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Parameterization of Translation-Invariant Two-Dimensional Two-State Quantum Walks

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Abstract

This study investigates unitary equivalence classes of translation-invariant two-dimensional two-state quantum walks. We show that unitary equivalence classes of such quantum walks are essentially parameterized by two real parameters.

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References

  1. Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of 33Th Annual ACM Symposium on Theory of Computing, pp 37–49. ACM, New York (2001)

  2. Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: One-dimensional quantum walks with one defect. Rev. Math. Phys. 24(2), 52 (2012)

    Article  MathSciNet  Google Scholar 

  3. Chen, T., Zhang, X., Zhang, X.: Quantum sensing of noises in one and two dimensional quantum walks. Sci. Rep. 7, 4962 (2017)

    Article  Google Scholar 

  4. Di Franco, C., Paternostro, M.: Localizationlike effect in two-dimensional alternate quantum walks with periodic coin operations. Phys. Rev. A 91(1), 012328 (2015)

    Article  MathSciNet  Google Scholar 

  5. Goyal, S.K., Konrad, T., Diósi, L.: Unitary equivalence of quantum walks. Phys. Lett. A 379(3), 100–104 (2015)

    Article  MathSciNet  Google Scholar 

  6. Gudder, S.P.: Quantum Probability. Academic Press, Inc, Boston (1988)

    MATH  Google Scholar 

  7. Konno, N.: Quantum random walks in one dimensional. Quantum Inf. Process. 1(5), 345–354 (2003)

    Article  Google Scholar 

  8. Ohno, H.: Unitary equivalent classes of one-dimensional quantum walks. Quantum Inf. Process. 15(9), 3599–3617 (2016)

    Article  MathSciNet  Google Scholar 

  9. Ohno, H.: Unitary equivalence classes of one-dimensional quantum walks II. Quantum Inf. Process. 16(12), 17 (2017)

    Article  MathSciNet  Google Scholar 

  10. Segawa, E., Suzuki, A.: Generator of an abstract quantum walk. Quantum Stud. Math. Found. 3(1), 11–30 (2016)

    Article  MathSciNet  Google Scholar 

  11. Szegedy, M.: Quantum speed-up of Markov chain based algorithms. Proceedings of 45th Annual, IEEE Symposium Foundations Computer Science, pp. 32–41 (2004)

  12. Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11(5), 1015–1106 (2012)

    Article  MathSciNet  Google Scholar 

  13. Verga, A.D.: Edge states in a two-dimensional quantum walk with disorder. Eur. Phys. J. B 90, 41 (2017)

    Article  Google Scholar 

Download references

Funding

This work was supported by JSPS KAKENHI Grant Numbers 17K05274.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano, 380-8553, Japan

    Hiromichi Ohno

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  1. Hiromichi Ohno
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Correspondence to Hiromichi Ohno.

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Ohno, H. Parameterization of Translation-Invariant Two-Dimensional Two-State Quantum Walks. Acta Math Vietnam 43, 737–747 (2018). https://doi.org/10.1007/s40306-018-0278-8

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  • DOI: https://doi.org/10.1007/s40306-018-0278-8

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