Partition-based discrete-time quantum walks

  • Norio Konno
  • Renato Portugal
  • Iwao Sato
  • Etsuo Segawa


We introduce a family of discrete-time quantum walks, called two-partition model, based on two equivalence-class partitions of the computational basis, which establish the notion of local dynamics. This family encompasses most versions of unitary discrete-time quantum walks driven by two local operators studied in literature, such as the coined model, Szegedy’s model, and the 2-tessellable staggered model. We also analyze the connection of those models with the two-step coined model, which is driven by the square of the evolution operator of the standard discrete-time coined walk. We prove formally that the two-step coined model, an extension of Szegedy model for multigraphs, and the two-tessellable staggered model are unitarily equivalent. Then, selecting one specific model among those families is a matter of taste not generality.


Quantum walk Coined walk Szegedy’s walk Staggered walk Graph tessellation Hypergraph walk Unitary equivalence Intersection graph Bipartite graph 


  1. 1.
    Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507–518 (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithm, pp. 1099–1108 (2005)Google Scholar
  3. 3.
    Ambainis, A., Portugal, R., Nahimov, N.: Spatial search on grids with minimum memory. Quantum Inf. Comput. 15, 1233–1247 (2015)MathSciNetGoogle Scholar
  4. 4.
    Asboth, J.K., Oroszlany, L., Palyi, A.: A Short Course on Topological Insulators: Band-Structure Topology and Edge States in One and Two Dimensions, Lecture Notes in Physics, vol. 919. Springer, Berlin (2016)zbMATHGoogle Scholar
  5. 5.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of the 33rd ACM Symposium on Theory of Computing, pp. 50–59 (2000)Google Scholar
  6. 6.
    Cantero, M.J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362, 29–56 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: The CGMV method for quantum walks. Quantum Inf. Process. 11, 1149–1192 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carteret, H.A., Ismail, M.E.H., Richmond, B.: Three routes to the exact asymptotics for the one-dimensional quantum walk. J. Phys. A Math. Gen. 36, 8775 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Feldman, E., Hillery, M.: Scattering theory and discrete-time quantum walks. Phys. Lett. A 324, 277 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feynman, R.F., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill Inc, New York (1965)zbMATHGoogle Scholar
  11. 11.
    Gudder, S.: Quantum Probability. Academic Press Inc., New York (1988)zbMATHGoogle Scholar
  12. 12.
    Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: Quantum graph walks I: mapping to quantum walks. Yokohama Math. J. 59, 33–55 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Higuchi, Y., Konno, N., Sato, I., Segawa, E.: A remark on zeta functions of finite graphs via quantum walks. Pac. J. Math Ind. 6, 73–84 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Higuchi, Y., Segawa, E., Suzuki, A.: arXiv:1506.06457
  15. 15.
    Kitagawa, T., Rudner, M.S., Berg, E., Demler, E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    Konno, N.: Quantum Walks, Lecture Notes in Mathematics. Springer, Berlin, Heidelberg (2008)Google Scholar
  17. 17.
    Loke, T., Wang, J.B.: Efficient quantum circuits for Szegedy quantum walks. Ann. Phys. 382, 64–84 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Matsue, K., Ogurisu, O., Segawa, E.: A note on the spectral mapping theorem of quantum walk models. Interdiscip. Inf. Sci. 23, 105–114 (2017)MathSciNetGoogle Scholar
  19. 19.
    Matsuoka, L., Yokoyama, K.: Physical implementation of quantum cellular automaton in a diatomic molecule. J. Comput. Theor. Nanosci. 10, 1617–1620 (2013)CrossRefGoogle Scholar
  20. 20.
    Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  21. 21.
    Ohno, H.: Unitary equivalent classes of one-dimensional quantum walks. Quantum Inf. Process. 15, 3599–3617 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Parthasarathy, K.R.: The passage from random walk to diffusion in quantum probability. J. Appl. Probab. 25, 151–166 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Peterson, D.: Gridline graphs: a review in two dimensions and an extension to higher dimensions. Discrete Appl. Math. 126, 223 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Portugal, R.: Establishing the equivalence between Szegedy’s and coined quantum walks using the staggered model. Quantum Inf. Process. 15, 1387–1409 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Portugal, R.: Staggered quantum walks on graphs. Phys. Rev. A 93, 062335 (2016)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Portugal, R., Oliveira, M.C., Moqadam, J.K.: Staggered quantum walks with Hamiltonians. Phys. Rev. A 95, 012328 (2017)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Portugal, R., Santos, R.A.M., Fernandes, T.D., Gonçalves, D.N.: The staggered quantum walk model. Quantum Inf. Process. 15, 85–101 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Portugal, R., Segawa, E.: Connecting coined quantum walks with Szegedy’s model. Interdiscip. Inf. Sci. 23, 119–125 (2017)MathSciNetGoogle Scholar
  29. 29.
    Segawa, E.: Localization of quantum qalks induced by recurrence properties of random walks. J. Comput. Theor. Nanosci. 10, 1583–1590 (2013)CrossRefGoogle Scholar
  30. 30.
    Segawa, E., Suzuki, A.: Generator of an abstract quantum walk. Quantum Stud. Math. Found. 3, 11–30 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Shenvi, N., Kempe, J., Whaley, K.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)ADSCrossRefGoogle Scholar
  32. 32.
    Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, pp. 32–41 (2004)Google Scholar

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Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of EngineeringYokohama National UniversityYokohamaJapan
  2. 2.National Laboratory of Scientific Computing - LNCCPetrópolisBrazil
  3. 3.Oyama National College of TechnologyOyamaJapan
  4. 4.Graduate School of Information SciencesTohoku UniversitySendaiJapan

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