### Abstract

In this theoretical study, we analyze quantum walks on complex networks, which model network-based processes ranging from quantum computing to biology and even sociology. Specifically, we analytically relate the average long-time probability distribution for the location of a unitary quantum walker to that of a corresponding classical walker. The distribution of the classical walker is proportional to the distribution of degrees, which measures the connectivity of the network nodes and underlies many methods for analyzing classical networks, including website ranking. The quantum distribution becomes exactly equal to the classical distribution when the walk has zero energy, and at higher energies, the difference, the socalled quantumness, is bounded by the energy of the initial state. We give an example for which the quantumness equals a Rényi entropy of the normalized weighted degrees, guiding us to regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate.

Original language | English |
---|---|

Article number | 041007 |

Journal | Physical Review X |

Volume | 3 |

Issue number | 4 |

DOIs | |

Publication status | Published - 13 Feb 2014 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Physical Review X*,

*3*(4), [041007]. https://doi.org/10.1103/PhysRevX.3.041007

**Degree distribution in quantum walks on complex networks.** / Faccin, Mauro; Johnson, Tomi; Biamonte, Jacob; Kais, Sabre; Migdal, Piotr.

Research output: Contribution to journal › Article

*Physical Review X*, vol. 3, no. 4, 041007. https://doi.org/10.1103/PhysRevX.3.041007

}

TY - JOUR

T1 - Degree distribution in quantum walks on complex networks

AU - Faccin, Mauro

AU - Johnson, Tomi

AU - Biamonte, Jacob

AU - Kais, Sabre

AU - Migdal, Piotr

PY - 2014/2/13

Y1 - 2014/2/13

N2 - In this theoretical study, we analyze quantum walks on complex networks, which model network-based processes ranging from quantum computing to biology and even sociology. Specifically, we analytically relate the average long-time probability distribution for the location of a unitary quantum walker to that of a corresponding classical walker. The distribution of the classical walker is proportional to the distribution of degrees, which measures the connectivity of the network nodes and underlies many methods for analyzing classical networks, including website ranking. The quantum distribution becomes exactly equal to the classical distribution when the walk has zero energy, and at higher energies, the difference, the socalled quantumness, is bounded by the energy of the initial state. We give an example for which the quantumness equals a Rényi entropy of the normalized weighted degrees, guiding us to regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate.

AB - In this theoretical study, we analyze quantum walks on complex networks, which model network-based processes ranging from quantum computing to biology and even sociology. Specifically, we analytically relate the average long-time probability distribution for the location of a unitary quantum walker to that of a corresponding classical walker. The distribution of the classical walker is proportional to the distribution of degrees, which measures the connectivity of the network nodes and underlies many methods for analyzing classical networks, including website ranking. The quantum distribution becomes exactly equal to the classical distribution when the walk has zero energy, and at higher energies, the difference, the socalled quantumness, is bounded by the energy of the initial state. We give an example for which the quantumness equals a Rényi entropy of the normalized weighted degrees, guiding us to regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate.

UR - http://www.scopus.com/inward/record.url?scp=84893547709&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893547709&partnerID=8YFLogxK

U2 - 10.1103/PhysRevX.3.041007

DO - 10.1103/PhysRevX.3.041007

M3 - Article

VL - 3

JO - Physical Review X

JF - Physical Review X

SN - 2160-3308

IS - 4

M1 - 041007

ER -