### Abstract

Discrete-time quantum walks are defined as a non-commutative analogue of the usual random walks on standard lattices and have been formulated in computer sciences. They are new objects in mathematics and are investigated in various areas, such as computer sciences, quantum physics, probability theory, and discrete geometric analysis. In this article, recent works on point-wise asymptotic behavior and an effective formula for nth power of the discrete-time quantum walks in one dimension are surveyed. The idea to obtain the formula for the nth power in one dimension is applied in this paper to compute the nth power of certain two-dimensional quantum walk, called the Grover walk to obtain a new formula for the two-dimensional Grover walk. The formula for nth power in one dimension has been used to prove a weak limit theorem. In this paper, the large deviation asymptotics, in one dimension, is deduced by using this formula which is a new proof of a previously obtained result.

Original language | English |
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Title of host publication | Geometric Methods in Physics - 34th Workshop |

Editors | Piotr Kielanowski, S. Twareque Ali, Pierre Bieliavsky, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov |

Publisher | Springer International Publishing |

Pages | 261-278 |

Number of pages | 18 |

ISBN (Print) | 9783319317557 |

DOIs | |

Publication status | Published - 2016 |

Event | 34th Workshop on Geometric Methods in Physics, 2015 - >Białowieża, Poland Duration: 2015 Jun 28 → 2015 Jul 4 |

### Publication series

Name | Trends in Mathematics |
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Volume | 0 |

ISSN (Print) | 2297-0215 |

ISSN (Electronic) | 2297-024X |

### Other

Other | 34th Workshop on Geometric Methods in Physics, 2015 |
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Country | Poland |

City | >Białowieża |

Period | 15/6/28 → 15/7/4 |

### Keywords

- Asymptotics
- One-dimensional quantum walks
- Semi-direct products
- Two-dimensional Grover walk

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Geometric Methods in Physics - 34th Workshop*(pp. 261-278). (Trends in Mathematics; Vol. 0). Springer International Publishing. https://doi.org/10.1007/978-3-319-31756-4_21