### Abstract

The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given l self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of N vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Shenvi, Kempe, and Whaley (2003) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when l scales less than N, while the second requires that l scale less than N ^{2}. Finally, continuous-time quantum walks differ from both of these discrete-time examples - the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.

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

Article number | 435304 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 48 |

Issue number | 43 |

DOIs | |

State | Published - Oct 7 2015 |

Externally published | Yes |

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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

**Grover search with lackadaisical quantum walks.** / Wong, Thomas.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 48, no. 43, 435304. https://doi.org/10.1088/1751-8113/48/43/435304

}

TY - JOUR

T1 - Grover search with lackadaisical quantum walks

AU - Wong, Thomas

PY - 2015/10/7

Y1 - 2015/10/7

N2 - The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given l self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of N vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Shenvi, Kempe, and Whaley (2003) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when l scales less than N, while the second requires that l scale less than N 2. Finally, continuous-time quantum walks differ from both of these discrete-time examples - the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.

AB - The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given l self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of N vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Shenvi, Kempe, and Whaley (2003) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when l scales less than N, while the second requires that l scale less than N 2. Finally, continuous-time quantum walks differ from both of these discrete-time examples - the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.

UR - http://www.scopus.com/inward/record.url?scp=84946051064&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84946051064&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/48/43/435304

DO - 10.1088/1751-8113/48/43/435304

M3 - Article

VL - 48

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 43

M1 - 435304

ER -