### 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) |
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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)