Hitting Time

  • Renato Portugal
Part of the Quantum Science and Technology book series (QST)


In this chapter we define a new model of discrete-time quantum walks on graphs without using an explicit coin operator. Instead, the model uses an auxiliary space by duplicating the original graph and forcing the walker to jump systematically from the original graph to the copy and vice versa. The graph and its copy form a bipartite graph. At the end, the copy is discarded. This model has many advantages over the standard quantum walk model and it allows to define the quantum hitting time in a natural and elegant way.


Quantum Walk Original Undirected Graph Marked Vertices Marked Elements Arrival Vertex 
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. 6.
    Aldous, D.J., Fill, J.A.: Reversible Markov Chains and Random Walks on Graphs. Book in preparation, (200X)
  2. 9.
    Ambainis, A.: Quantum walk algorithm for element distinctness. In: FOCS ’04: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 22–31. IEEE Computer Society, Washington, DC (2004)Google Scholar
  3. 40.
    Itakura, Y.K.: Quantum algorithm for commutativity testing of a matrix set. Master’s thesis, University of Waterloo, Waterloo (2005)Google Scholar
  4. 46.
    Krovi, H., Magniez, F., Ozols, M., Roland, J.: Finding is as easy as detecting for quantum walks. In: Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 6198, pp. 540–551. Springer, Berlin (2010)Google Scholar
  5. 47.
    Lovász, L.: Random walks on graphs: a survey. Combinatorics, Paul Erdös is Eighty, vol. 2), pp. 1–46 (1993)Google Scholar
  6. 50.
    Magniez, F., Nayak, A.: Quantum complexity of testing group commutativity. Algorithmica 48(3), 221–232 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 51.
    Magniez, F., Nayak, A., Richter, P., Santha, M.: On the hitting times of quantum versus random walks. In: Proceedings of 20th ACM-SIAM Symposium on Discrete Algorithms (2009)Google Scholar
  8. 52.
    Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: Proceedings of 39th ACM Symposium on Theory of Computing, pp. 575–584 (2007)Google Scholar
  9. 53.
    Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. SIAM J. Comput. 37(2), 413–424 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 58.
    Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2001)Google Scholar
  11. 62.
    Motwani, R., Raghavan, P.: Randomized algorithms. ACM Comput. Surv. 28(1), 33–37 (1996)CrossRefGoogle Scholar
  12. 70.
    Santos, R.A.M., Portugal, R.: Quantum hitting time on the complete graph. Int. J. Quant. Inform. 8(5), 881–894 (2010), arXiv:0912.1217Google Scholar
  13. 74.
    Szegedy, M.: Quantum speed-up of markov chain based algorithms. Annual IEEE symposium on foundations of computer science, pp. 32–41 (2004)Google Scholar
  14. 75.
    Szegedy, M.: Spectra of Quantized Walks and a δε Rule. (2004), quant-ph/0401053Google Scholar
  15. 78.
    Tulsi, A.: Faster quantum-walk algorithm for the two-dimensional spatial search. Phys. Rev. A 78(1), 012310 (2008)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Renato Portugal
    • 1
  1. 1.Department of Computer ScienceNational Laboratory of Scientific Computing (LNCC)PetrópolisBrazil

Personalised recommendations