skip to main content
research-article
Open Access

Semantics for variational Quantum programming

Published:12 January 2022Publication History
Skip Abstract Section

Abstract

We consider a programming language that can manipulate both classical and quantum information. Our language is type-safe and designed for variational quantum programming, which is a hybrid classical-quantum computational paradigm. The classical subsystem of the language is the Probabilistic FixPoint Calculus (PFPC), which is a lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. The quantum subsystem is a first-order linear type system that can manipulate quantum information. The two subsystems are related by mixed classical/quantum terms that specify how classical probabilistic effects are induced by quantum measurements, and conversely, how classical (probabilistic) programs can influence the quantum dynamics. We also describe a sound and computationally adequate denotational semantics for the language. Classical probabilistic effects are interpreted using a recently-described commutative probabilistic monad on DCPO. Quantum effects and resources are interpreted in a category of von Neumann algebras that we show is enriched over (continuous) domains. This strong sense of enrichment allows us to develop novel semantic methods that we use to interpret the relationship between the quantum and classical probabilistic effects. By doing so we provide a very detailed denotational analysis that relates domain-theoretic models of classical probabilistic programming to models of quantum programming.

Skip Supplemental Material Section

Supplemental Material

Auxiliary Presentation Video

Short talk to advertise our paper before the POPL'22 conference.

