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.
Supplemental Material
- Samson Abramsky, Radha Jagadeesan, and Pasquale Malacaria. 2000. Full Abstraction for PCF. Inf. Comput. 163, 2 (2000), 409–470. Google Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- Gavin Bierman. 1993. On intuitionistic linear logic. Ph.D. Dissertation. University of Cambridge Computer Laboratory.Google Scholar
- 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 Scholar
Digital Library
- Ana C. Calderon and Guy McCusker. 2010. Understanding Game Semantics Through Coherence Spaces. Electr. Notes Theor. Comput. Sci. 265 (2010), 231–244. Google Scholar
Digital Library
- 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 Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- Simon Castellan, Pierre Clairambault, Silvain Rideau, and Glynn Winskel. 2017. Games and Strategies as Event Structures. LMCS 13, 3 (2017).Google Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
- 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 Scholar
Digital Library
- Pierre Clairambault, Marc de Visme, and Glynn Winskel. 2019. Game semantics for quantum programming. PACMPL 3, POPL (2019), 32:1–32:29. Google Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- Yannick Delbecque. 2011. Game Semantics for Quantum Data. Electr. Notes Theor. Comput. Sci. 270, 1 (2011), 41–57. Google Scholar
Digital Library
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, and Hugo Zbinden. 2002. Quantum cryptography. Reviews of modern physics 74, 1 (2002), 145.Google Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- Martin Hyland and Andrea Schalk. 1999. Abstract Games for Linear Logic. Electr. Notes Theor. Comput. Sci. 29 (1999), 127–150. Google Scholar
Cross Ref
- Martin Hyland and Andrea Schalk. 2003. Glueing and orthogonality for models of linear logic. Theor. Comput. Sci. 294, 1/2 (2003), 183–231. Google Scholar
Digital Library
- 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 Scholar
Cross Ref
- Jim Laird, Giulio Manzonetto, and Guy McCusker. 2013. Constructing differential categories and deconstructing categories of games. Inf. Comput. 222 (2013), 247–264. Google Scholar
Digital Library
- Octavio Malherbe. 2013. Categorical models of computation: partially traced categories and presheaf models of quantum computation. Ph.D. Dissertation. University of Ottawa.Google Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- Paul-André Melliès. 2006. Asynchronous games 2: The true concurrency of innocence. Theor. Comput. Sci. 358, 2-3 (2006), 200–228. Google Scholar
Digital Library
- Paul-André Melliès and Nicolas Tabareau. 2010. Resource modalities in tensor logic. Ann. Pure Appl. Logic 161, 5 (2010), 632–653. Google Scholar
Cross Ref
- Paul-André Melliès. 2009. Categorical semantics of linear logic. Panoramas et syntheses 27 (2009), 15–215.Google Scholar
- Robin Milner. 1977. Fully Abstract Models of Typed lambda-Calculi. Theor. Comput. Sci. 4, 1 (1977), 1–22.Google Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- John Power and Edmund Robinson. 1997. Premonoidal Categories and Notions of Computation. Mathematical Structures in Computer Science 7, 5 (1997), 453–468. Google Scholar
Digital Library
- John Power and Hayo Thielecke. 1999. Closed Freyd- and kappa-categories. In ICALP’99 (LNCS), Vol. 1644. Springer.Google Scholar
Digital Library
- Silvain Rideau and Glynn Winskel. 2011. Concurrent Strategies. In LICS ’11, June 21-24, 2011, Toronto, Canada. 409–418.Google Scholar
Digital Library
- Peter Selinger. 2004. Towards a quantum programming language. Mathematical Structures in Computer Science 14, 4 (2004), 527–586. Google Scholar
Digital Library
- Peter Selinger. 2007. Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract). Electr. Notes Theor. Comput. Sci. 170 (2007), 139–163. Google Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- Glynn Winskel. 2007. Event Structures with Symmetry. Electr. Notes Theor. Comput. Sci. 172 (2007), 611–652. Google Scholar
Digital Library
- 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 Scholar
- Dongsheng Zhao and Taihe Fan. 2010. Dcpo-completion of posets. Theor. Comput. Sci. 411, 22-24 (2010), 2167–2173. Google Scholar
Digital Library
Index Terms
Full abstraction for the quantum lambda-calculus
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