2011. Concurrent Strategies. In LICS ’11, June 21-24, 2011, Toronto, Canada. 409–418.Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Peter Selinger. 2004. Towards a quantum programming language. Mathematical Structures in Computer Science 14, 4 (2004), 527–586. Google ScholarGoogle ScholarDigital LibraryDigital Library
  46. Peter Selinger. 2007. Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract). Electr. Notes Theor. Comput. Sci. 170 (2007), 139–163. Google ScholarGoogle ScholarDigital LibraryDigital Library
  47. Peter Selinger and Benoît Valiron. 2006. A lambda calculus for quantum computation with classical control. Mathematical Structures in Computer Science 16, 3 (2006), 527–552. Google ScholarGoogle ScholarDigital LibraryDigital Library
  48. Peter Selinger and Benoît Valiron. 2008. On a Fully Abstract Model for a Quantum Linear Functional Language: (Extended Abstract). Electr. Notes Theor. Comput. Sci. 210 (2008), 123–137. Google ScholarGoogle ScholarDigital LibraryDigital Library
  49. Peter W. Shor. 1997. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM J. Comput. 26, 5 (1997), 1484–1509. Google ScholarGoogle ScholarDigital LibraryDigital Library
  50. Takeshi Tsukada and C.-H. Luke Ong. 2015. Nondeterminism in Game Semantics via Sheaves. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015. 220–231.Google ScholarGoogle ScholarDigital LibraryDigital Library
  51. Glynn Winskel. 2007. Event Structures with Symmetry. Electr. Notes Theor. Comput. Sci. 172 (2007), 611–652. Google ScholarGoogle ScholarDigital LibraryDigital Library
  52. Glynn Winskel. 2015. On Probabilistic Distributed Strategies. In Theoretical Aspects of Computing - ICTAC 2015 - 12th International Colloquium Cali, Colombia, October 29-31, 2015, Proceedings. 69–88.Google ScholarGoogle Scholar
  53. Dongsheng Zhao and Taihe Fan. 2010. Dcpo-completion of posets. Theor. Comput. Sci. 411, 22-24 (2010), 2167–2173. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Full abstraction for the quantum lambda-calculus

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image Proceedings of the ACM on Programming Languages
        Proceedings of the ACM on Programming Languages  Volume 4, Issue POPL
        January 2020
        1984 pages
        EISSN:2475-1421
        DOI:10.1145/3377388
        Issue’s Table of Contents

        Copyright © 2019 Owner/Author

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 20 December 2019
        Published in pacmpl Volume 4, Issue POPL

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!