Techniques for Formal Modelling and Analysis of Quantum Systems

  • Simon J. Gay
  • Rajagopal Nagarajan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7860)


Quantum communication and cryptographic protocols are well on the way to becoming an important practical technology. Although a large amount of successful research has been done on proving their correctness, most of this work does not make use of familiar techniques from formal methods such as formal logics for specification, formal modelling languages, separation of levels of abstraction, and compositional analysis. We argue that these techniques will be necessary for the analysis of large-scale systems that combine quantum and classical components. We summarize the results of our investigation using different approaches: behavioural equivalence in process calculus, model-checking and equivalence checking. Quantum teleportation is used as an example to illustrate our techniques.


Quantum Teleportation Quantum Information Processing Equivalence Check Unknown Quantum State Quantum Protocol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aaronson, S., Gottesman, D.: Improved simulation of stabilizer circuits. Physical Review A 70, 052328 (2004)CrossRefGoogle Scholar
  2. 2.
    Abramsky, S., Gay, S.J., Nagarajan, R.: Interaction categories and the foundations of typed concurrent programming. In: Broy, M. (ed.) Deductive Program Design: Proceedings of the 1994 Marktoberdorf International Summer School. NATO ASI Series F: Computer and Systems Sciences. Springer (1995)Google Scholar
  3. 3.
    Abramsky, S.: Interaction Categories (Extended Abstract). In: Burn, G.L., Gay, S.J., Ryan, M.D. (eds.) Theory and Formal Methods 1993: Proceedings of the First Imperial College Department of Computing Workshop on Theory and Formal Methods. Workshops in Computer Science, pp. 57–70. Springer (1993)Google Scholar
  4. 4.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004), pp. 415–425. IEEE Computer Society (2004); Also arXiv:quant-ph/0402130Google Scholar
  5. 5.
    Ardeshir-Larijani, E., Gay, S.J., Nagarajan, R.: Equivalence checking of quantum protocols. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013 (ETAPS 2013). LNCS, vol. 7795, pp. 478–492. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Baltazar, P., Chadha, R., Mateus, P.: Quantum computation tree logic – model checking and complete calculus. International Journal of Quantum Information 6(2), 219–236 (2008)CrossRefMATHGoogle Scholar
  7. 7.
    Baltazar, P., Chadha, R., Mateus, P., Sernadas, A.: Towards model-checking quantum security protocols. In: First International Conference on Quantum, Nano, and Micro Technologies, ICQNM. IEEE Computer Society (2007)Google Scholar
  8. 8.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters 70, 1895–1899 (1993)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Canetti, R.: Universally composable security: A new paradigm for cryptographic protocols. In: 42nd IEEE Symposium on Foundations of Computer Science, FOCS, pp. 136–145. IEEE Computer Society (2001)Google Scholar
  10. 10.
    Davidson, T., Gay, S.J., Mlnařík, H., Nagarajan, R., Papanikolaou, N.: Model checking for Communicating Quantum Processes. International Journal of Unconventional Computing 8(1), 73–98 (2012)Google Scholar
  11. 11.
    Davidson, T.A.S.: Formal Verification Techniques using Quantum Process Calculus. PhD thesis, University of Warwick (2011)Google Scholar
  12. 12.
    Emerson, E.A.: Temporal and modal logic, vol. B: Formal Models and Semantics, pp. 995–1072. MIT Press (1990)Google Scholar
  13. 13.
    Feng, Y., Duan, R., Ying, M.: Bisimulation for quantum processes. In: 38th ACM Symposium on Principles of Programming Languages, POPL. ACM (2011)Google Scholar
  14. 14.
    Feng, Y., Yu, N., Ying, M.: Model checking quantum Markov chains. arXiv:1205.2187 [quant-ph] (2012)Google Scholar
  15. 15.
    Feynman, R.P.: Simulating physics with computers. International Journal of Theoretical Physics 21(6-7), 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gay, S.J., Mackie, I.C. (eds.): Semantic Techniques in Quantum Computation. Cambridge University Press (2010)Google Scholar
  17. 17.
    Gay, S.J., Nagarajan, R.: Communicating quantum processes. In: 32nd ACM Symposium on Principles of Programming Languages, POPL, pp. 145–157 (2005); Also arXiv:quant-ph/0409052Google Scholar
  18. 18.
    Gay, S.J.: Stabilizer states as a basis for density matrices. arXiv:1112.2156 (2011)Google Scholar
  19. 19.
    Gay, S.J., Nagarajan, R.: Types and typechecking for Communicating Quantum Processes. Mathematical Structures in Computer Science 16(3), 375–406 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gay, S.J., Nagarajan, R., Papanikolaou, N.: QMC: A model checker for quantum systems. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 543–547. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Gay, S.J., Papanikolaou, N., Nagarajan, R.: Specification and verification of quantum protocols. In: Semantic Techniques in Quantum Computation. Cambridge University Press (2010)Google Scholar
  22. 22.
    Gottesman, D.: Class of quantum error-correcting codes saturating the quantum Hamming bound. Physical Review A 54, 1862 (1996)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Grover, L.: A fast quantum mechanical algorithm for database search. In: 28th ACM Symposium on the Theory of Computation, STOC, pp. 212–219. ACM Press (1996)Google Scholar
  24. 24.
    Lalire, M.: Relations among quantum processes: bisimilarity and congruence. Mathematical Structures in Computer Science 16(3), 407–428 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mateus, P., Sernadas, A.: Weakly complete axiomatization of exogenous quantum propositional logic. Information and Computation 204(5), 771–794 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Mayers, D.: Unconditional Security in Quantum Cryptography. Journal of the ACM 48(3), 351–406 (2001)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Nagarajan, R., Gay, S.J.: Formal verification of quantum protocols. arXiv:quant-ph/0203086 (March 2002)Google Scholar
  28. 28.
    Papanikolaou, N.K.: Model Checking Quantum Protocols. PhD thesis, University of Warwick (2009)Google Scholar
  29. 29.
    Selinger, P.: Towards a quantum programming language. Mathematical Structures in Computer Science 14(4), 527–586 (2004)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: 35th IEEE Symposium on Foundations of Computer Science, FOCS (1994)Google Scholar
  31. 31.
    Trčka, N., Georgievska, S.: Branching bisimulation congruence for probabilistic systems. Electronic Notes in Theoretical Computer Science 220(3), 129–143 (2008)CrossRefMATHGoogle Scholar
  32. 32.
    Ying, M., Feng, Y., Duan, R., Ji, Z.: An algebra of quantum processes. ACM Transactions on Computational Logic 10(3), 19 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Simon J. Gay
    • 1
  • Rajagopal Nagarajan
    • 2
  1. 1.School of Computing ScienceUniversity of GlasgowUK
  2. 2.Department of Computer Science, School of Science and TechnologyMiddlesex UniversityLondonUK

Personalised recommendations