### Abstract

In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such Kronecker graphs can be searched by a continuous-time quantum walk in optimal ΘN time. First, if the initiator is regular and its adjacency matrix has a dominant principal eigenvalue, meaning its unique largest eigenvalue asymptotically dominates the other eigenvalues in magnitude, then the Kronecker graphs generated by this initiator can be quantum searched with probability 1 in πN/2 time, asymptotically, and we give the critical jumping rate of the walk that enables this. Second, for any fixed initiator that is regular, nonbipartite, and connected, the Kronecker graphs generated by it are quantum searched in ΘN time. This greatly extends the number of Kronecker graphs on which quantum walks are known to optimally search. If the fixed, regular, connected initiator is bipartite, however, then search on its Kronecker powers is not optimal, but is still better than a classical computer's O(N) runtime if the initiator has more than two vertices.

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

Article number | 062334 |

Journal | Physical Review A |

Volume | 98 |

Issue number | 6 |

DOIs | |

State | Published - Dec 27 2018 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics

### Cite this

**Optimal quantum-walk search on Kronecker graphs with dominant or fixed regular initiators.** / Glos, Adam; Wong, Thomas.

Research output: Contribution to journal › Article

*Physical Review A*, vol. 98, no. 6, 062334. https://doi.org/10.1103/PhysRevA.98.062334

}

TY - JOUR

T1 - Optimal quantum-walk search on Kronecker graphs with dominant or fixed regular initiators

AU - Glos, Adam

AU - Wong, Thomas

PY - 2018/12/27

Y1 - 2018/12/27

N2 - In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such Kronecker graphs can be searched by a continuous-time quantum walk in optimal ΘN time. First, if the initiator is regular and its adjacency matrix has a dominant principal eigenvalue, meaning its unique largest eigenvalue asymptotically dominates the other eigenvalues in magnitude, then the Kronecker graphs generated by this initiator can be quantum searched with probability 1 in πN/2 time, asymptotically, and we give the critical jumping rate of the walk that enables this. Second, for any fixed initiator that is regular, nonbipartite, and connected, the Kronecker graphs generated by it are quantum searched in ΘN time. This greatly extends the number of Kronecker graphs on which quantum walks are known to optimally search. If the fixed, regular, connected initiator is bipartite, however, then search on its Kronecker powers is not optimal, but is still better than a classical computer's O(N) runtime if the initiator has more than two vertices.

AB - In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such Kronecker graphs can be searched by a continuous-time quantum walk in optimal ΘN time. First, if the initiator is regular and its adjacency matrix has a dominant principal eigenvalue, meaning its unique largest eigenvalue asymptotically dominates the other eigenvalues in magnitude, then the Kronecker graphs generated by this initiator can be quantum searched with probability 1 in πN/2 time, asymptotically, and we give the critical jumping rate of the walk that enables this. Second, for any fixed initiator that is regular, nonbipartite, and connected, the Kronecker graphs generated by it are quantum searched in ΘN time. This greatly extends the number of Kronecker graphs on which quantum walks are known to optimally search. If the fixed, regular, connected initiator is bipartite, however, then search on its Kronecker powers is not optimal, but is still better than a classical computer's O(N) runtime if the initiator has more than two vertices.

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U2 - 10.1103/PhysRevA.98.062334

DO - 10.1103/PhysRevA.98.062334

M3 - Article

AN - SCOPUS:85059439924

VL - 98

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

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

SN - 1050-2947

IS - 6

M1 - 062334

ER -