Abstract
Quantum walks are considered to be quantum counterparts of random walks. They show us impressive probability distributions which are different from those of random walks. That fact has been precisely proved in terms of mathematics and some of the results were reported as limit theorems. When we analyze quantum walks, some conventional methods are used for the computations; especially, the Fourier analysis has played a role to do that. It is, however, compatible with some types of quantum walks (e.g., quantum walks on the line with a spatially homogeneous dynamics) and cannot well work on the derivation of limit theorems for all the quantum walks. In this paper, we try to obtain a limit theorem for a quantum walk on the half line by the usage of the Fourier analysis. Substituting a quantum walk on the line for it, we will lead to a possibility that the Fourier analysis is useful to compute a limit distribution of the quantum walk on the half line.
Similar content being viewed by others
References
Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11(5), 1015 (2012)
Di Franco, C., McGettrick, M., Busch, T.: Mimicking the probability distribution of a two-dimensional Grover walk with a single–qubit coin. Phys. Rev. Lett. 106(8), 080502 (2011)
Di Franco, C., McGettrick, M., Machida, T., Busch, T.: Alternate two-dimensional quantum walk with a single–qubit coin. Phys. Rev. A 84(4), 042337 (2011)
Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1(5), 345 (2002)
Liu, C., Petulante, N.: Weak limits for quantum walks on the half-line. Int. J. Quantum Inf. 11(06), 1350054 (2013)
Konno, N., Segawa, E.: Localization of discrete-time quantum walks on a half line via the CGMV method. Quantum Inf. Comput. 11(5 & 6), 485 (2011)
Konno, N., Segawa, E.: One-dimensional quantum walks via generating function and the CGMV method. Quantum Inf. Comput. 14(13 & 14), 1165 (2014)
Cantero, M.J., Moral, L., Grünbaum, F.A., Velázquez, L.: Matrix-valued Szegö polynomials and quantum random walks. Commun. Pure Appl. Math. 63(4), 464 (2010)
Grimmett, G., Janson, S., Scudo, P.F.: Weak limits for quantum random walks. Phys. Rev. E 69(2), 026119 (2004)
Acknowledgments
The author is supported by JSPS Grant-in-Aid for Young Scientists (B) (No. 16K17648).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Machida, T. A quantum walk on the half line with a particular initial state. Quantum Inf Process 15, 3101–3119 (2016). https://doi.org/10.1007/s11128-016-1351-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-016-1351-7