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Equivalence Checking of Quantum Protocols

  • Ebrahim Ardeshir-Larijani
  • Simon J. Gay
  • Rajagopal Nagarajan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7795)

Abstract

Quantum Information Processing (QIP) is an emerging area at the intersection of physics and computer science. It aims to establish the principles of communication and computation for systems based on the theory of quantum mechanics. Interesting QIP protocols such as quantum key distribution, teleportation, and blind quantum computation have already been realised in the laboratory and are now in the realm of mainstream industrial applications. The complexity of these protocols, along with possible inaccuracies in implementation, demands systematic and formal analysis. In this paper, we present a new technique and a tool, with a high-level interface, for verification of quantum protocols using equivalence checking. Previous work by Gay, Nagarajan and Papanikolaou used model-checking to verify quantum protocols represented in the stabilizer formalism, a restricted model which can be simulated efficiently on classical computers. Here, we are able to go beyond stabilizer states and verify protocols efficiently on all input states.

Keywords

quantum protocols equivalence checking model checking stabilizers 

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References

  1. 1.
    Aaronson, S., Gottesman, D.: Improved simulation of stabilizer circuits. Phys. Rev. A 70, 052328 (2004)Google Scholar
  2. 2.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425 (2004)Google Scholar
  3. 3.
    Audenaert, K.M.R., Plenio, M.B.: Entanglement on mixed stabilizer states: normal forms and reduction procedures. New Journal of Physics 7(1), 170 (2005)CrossRefGoogle Scholar
  4. 4.
    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)zbMATHCrossRefGoogle Scholar
  5. 5.
    Belardinelli, F., Gonzalez, P., Lomuscio, A.: Automated verification of quantum protocols using MCMAS. In: Proc. QAPL. EPTCS, vol. 85, pp. 48–62 (2012)Google Scholar
  6. 6.
    Bennett, C.H., Brassard, G.: Quantum cryptography: Public key distribution and coin tossing. In: Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, pp. 175–179 (1984)Google Scholar
  7. 7.
    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. Phys. Rev. Lett. 70, 1895–1899 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bryant, R.: Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers C-35(8), 677–691 (1986)zbMATHCrossRefGoogle Scholar
  10. 10.
    Davidson, T.A.S.: Formal Verification Techniques Using Quantum Process Calculus. PhD thesis, University of Warwick (2011)Google Scholar
  11. 11.
    D’Hondt, E., Panangaden, P.: Reasoning About Quantum Knowledge. In: Sarukkai, S., Sen, S. (eds.) FSTTCS 2005. LNCS, vol. 3821, pp. 553–564. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Dixon, L., Duncan, R.: Graphical reasoning in compact closed categories for quantum computation. Annals of Mathematics and Artificial Intelligence 56(1), 23–42 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Feng, Y., Yu, N., Ying, M.: Model checking quantum Markov chains. arXiv:1205.2187 (2012)Google Scholar
  14. 14.
    Feng, Y., Duan, R., Ying, M.: Bisimulation for quantum processes. In: Proceedings of the 38th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 523–534. ACM (2011)Google Scholar
  15. 15.
    Gagnon, E.: SableCC, an object-oriented compiler framework. Master’s thesis, School of Computer Science, McGill University (1998)Google Scholar
  16. 16.
    Gay, S.J.: Quantum programming languages: survey and bibliography. Mathematical Structures in Computer Science 16(4), 581–600 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gay, S.J.: Stabilizer states as a basis for density matrices. arXiv:1112.2156 (2011)Google Scholar
  18. 18.
    Gay, S.J., Nagarajan, R.: Communicating Quantum Processes. In: Proceedings of the 32nd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 145–157. ACM (2005)Google Scholar
  19. 19.
    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
  20. 20.
    Knill, E.: Conventions for quantum pseudocode. Technical Report LAUR-96-2724, Los Alamos National Laboratory (1996)Google Scholar
  21. 21.
    Lomuscio, A., Qu, H., Raimondi, F.: MCMAS: A Model Checker for the Verification of Multi-Agent Systems. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 682–688. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. 22.
    Mateus, P., Sernadas, A.: Weakly complete axiomatization of exogenous quantum propositional logic. Information and Computation 204(5), 771–794 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000)Google Scholar
  24. 24.
    Papanikolaou, N.: Model Checking Quantum Protocols. PhD thesis, University of Warwick (2009)Google Scholar
  25. 25.
    Selinger, P.: Towards a quantum programming language. Mathematical Structures in Computer Science 14(4), 527–586 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Viamontes, G.F., Markov, I.L., Hayes, J.P.: Quantum Circuit Simulation. Springer (2009)Google Scholar
  27. 27.
    Ying, M., Feng, Y., Duan, R., Ji, Z.: An algebra of quantum processes. ACM Trans. Comput. Logic 10(3), 19:1–19:36 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ebrahim Ardeshir-Larijani
    • 1
  • Simon J. Gay
    • 2
  • Rajagopal Nagarajan
    • 3
  1. 1.Department of Computer ScienceUniversity of WarwickUK
  2. 2.School of Computing ScienceUniversity of GlasgowUK
  3. 3.Department of Computer Science, School of Science and TechnologyMiddlesex UniversityUK

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