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Quantum information effects

Published:12 January 2022Publication History
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Abstract

We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguage with choice, starting with any rig groupoid interpreting the reversible base language. Several properties of quantum measurement follow in general, and we translate (noniterative) quantum flow charts into our language. The semantic constructions turn the category of unitaries between Hilbert spaces into the category of completely positive trace-preserving maps, and they turn the category of bijections between finite sets into the category of functions with chosen garbage. Thus they capture the fundamental theorems of classical and quantum reversible computing of Toffoli and Stinespring.

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We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguage with choice, starting with any rig groupoid interpreting the reversible base language. Several properties of quantum measurement follow in general, and we translate (noniterative) quantum flow charts into our language. The semantic constructions turn the category of unitaries between Hilbert spaces into the category of completely positive trace-preserving maps, and they turn the category of bijections between finite sets into the category of functions with chosen garbage. Thus they capture the fundamental theorems of classical and quantum reversible computing of Toffoli and Stinespring.

References

  1. T. Altenkirch and A. S. Green. 2010. The Quantum IO Monad. Semantic Techniques in Quantum Computation, 173–205. https://doi.org/10.1017/CBO9781139193313.006 Google ScholarGoogle ScholarCross RefCross Ref
  2. C. H. Bennett. 1973. Logical Reversibility of Computation. IBM Journal of Research and Development, 17, 6 (1973), 525–532. https://doi.org/10.1147/rd.176.0525 Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. B. Bichsel, M. Baader, T. Gehr, and M. Vechev. 2020. Silq: A High-Level Quantum Language with Safe Uncomputation and Intuitive Semantics. In Proceedings of the 41st ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 2020) (PLDI 2020). ACM, 286–300. https://doi.org/10.1145/3385412.3386007 Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. W. J. Bowman, R. P. James, and A. Sabry. 2011. Dagger Traced Symmetric Monoidal Categories and Reversible Programming. Work-in-progress report presented at the 3rd International Workshop on Reversible Computation.Google ScholarGoogle Scholar
  5. J. Carette and A. Sabry. 2016. Computing with Semirings and Weak Rig Groupoids. In Proceedings of the 25th European Symposium on Programming (ESOP 2016). Springer, 123–148. https://doi.org/10.1007/978-3-662-49498-1_6 Google ScholarGoogle ScholarCross RefCross Ref
  6. C.-H. Chen, V. Choudhury, J. Carette, and A. Sabry. 2020. Fractional Types: Expressive and Safe Space Management for Ancilla Bits. In Proceedings of the 12th International Conference on Reversible Computation (RC 2020). Springer, 169–186. https://doi.org/10.1007/978-3-030-52482-1_10 Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. C.-H. Chen and A. Sabry. 2021. A Computational Interpretation of Compact Closed Categories: Reversible Programming with Negative and Fractional Types. Proc. ACM Program. Lang., 5, POPL (2021), Article 9, jan, 29 pages. https://doi.org/10.1145/3434290 Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. K. Cho and A. Westerbaan. 2016. Von Neumann Algebras Form a Model for the Quantum Lambda Calculus. arxiv:1603.02133.Google ScholarGoogle Scholar
  9. G. S. H. Cruttwell, B. Gavranović, N. Ghani, P. Wilson, and F. Zanasi. 2021. Categorical Foundations of Gradient-Based Learning. arxiv:2103.01931.Google ScholarGoogle Scholar
  10. B. Fong, D. Spivak, and R. Tuyéras. 2019. Backprop as Functor: A Compositional Perspective on Supervised Learning. In 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2019). 1–13. https://doi.org/10.1109/LICS.2019.8785665 Google ScholarGoogle ScholarCross RefCross Ref
  11. A. S. Green and T. Altenkirch. 2008. From Reversible to Irreversible Computations. In Proceedings of the 4th International Workshop on Quantum Programming Languages (QPL 2006) (Electronic Notes in Theoretical Computer Science, Vol. 210). Elsevier, 65–74. https://doi.org/10.1016/j.entcs.2008.04.018 Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. A. S. Green, P. L. Lumsdaine, N. J. Ross, P. Selinger, and B. Valiron. 2013. Quipper: A Scalable Quantum Programming Language. In Proceedings of the 34th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 2013). ACM, 333–342. https://doi.org/10.1145/2491956.2462177 Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. J. Hatcliff. 1998. An Introduction to Online and Offline Partial Evaluation Using a Simple Flowchart Language. In Partial Evaluation: Practice and Theory (Lecture Notes in Computer Science, Vol. 1706). 20–82. https://doi.org/10.1007/3-540-47018-2_2 Google ScholarGoogle ScholarCross RefCross Ref
  14. C. Hermida and R. D. Tennent. 2012. Monoidal Indeterminates and Categories of Possible Worlds. Theoretical Computer Science, 430 (2012), 3–22. https://doi.org/10.1016/j.tcs.2012.01.001 Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. C. Heunen and R. Kaarsgaard. 2021. Bennett and Stinespring, Together at Last. In Proceedings 18th International Conference on Quantum Physics and Logic (QPL 2021) (Electronic Proceedings in Theoretical Computer Science, Vol. 343). OPA, 102–118. https://doi.org/10.4204/EPTCS.343.5 Google ScholarGoogle ScholarCross RefCross Ref
  16. C. Heunen and J. Vicary. 2019. Categories for Quantum Theory. Oxford University Press. https://doi.org/10.1093/oso/9780198739623.001.0001 Google ScholarGoogle ScholarCross RefCross Ref
