Skip to main content
SpringerLink
Log in

An index theorem for split-step quantum walks

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Split-step quantum walks are models of supersymmetric quantum walk, and thus, their Witten indices can be defined. We prove that the Witten index of a split-step quantum walk coincides with the difference between the winding numbers of functions corresponding to the right limit of coins and the left limit of coins. As a corollary, we give an alternative derivation of the index formula for split-step quantum walks, which is recently obtained by Suzuki and Tanaka.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asbóth, J.K., Obuse, H.: Bulk-boundary correspondence for chiral symmetric quantum walks. Phys. Rev. B 88, 121406 (2013)

    Article  ADS  Google Scholar 

  2. Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: One-dimensional quantum walks with one defect. Rev. Math. Phys. 24, 1250002 (2012)

    Article  MathSciNet  Google Scholar 

  3. Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: The CGMV method for quantum walks. Quantum Inf. Process. 11, 1149–1192 (2012)

    Article  MathSciNet  Google Scholar 

  4. 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)

    Article  ADS  MathSciNet  Google Scholar 

  5. 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)

    Article  ADS  MathSciNet  Google Scholar 

  6. Fuda, T., Funakawa, D., Suzuki, A.: Weak limit theorem for a one-dimensional split-step quantum walk. Rev. Math. Pures Appl. 64, 157–165 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Kitagawa, T., Rudner, M.S., Berg, E., Demler, E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)

    Article  ADS  Google Scholar 

  8. Murphy, G.: \(C^*\)-algebras and operator theory. Academic Press, London (1990)

    MATH  Google Scholar 

  9. Richard, S., Suzuki, A., Tiedra de Aldecoa, R.: Quantum walks with an anisotropic coin I: spectral theory. Lett. Math. Phys. 108, 331–357 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  10. Segawa, E., Suzuki, A.: Generator of an abstract quantum walk. Quantum Stud. Math. Found. 3, 11–30 (2016)

    Article  MathSciNet  Google Scholar 

  11. Shikano, Y., Katsura, H.: Localization and fractality in inhomogeneous quantum walks with self-duality. Phys. Rev. E 82, 031122 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  12. Shikano, Y., Katsura, H.: Notes on inhomogeneous quantum walks. AIP Conf. Proc. E 1363, 151–154 (2011)

    Article  ADS  Google Scholar 

  13. Suzuki, A.: Supersymmetry for chiral symmetric quantum walks. Quantum Inf. Process. 18, 363 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  14. Suzuki, A., Tanaka, Y.: The Witten index for 1D supersymmetric quantum walks with anisotropic coins. Quantum Inf. Process. 18, 377 (2019)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to thank Akito Suzuki and Yohei Tanaka for helpful discussions on split-step quantum walks and their careful reading of the first draft of this paper. I also would like to thank Itaru Sasaki for informing me of the relation between the winding number of a holomorphic function and the number of zeros of it in the open unit disc, which was used in Sect. 4.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Shinshu University, 6-Ro, Nishi-nagano, Nagano, 380-8544, Japan

    Yasumichi Matsuzawa

Authors
  1. Yasumichi Matsuzawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yasumichi Matsuzawa.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matsuzawa, Y. An index theorem for split-step quantum walks. Quantum Inf Process 19, 227 (2020). https://doi.org/10.1007/s11128-020-02720-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-020-02720-7

Keywords

Search

Navigation