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.
Supplemental Material
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
- 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 Scholar
Cross Ref
- P. N. Benton and P. Wadler. 1996. Linear Logic, Monads and the Lambda Calculus. In LICS 1996.Google Scholar
Digital Library
- B. Blackadar. 2006. Operator Algebras: Theory of C*-algebras and von Neumann algebras. Springer-Verlag.Google Scholar
- 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 Scholar
Digital Library
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. 2014. A Quantum Approximate Optimization Algorithm. arxiv:1411.4028.Google Scholar
- Marcelo Fiore and Gordon Plotkin. 1994. An Axiomatization of Computationally Adequate Domain Theoretic Models of FPC. In LICS. IEEE Computer Society, 92–102.Google Scholar
- M. P. Fiore. 1994. Axiomatic domain theory in categories of partial maps. Ph.D. Dissertation. University of Edinburgh, UK.Google Scholar
Digital Library
- R. Furber. 2019. Continuous Dcpos in Quantum Computing. preprint, http://people.cs.aau.dk/~furber/papers/contawconf.pdfGoogle Scholar
- 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 Scholar
- J.-Y. Girard. 1987. Linear Logic. Theoretical Computer Science, 50 (1987), 1 – 101.Google Scholar
Digital Library
- Robert Harper. 2016. Practical Foundations for Programming Languages (2nd. Ed.). Cambridge University Press. isbn:9781107150300 https://www.cs.cmu.edu/%7Erwh/pfpl/index.htmlGoogle Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
Digital Library
- Xiaodong Jia, Michael W. Mislove, and Vladimir Zamdzhiev. 2021. The Central Valuations Monad. CoRR, abs/2111.10873 (2021), arXiv:2111.10873. arxiv:2111.10873Google Scholar
- Claire Jones. 1990. Probabilistic Non-determinism. Ph.D. Dissertation. University of Edinburgh, UK. http://hdl.handle.net/1842/413Google Scholar
- 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 Scholar
Cross Ref
- R.V. Kadison and J.R. Ringrose. 1997. Fundamentals of the Theory of Operator Algebra, Volume I: Elementary Theory. American Mathematical Society.Google Scholar
- 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 Scholar
Cross Ref
- John L. Kelley. 1975. General Topology (Graduate Texts in Mathematics). Springer-Verlag. issn:978-0-387-90125-1Google Scholar
- A. Kornell. 2020. Quantum Sets. J. Math. Phys., 61 (2020), 102202. https://doi.org/10.1063/1.5054128 Google Scholar
Cross Ref
- Klaas Landsman. 2017. Foundations of Quantum Theory - From Classical Concepts to Operator Algebras. Springer Open.Google Scholar
- 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 Scholar
Digital Library
- 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 Scholar
- Saunders Mac Lane. 1998. Categories for the Working Mathematician (2nd ed.). Springer.Google Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
- 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 Scholar
Digital Library
- J. Paykin, R. Rand, and S. Zdancewic. 2017. QWIRE: a core language for quantum circuits. In POPL. ACM, 846–858.Google Scholar
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- P. Selinger. 2004. Towards a quantum programming language. Mathematical Structures in Computer Science, 14, 4 (2004), 527–586.Google Scholar
Digital Library
- Peter Selinger. 2004. Towards a semantics for higher-order quantum computation. Proceedings of the 2nd International Workshop on Quantum Programming Languages, 127–143.Google Scholar
- M.B. Smyth and G.D. Plotkin. 1982. The Category-theoretic Solution of Recursive Domain Equations. Siam J. Comput..Google Scholar
- M. Takesaki. 2000. Theory of Operator Algebra I. Springer.Google Scholar
- 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 Scholar
Digital Library
- Abraham Westerbaan. 2019. The Category of von Neumann algebras, PhD Thesis. arxiv:arxiv:1804.02203.Google Scholar
- William K Wootters and Wojciech H Zurek. 1982. A single quantum cannot be cloned. Nature, 299, 5886 (1982), 802–803.Google Scholar
Index Terms
Semantics for variational Quantum programming
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