References

  1. M. Abadi and M. P. Fiore. 1996. Syntactic Considerations on Recursive Types. In Proceedings 11th Annual IEEE Symposium on Logic in Computer Science. 242–252. issn:1043-6871 https://doi.org/10.1109/LICS.1996.561324 Google ScholarGoogle ScholarCross RefCross Ref
  2. S. Abramsky and A. Jung. 1994. Domain Theory. In Handbook of Logic in Computer Science (Vol. 3). Oxford University Press, Oxford, UK. 1–168. isbn:0-19-853762-X http://dl.acm.org/citation.cfm?id=218742.218744Google ScholarGoogle Scholar
  3. Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. 1993. Teleporting an Unknown Quantum State via Dual Classical and EPR Channels.Google ScholarGoogle Scholar
  4. P.N. Benton. 1995. A mixed linear and non-linear logic: Proofs, terms and models. In Computer Science Logic: 8th Workshop, CSL ’94, Selected Papaers. https://doi.org/10.1007/BFb0022251 Google ScholarGoogle ScholarCross RefCross Ref
  5. P. N. Benton and P. Wadler. 1996. Linear Logic, Monads and the Lambda Calculus. In LICS 1996.Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. B. Blackadar. 2006. Operator Algebras: Theory of C*-algebras and von Neumann algebras. Springer-Verlag.Google ScholarGoogle Scholar
  7. K. Cho. 2016. Semantics for a Quantum Programming Language by Operator Algebras. New Generation Comput., 34, 1-2 (2016), 25–68. https://doi.org/10.1007/s00354-016-0204-3 Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Kenta Cho and Abraham Westerbaan. 2016. Von Neumann Algebras form a Model for the Quantum Lambda Calculus. CoRR, abs/1603.02133 (2016), arxiv:1603.02133. arxiv:1603.02133Google ScholarGoogle Scholar
  9. Pierre Clairambault and Marc de Visme. 2020. Full abstraction for the quantum lambda-calculus. Proc. ACM Program. Lang., 4, POPL (2020), 63:1–63:28. https://doi.org/10.1145/3371131 Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Pierre Clairambault, Marc de Visme, and Glynn Winskel. 2019. Game semantics for quantum programming. Proc. ACM Program. Lang., 3, POPL (2019), 32:1–32:29. https://doi.org/10.1145/3290345 Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. 2014. A Quantum Approximate Optimization Algorithm. arxiv:1411.4028.Google ScholarGoogle Scholar
  12. Marcelo Fiore and Gordon Plotkin. 1994. An Axiomatization of Computationally Adequate Domain Theoretic Models of FPC. In LICS. IEEE Computer Society, 92–102.Google ScholarGoogle Scholar
  13. M. P. Fiore. 1994. Axiomatic domain theory in categories of partial maps. Ph.D. Dissertation. University of Edinburgh, UK.Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. R. Furber. 2019. Continuous Dcpos in Quantum Computing. preprint, http://people.cs.aau.dk/~furber/papers/contawconf.pdfGoogle ScholarGoogle Scholar
  15. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott. 2003. Continuous Lattices and Domains. Cambridge University Press.Google ScholarGoogle Scholar
  16. J.-Y. Girard. 1987. Linear Logic. Theoretical Computer Science, 50 (1987), 1 – 101.Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Robert Harper. 2016. Practical Foundations for Programming Languages (2nd. Ed.). Cambridge University Press. isbn:9781107150300 https://www.cs.cmu.edu/%7Erwh/pfpl/index.htmlGoogle ScholarGoogle Scholar
  18. Bart Jacobs. 2016. Introduction to Coalgebra: Towards Mathematics of States and Observation (Cambridge Tracts in Theoretical Computer Science, Vol. 59). Cambridge University Press. isbn:9781316823187 https://doi.org/10.1017/CBO9781316823187 Google ScholarGoogle ScholarCross RefCross Ref
  19. Xiaodong Jia, Andre Kornell, Bert Lindenhovius, Michael W. Mislove, and Vladimir Zamdzhiev. 2021. Semantics for Variational Quantum Programming. CoRR, abs/2107.13347 (2021), arXiv:2107.13347. arxiv:2107.13347 Extended version of this POPL paper.Google ScholarGoogle Scholar
  20. Xiaodong Jia, Bert Lindenhovius, Michael Mislove, and Vladimir Zamdzhiev. 2021. Commutative Monads for Probabilistic Programming Languages. In Logic in Computer Science (LICS 2021). arxiv:2102.00510.Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Xiaodong Jia, Michael W. Mislove, and Vladimir Zamdzhiev. 2021. The Central Valuations Monad. CoRR, abs/2111.10873 (2021), arXiv:2111.10873. arxiv:2111.10873Google ScholarGoogle Scholar
  22. Claire Jones. 1990. Probabilistic Non-determinism. Ph.D. Dissertation. University of Edinburgh, UK. http://hdl.handle.net/1842/413Google ScholarGoogle Scholar
  23. C. Jones and Gordon D. Plotkin. 1989. A Probabilistic Powerdomain of Evaluations. In Proceedings of the Fourth Annual Symposium on Logic in Computer Science (LICS ’89), Pacific Grove, California, USA, June 5-8, 1989. IEEE Computer Society, 186–195. https://doi.org/10.1109/LICS.1989.39173 Google ScholarGoogle ScholarCross RefCross Ref
  24. R.V. Kadison and J.R. Ringrose. 1997. Fundamentals of the Theory of Operator Algebra, Volume I: Elementary Theory. American Mathematical Society.Google ScholarGoogle Scholar
