The Witten index for 1D supersymmetric quantum walks with anisotropic coins

  • Akito SuzukiEmail author
  • Yohei Tanaka


Chiral symmetric discrete-time quantum walks possess supersymmetry, and their associated Witten indices can be naturally defined. The Witten index is known to give a lower bound for the number of topologically protected bound states. The purpose of this paper is to give a complete classification of the Witten index for a one-dimensional split-step quantum walk. It turns out that the Witten index of this model exhibits a striking similarity to that of a Dirac particle model in supersymmetric quantum mechanics.


Quantum walks Supersymmetry Witten index Split-step quantum walks 



The authors are deeply indebted to the members of the Shinshu Mathematical Physics Group for extremely valuable discussions and comments. A. S. was supported by JSPS KAKENHI Grant Number JP18K03327. This work was also supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.


  1. 1.
    Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of 33th ACM Symposium of the Theory of Computing, pp. 37–49 (2001)Google Scholar
  2. 2.
    Arai, A.: Analysis on Fock Spaces and Mathematical Theory of Quantum Fields: An Introduction to Mathematical Analysis of Quantum Fields. World Scientific Publishing Company, Singapore (2017)zbMATHGoogle Scholar
  3. 3.
    Asbóth, J.K., Obuse, H.: Bulk-boundary correspondence for chiral symmetric quantum walks. Phys. Rev. B 88, 121406 (2013)ADSCrossRefGoogle Scholar
  4. 4.
    Barkhofen, S., Lorz, L., Nitsche, T., Silberhorn, C., Schomerus, H.: Supersymmetric polarization anomaly in photonic discrete-time quantum walks. Phys. Rev. Lett. 121, 260501 (2018)ADSCrossRefGoogle Scholar
  5. 5.
    Bolle, D., Gesztesy, F., Grosse, H., Schweiger, W., Simon, B.: Witten index, axial anomaly, and Krein’s spectral shift function in supersymmetric quantum mechanics. J. Math. Phys. 28, 1512–1525 (1987)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bracken, A.J., Ellinas, D., Smyrnakis, I.: Free-dirac-particle evolution as a quantum random walk. Phys. Rev. A 75, 022322 (2007)ADSCrossRefGoogle Scholar
  7. 7.
    Cedzich, C., Geib, T., Stahl, C., VelÃązquez, L., Werner, A.H., Werner, R.F.: Complete homotopy invariants for translation invariant symmetric quantum walks on a chain. arXiv:1804.04520
  8. 8.
    Cedzich, C., Geib, T., Grünbaum, F.A., Stahl, C., Velázquez, L., Werner, A.H., Werner, R.F.: The topological classification of one-dimensional symmetric quantum walks. Ann. Henri Poincaré 19, 325–383 (2018)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Elaydi, S.: An Introduction to Difference Equations. Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2005)Google Scholar
  10. 10.
    Endo, S., Endo, T., Konno, N., Segawa, E., Takei, M.: Limit theorems of a two-phase quantum walk with one defect. Quantum Inf. Comput. 15, 1373–1396 (2015)MathSciNetGoogle Scholar
  11. 11.
    Fuda, T., Funakawa, D., Suzuki, A.: Weak limit theorem for a one-dimensional split-step quantum walk. Rev. Math. Pures Appl. 64(2–3), 157–165 (2019)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Fuda, T., Funakawa, D., Suzuki, A.: Localization of a multi-dimensional quantum walk with one defect. Quantum Inf. Process. 16, 203 (2017)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Fuda, T., Funakawa, D., Suzuki, A.: Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations. J. Math. Phys. 59, 082201 (2018). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gesztesy, F., Simon, B.: Topological invariance of the witten index. J. Funct. Anal. 79, 91–102 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grimmett, G., Janson, S., Scudo, P.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)ADSCrossRefGoogle Scholar
  16. 16.
    Gross, D., Nesme, V., Vogts, H., Werner, R.F.: Index theory of one dimensional quantum walks and cellular automata. Commun. Math. Phys. 310, 419–454 (2012)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Grover, L.: A fast quantum mechanical algorithm for database search. In: Proceeding of the 28th ACM Symposium on Theory of Computing, pp. 212–219 (1996)Google Scholar
  18. 18.
    Higuchi, Yu., Segawa, E.: The spreading behavior of quantum walks induced by drifted random walks on some magnifier graph. Quantum Inf. Comput. 17, 0399–0414 (2017)MathSciNetGoogle Scholar
  19. 19.
    Higuchi, Yu., Segawa, E.: Quantum walks induced by Dirichlet random walks on infinite trees. J. Phys. A Math. Theor. 51, 075303 (2018)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: Spectral and asymptotic properties of Grover walks on crystal lattices. J. Funct. Anal. 267, 4197–4235 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kitagawa, T.: Topological phenomena in quantum walks: elementary introduction to the physics of topological phases. Quantum Inf. Process. 11, 1107–1148 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kitagawa, T., Rudner, M.S., Berg, E., Demler, E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    Kitagawa, T., Broome, M.A., Fedrizzi, A., Rudner, M.S., Berg, E., Kassal, I., Aspuru-Guzik, A., Demler, E., White, A.G.: Observation of topologically protected bound states in photonic quantum walks. Nat. Commun. 3, 882 (2012)ADSCrossRefGoogle Scholar
