### Abstract

We set the ground for a theory of quantum walks on graphs-the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.

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

Pages (from-to) | 50-59 |

Number of pages | 10 |

Journal | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

State | Published - Jan 1 2001 |

Event | 33rd Annual ACM Symposium on Theory of Computing - Creta, Greece Duration: Jul 6 2001 → Jul 8 2001 |

### ASJC Scopus subject areas

- Software

### Cite this

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*, 50-59.

**Quantum walks on graphs.** / Aharonov, D.; Ambainis, A.; Kempe, Julia; Vazirani, U.

Research output: Contribution to journal › Conference article

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*, pp. 50-59.

}

TY - JOUR

T1 - Quantum walks on graphs

AU - Aharonov, D.

AU - Ambainis, A.

AU - Kempe, Julia

AU - Vazirani, U.

PY - 2001/1/1

Y1 - 2001/1/1

N2 - We set the ground for a theory of quantum walks on graphs-the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.

AB - We set the ground for a theory of quantum walks on graphs-the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.

UR - http://www.scopus.com/inward/record.url?scp=0034819347&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034819347&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:0034819347

SP - 50

EP - 59

JO - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

JF - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

SN - 0734-9025

ER -