### Abstract

We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy's quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has l integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight l. We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we generalize Grover's algorithm, or search on the complete graph, to have a weighted self-loop at each vertex, yielding an improved success probability when 1 < 3+2 √2 ≈ 5.828.

Original language | English (US) |
---|---|

Article number | 475301 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 50 |

Issue number | 47 |

DOIs | |

State | Published - Oct 25 2017 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

**Coined quantum walks on weighted graphs.** / Wong, Thomas.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 50, no. 47, 475301. https://doi.org/10.1088/1751-8121/aa8c17

}

TY - JOUR

T1 - Coined quantum walks on weighted graphs

AU - Wong, Thomas

PY - 2017/10/25

Y1 - 2017/10/25

N2 - We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy's quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has l integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight l. We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we generalize Grover's algorithm, or search on the complete graph, to have a weighted self-loop at each vertex, yielding an improved success probability when 1 < 3+2 √2 ≈ 5.828.

AB - We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy's quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has l integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight l. We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we generalize Grover's algorithm, or search on the complete graph, to have a weighted self-loop at each vertex, yielding an improved success probability when 1 < 3+2 √2 ≈ 5.828.

UR - http://www.scopus.com/inward/record.url?scp=85034219808&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85034219808&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aa8c17

DO - 10.1088/1751-8121/aa8c17

M3 - Article

VL - 50

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 47

M1 - 475301

ER -