  17. N. G. Houghton-Larsen. 2021. A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing. Ph.D. Dissertation. Department of Mathematical Sciences, University of Copenhagen.Google ScholarGoogle Scholar
  18. J. Hughes. 2005. Programming with Arrows. In Advanced Functional Programming (Lecture Notes in Computer Science, Vol. 3622). Springer, 73–129. https://doi.org/10.1007/11546382_2 Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. M. Huot and S. Staton. 2018. Universal Properties in Quantum Theory. In Proceedings of the 15th International Conference on Quantum Physics and Logic (QPL 2018) (Electronic Proceedings in Theoretical Computer Science, Vol. 287). OPA, 213–224. https://doi.org/10.4204/EPTCS.287.12 Google ScholarGoogle ScholarCross RefCross Ref
  20. M. Huot and S. Staton. 2019. Quantum Channels as a Categorical Completion. In Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2019). IEEE, 1–13. https://doi.org/10.1109/LICS.2019.8785700 Google ScholarGoogle ScholarCross RefCross Ref
  21. B. Jacobs, C. Heunen, and I. Hasuo. 2009. Categorical Semantics for Arrows. Journal of Functional Programming, 19, 3–4 (2009), 403–438. https://doi.org/10.1017/S0956796809007308 Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. R. P. James and A. Sabry. 2012. Information Effects. In POPL ’12: Proceedings of the 39th Annual ACM SIGPLAN-SIGACT Symposium on Principles of programming languages. ACM, 73–84. https://doi.org/10.1145/2103656.2103667 Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. R. P. James and A. Sabry. 2014. Theseus: A High Level Language for Reversible Computing.Google ScholarGoogle Scholar
  24. N. D. Jones, C. K. Gomard, and P. Sestoft. 1993. Partial Evaluation and Automatic Program Generation. Prentice Hall International.Google ScholarGoogle Scholar
  25. R. Kaarsgaard and N. Veltri. 2019. En Garde! Unguarded Iteration for Reversible Computation in the Delay Monad. In Proceedings of the 13th International Conference on Mathematics of Program Construction (MPC 2019). Springer, 366–384. https://doi.org/10.1007/978-3-030-33636-3_13 Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. G. Kelly. 1974. Coherence Theorems for Lax Algebras and Distributive Laws. Lecture Notes in Mathematics, 420 (1974), 281–375. https://doi.org/10.1007/BFb0063106 Google ScholarGoogle ScholarCross RefCross Ref
  27. R. Landauer. 1961. Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development, 5, 3 (1961), 261–269. https://doi.org/10.1147/rd.53.0183 Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. M. Laplaza. 1972. Coherence for Distributivity. Lecture Notes in Mathematics, 281 (1972), 29–72. https://doi.org/10.1007/BFb0059555 Google ScholarGoogle ScholarCross RefCross Ref
  29. T. Leinster. 2016. Basic Category Theory. Cambridge University Press.Google ScholarGoogle Scholar
  30. M. A. Nielsen and I. Chuang. 2002. Quantum Computation and Quantum Information. American Association of Physics Teachers.Google ScholarGoogle Scholar
  31. J. Paykin, R. Rand, and S. Zdancewic. 2017. QWIRE: A Core Language for Quantum Circuits. POPL 2017: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages, 846–858. https://doi.org/10.1145/3009837.3009894 Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. R. Péchoux, S. Perdrix, M. Rennela, and V. Zamdzhiev. 2020. Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory. In Foundations of Software Science and Computation Structures (FOSSACS 2020) (Lecture Notes in Computer Science, Vol. 12077). 562–581. https://doi.org/10.1007/978-3-030-45231-5_29 Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. J. Power and E. Robinson. 1997. Premonoidal Categories and Notions of Computation. Mathematical Structures in Computer Science, 7, 5 (1997), https://doi.org/10.1017/S0960129597002375 Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. M. Rennela and S. Staton. 2020. Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory. Logical Methods in Computer Science, 16 (2020), 6192. https://doi.org/10.23638/LMCS-16(1:30)2020 Google ScholarGoogle ScholarCross RefCross Ref
  35. A. Sabry, B. Valiron, and J. K. Vizzotto. 2018. From Symmetric Pattern-Matching to Quantum Control. In International Conference on Foundations of Software Science and Computation Structures (FOSSACS 2018). 348–364. https://doi.org/10.1007/978-3-319-89366-2_19 Google ScholarGoogle ScholarCross RefCross Ref
  36. P. Selinger. 2004. Towards a Quantum Programming Language. Mathematical Structures in Computer Science, 14, 4 (2004), 527–586. https://doi.org/10.1017/S0960129504004256 Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. T. Toffoli. 1980. Reversible Computing. In Proceedings of the 7th Colloquium on Automata, Languages, and Programming (ICALP 1980). Springer, 632–644.Google ScholarGoogle ScholarCross RefCross Ref
  38. J. Vizzotto, T. Altenkirch, and A. Sabry. 2006. Structuring Quantum Effects: Superoperators as Arrows. Mathematical Structures in Computer Science, 16, 3 (2006), 453–468. https://doi.org/10.1017/S0960129506005287 Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. J. K. Vizzotto, A. R. Du Bois, and A. Sabry. 2009. The Arrow Calculus as a Quantum Programming Language. In International Workshop on Logic, Language, Information, and Computation (WoLLIC 2009). 379–393. https://doi.org/10.1007/978-3-642-02261-6_30 Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. A. Westerbaan. 2017. Quantum Programs as Kleisli Maps. In Proceedings 13th International Conference on Quantum Physics and Logic (QPL 2016) (Electronic Proceedings in Theoretical Computer Science, Vol. 236). 215–228. https://doi.org/10.4204/EPTCS.236.14 Google ScholarGoogle ScholarCross RefCross Ref
  41. N. Yanofsky and M. A. Mannucci. 2008. Quantum Computing for Computer Scientists. Cambridge University Press.Google ScholarGoogle Scholar

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