  24. 24.
    Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1, 345–354 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Konno, N.: One-dimensional discrete-time quantum walks on random environment. Quantum Inf. Process. 8, 387–399 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Konno, N.: Localization of an inhomogeneous discrete-time quantum walk on the line. Quantum Inf. Process. 9, 405–418 (2010)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Konno, N., Łuczak, T., Segawa, E.: Limit measures of inhomogeneous discrete-time quantum walks in one dimensional. Quantum Inf. Process. 12, 33–53 (2013)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Konno, K., Portugal, R., Sato, I., Segawa, E.: Partition-based discrete-time quantum walks. Qunatum Inf. Process. 17, 100 (2018)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Kurzyński, P.: Relativistic effects in quantum walks: Klein’s paradox and zitterbewegung. Phys. Lett. A 372, 6125–6129 (2008)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Maeda, M., Suzuki, A.: Continuous limits of linear and nonlinear quantum walks. Rev. Math. Phys.
  31. 31.
    Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: ACM Symposium on Theory of Computing, pp. 575–584 (2007)Google Scholar
  32. 32.
    Magniez, F., Nayak, A., Richter, P., Santha, M.: On the hitting times of quantum versus random walks. Algorithmica 63, 91–116 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Meyer, D.A.: Quantum lattice gases and their invariants. Int. J. Mod. Phys. C 8, 717–735 (1997)ADSCrossRefGoogle Scholar
  34. 34.
    Mochizuki, K., Kim, D., Obuse, H.: Explicit definition of PT symmetry for nonunitary quantum walks with gain and loss. Phys. Rev. A 93, 062116 (2016)ADSCrossRefGoogle Scholar
  35. 35.
    Obuse, H., Kawakami, N.: Topological phases and delocalization of quantum walks in random environments. Phys. Rev. B 84, 195139 (2011)ADSCrossRefGoogle Scholar
  36. 36.
    Obuse, H., Asbóth, J.K., Nishimura, Y., Kawakami, N.: Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk. Phys. Rev. B 92, 045424 (2015)ADSCrossRefGoogle Scholar
  37. 37.
    Ohno, H.: Unitary equivalent classes of one-dimensional quantum walks. Quantum Inf. Process. 15, 3599–3617 (2016)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Portugal, R., Santos, R.A.M., Fernandes, T.D., GonÃğalves, D.N.: The staggered quantum walk model. Quantum Inf. Process. 15, 85 (2016)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Richard, S., Suzuki, A., Tiedra de Aldecoa, R.: Quantum walks with an anisotropic coin I: spectral theory. Lett. Math. Phys. 108, 331–357 (2018)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Richard, S., Suzuki, A., Tiedra de Aldecoa, R.: Quantum walks with an anisotropic coin II: scattering theory. Lett. Math. Phys. 109, 61–88 (2019)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Segawa, E., Suzuki, A.: Spectral mapping theorem of an abstract quantum walk. Quantum Inf. Process.
  42. 42.
    Segawa, E.: Localization of quantum walks induced by recurrence properties of random walks. J. Comput. Theor. Nanosci. 10, 1583–1590 (2013)CrossRefGoogle Scholar
  43. 43.
    Shikano, Y., Katsura, H.: Localization and fractality in inhomogeneous quantum walks with self-duality. Phys. Rev. E 82, 031122 (2010)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Strauch, F.W.: Relativistic quantum walks. Phys. Rev. A 73, 054302 (2006). Erratum Phys. Rev. A 73, 069908, 2006ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Strauch, F.W.: Relativistic effects and rigorous limits for discrete- and continuous-time quantum walks. J. Math. Phys. 48, 082102 (2007)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Suzuki, A.: Supersymmetry for chiral symmetric quantum walks. Quantum Inf. Process. 18, 363 (2019)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Suzuki, A.: Asymptotic velocity of a position-dependent quantum walk. Quantum Inf. Process. 15, 103–119 (2016)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proceedings of 45th IEEE Symposium on Foundations of Computer Science, pp. 32–41 (2004)Google Scholar
  49. 49.
    Thaller, B.: The Dirac equation. Texts and Monographs in Physics. Springer, Berlin (1992)Google Scholar
  50. 50.
    Xiao, L., Zhan, X., Bian, Z.H., Wang, K.K., Zhang, X., Wang, X.P., Li, J., Mochizuki, K., Kim, D., Kawakami, N., Yi, W., Obuse, H., Sanders, B.C., Xue, P.: Observation of topological edge states in parity-time-symmetric quantum walks. Nat. Phys. 13, 1117–1123 (2017)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Division of Mathematics and Physics, Faculty of EngineeringShinshu UniversityWakasatoJapan
  2. 2.School of Computer Science, Engineering and MathematicsFlinders UniversityClovelly ParkAustralia

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