  25. Klaus Keimel and Gordon D. Plotkin. 2017. Mixed powerdomains for probability and nondeterminism. Logical Methods in Computer Science, 13, Issue 1 (2017), Jan., https://doi.org/10.23638/LMCS-13(1:2)2017 Google ScholarGoogle ScholarCross RefCross Ref
  26. John L. Kelley. 1975. General Topology (Graduate Texts in Mathematics). Springer-Verlag. issn:978-0-387-90125-1Google ScholarGoogle Scholar
  27. A. Kornell. 2020. Quantum Sets. J. Math. Phys., 61 (2020), 102202. https://doi.org/10.1063/1.5054128 Google ScholarGoogle ScholarCross RefCross Ref
  28. Klaas Landsman. 2017. Foundations of Quantum Theory - From Classical Concepts to Operator Algebras. Springer Open.Google ScholarGoogle Scholar
  29. Bert Lindenhovius, Michael Mislove, and Vladimir Zamdzhiev. 2019. Mixed Linear and Non-linear Recursive Types. Proc. ACM Program. Lang., 3, ICFP (2019), Article 111, Aug., 29 pages. issn:2475-1421 https://doi.org/10.1145/3341715 Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. Bert Lindenhovius, Michael Mislove, and Vladimir Zamdzhiev. 2021. LNL-FPC: The Linear/Non-linear Fixpoint Calculus. Logical Methods in Computer Science, Volume 17, Issue 2 (2021), April, https://lmcs.episciences.org/7390Google ScholarGoogle Scholar
  31. Saunders Mac Lane. 1998. Categories for the Working Mathematician (2nd ed.). Springer.Google ScholarGoogle Scholar
  32. Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. 2016. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18, 2 (2016), 023023.Google ScholarGoogle ScholarCross RefCross Ref
  33. Eugenio Moggi. 1991. Notions of Computation and Monads. Inf. Comput., 93, 1 (1991), 55–92. https://doi.org/10.1016/0890-5401(91)90052-4 Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. Adam Paetznick and Krysta M. Svore. 2014. Repeat-until-Success: Non-Deterministic Decomposition of Single-Qubit Unitaries. Quantum Info. Comput., 14, 15–16 (2014), Nov., 1277–1301. issn:1533-7146Google ScholarGoogle Scholar
  35. Michele Pagani, Peter Selinger, and Benoît Valiron. 2014. Applying quantitative semantics to higher-order quantum computing. In POPL ’14. ACM, 647–658. https://doi.org/10.1145/2535838.2535879 Google ScholarGoogle ScholarDigital LibraryDigital Library
  36. J. Paykin, R. Rand, and S. Zdancewic. 2017. QWIRE: a core language for quantum circuits. In POPL. ACM, 846–858.Google ScholarGoogle Scholar
  37. Romain Péchoux, Simon Perdrix, Mathys Rennela, and Vladimir Zamdzhiev. 2020. Quantum Programming with Inductive Datatypes. https://homepages.loria.fr/VZamdzhiev/papers/qpl-inductive.pdf Preprint.Google ScholarGoogle Scholar
  38. Romain Péchoux, Simon Perdrix, Mathys Rennela, and Vladimir Zamdzhiev. 2020. Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory. In Foundations of Software Science and Computation Structures - 23rd International Conference, FOSSACS 2020 (Lecture Notes in Computer Science, Vol. 12077). Springer, 562–581. https://doi.org/10.1007/978-3-030-45231-5_29 Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O’brien. 2014. A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5, 1 (2014), 1–7.Google ScholarGoogle Scholar
  40. John Power and Edmund Robinson. 1997. Premonoidal Categories and Notions of Computation. Math. Struct. Comput. Sci., 7, 5 (1997), 453–468. https://doi.org/10.1017/S0960129597002375 Google ScholarGoogle ScholarDigital LibraryDigital Library
  41. Mathys Rennela and Sam Staton. 2020. Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory. Log. Methods Comput. Sci., 16, 1 (2020), https://doi.org/10.23638/LMCS-16(1:30)2020 Google ScholarGoogle ScholarCross RefCross Ref
  42. P. Selinger. 2004. Towards a quantum programming language. Mathematical Structures in Computer Science, 14, 4 (2004), 527–586.Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. Peter Selinger. 2004. Towards a semantics for higher-order quantum computation. Proceedings of the 2nd International Workshop on Quantum Programming Languages, 127–143.Google ScholarGoogle Scholar
  44. M.B. Smyth and G.D. Plotkin. 1982. The Category-theoretic Solution of Recursive Domain Equations. Siam J. Comput..Google ScholarGoogle Scholar
  45. M. Takesaki. 2000. Theory of Operator Algebra I. Springer.Google ScholarGoogle Scholar
  46. Takeshi Tsukada, Kazuyuki Asada, and C.-H. Luke Ong. 2018. Species, Profunctors and Taylor Expansion Weighted by SMCC: A Unified Framework for Modelling Nondeterministic, Probabilistic and Quantum Programs. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, Anuj Dawar and Erich Grädel (Eds.). ACM, 889–898. https://doi.org/10.1145/3209108.3209157 Google ScholarGoogle ScholarDigital LibraryDigital Library
  47. Abraham Westerbaan. 2019. The Category of von Neumann algebras, PhD Thesis. arxiv:arxiv:1804.02203.Google ScholarGoogle Scholar
  48. William K Wootters and Wojciech H Zurek. 1982. A single quantum cannot be cloned. Nature, 299, 5886 (1982), 802–803.Google ScholarGoogle Scholar

Index Terms

  1. Semantics for variational Quantum programming

      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

      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!