### Abstract

When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if nonbipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.

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

Article number | 032334 |

Journal | Physical Review A |

Volume | 94 |

Issue number | 3 |

DOIs | |

State | Published - Sep 29 2016 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics

### Cite this

*Physical Review A*,

*94*(3), [032334]. https://doi.org/10.1103/PhysRevA.94.032334

**Stationary states in quantum walk search.** / Prusis, Krišjanis; Vihrovs, Jevgenijs; Wong, Thomas.

Research output: Contribution to journal › Article

*Physical Review A*, vol. 94, no. 3, 032334. https://doi.org/10.1103/PhysRevA.94.032334

}

TY - JOUR

T1 - Stationary states in quantum walk search

AU - Prusis, Krišjanis

AU - Vihrovs, Jevgenijs

AU - Wong, Thomas

PY - 2016/9/29

Y1 - 2016/9/29

N2 - When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if nonbipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.

AB - When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if nonbipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.

UR - http://www.scopus.com/inward/record.url?scp=84991687884&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84991687884&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.94.032334

DO - 10.1103/PhysRevA.94.032334

M3 - Article

AN - SCOPUS:84991687884

VL - 94

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

SN - 1050-2947

IS - 3

M1 - 032334

ER -