Verification of Concurrent Quantum Protocols by Equivalence Checking

  • Ebrahim Ardeshir-Larijani
  • Simon J. Gay
  • Rajagopal Nagarajan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8413)


We present a tool which uses a concurrent language for describing quantum systems, and performs verification by checking equivalence between specification and implementation. In general, simulation of quantum systems using current computing technology is infeasible. We restrict ourselves to the stabilizer formalism, in which there are efficient simulation algorithms. In particular, we consider concurrent quantum protocols that behave functionally in the sense of computing a deterministic input-output relation for all interleavings of the concurrent system. Crucially, these input-output relations can be abstracted by superoperators, enabling us to take advantage of linearity. This allows us to analyse the behaviour of protocols with arbitrary input, by simulating their operation on a finite basis set consisting of stabilizer states. Despite the limitations of the stabilizer formalism and also the range of protocols that can be analysed using this approach, we have applied our equivalence checking tool to specify and verify interesting and practical quantum protocols from teleportation to secret sharing.


Quantum State Model Check Density Operator Secret Sharing Equivalence Check 
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.


  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.
    Ardeshir-Larijani, E.: Quantum equivalence checker (2013),
  4. 4.
    Ardeshir-Larijani, E., Gay, S.J., Nagarajan, R.: Equivalence checking of quantum protocols. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 478–492. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    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
  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.
    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)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996)CrossRefGoogle Scholar
  9. 9.
    Davidson, T.A.S., Gay, S.J., Nagarajan, R., Puthoor, I.V.: Analysis of a quantum error correcting code using quantum process calculus. EPTCS 95, 67–80 (2012)CrossRefGoogle Scholar
  10. 10.
    Davidson, T.A.S.: Formal Verification Techniques Using Quantum Process Calculus. PhD thesis, University of Warwick (2011)Google Scholar
  11. 11.
    de Riedmatten, H., Marcikic, I., Tittel, W., Zbinden, H., Collins, D., Gisin, N.: Long distance quantum teleportation in a quantum relay configuration. Physical Review Letters 92(4), 047904 (2004)Google 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)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Duncan, R., Lucas, M.: Verifying the Steane code with quantomatic. arXiv:1306.4532 (2013)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.: Stabilizer states as a basis for density matrices. arXiv:1112.2156 (2011)Google Scholar
  17. 17.
    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
  18. 18.
    Harel, D., Kupferman, O., Vardi, M.Y.: On the complexity of verifying concurrent transition systems. Information and Computation 173(2), 143–161 (2002)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829–1834 (1999)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Mayers, D.: Unconditional Security in Quantum Cryptography. Journal of the ACM 48(3), 351–406 (2001)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Milner, R.: Communication and concurrency. Prentice Hall (1989)Google Scholar
  22. 22.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000)Google Scholar
  23. 23.
    Selinger, P.: Towards a quantum programming language. Mathematical Structures in Computer Science 14(4), 527–586 (2004)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Shor, P.W.: Fault-tolerant quantum computation. In: Proceedings of the 37th Annual Symposium on Foundations of Computer Science, FOCS 1996. IEEE Computer Society, Washington, DC (1996)Google Scholar
  25. 25.
    Viamontes, G.F., Markov, I.L., Hayes, J.P.: Quantum Circuit Simulation. Springer (2009)Google Scholar
  26. 26.
    Wille, R., Grosse, D., Miller, D., Drechsler, R.: Equivalence checking of reversible circuits. In: 39th International Symposium on Multiple-Valued Logic, pp. 324–330 (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 2014

Authors and Affiliations

  • Ebrahim Ardeshir-Larijani
    • 1
    • 2
  • 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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