### Abstract

A quantum particle evolving by Schrödinger’s equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace’s operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.

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

Pages (from-to) | 4029-4048 |

Number of pages | 20 |

Journal | Quantum Information Processing |

Volume | 15 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1 2016 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Electronic, Optical and Magnetic Materials
- Statistical and Nonlinear Physics
- Theoretical Computer Science
- Signal Processing
- Modeling and Simulation
- Electrical and Electronic Engineering

### Cite this

*Quantum Information Processing*,

*15*(10), 4029-4048. https://doi.org/10.1007/s11128-016-1373-1

**Laplacian versus adjacency matrix in quantum walk search.** / Wong, Thomas; Tarrataca, Luís; Nahimov, Nikolay.

Research output: Contribution to journal › Article

*Quantum Information Processing*, vol. 15, no. 10, pp. 4029-4048. https://doi.org/10.1007/s11128-016-1373-1

}

TY - JOUR

T1 - Laplacian versus adjacency matrix in quantum walk search

AU - Wong, Thomas

AU - Tarrataca, Luís

AU - Nahimov, Nikolay

PY - 2016/10/1

Y1 - 2016/10/1

N2 - A quantum particle evolving by Schrödinger’s equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace’s operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.

AB - A quantum particle evolving by Schrödinger’s equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace’s operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.

UR - http://www.scopus.com/inward/record.url?scp=84976476156&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976476156&partnerID=8YFLogxK

U2 - 10.1007/s11128-016-1373-1

DO - 10.1007/s11128-016-1373-1

M3 - Article

VL - 15

SP - 4029

EP - 4048

JO - Quantum Information Processing

JF - Quantum Information Processing

SN - 1570-0755

IS - 10

ER -