 
Summary: Quantum Walks On Graphs
Dorit Aharonov , Andris Ambainis y , Julia Kempe z and Umesh Vazirani x
December 17, 2000
Abstract
We initiate the study of the generalization of random walks on nite 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 denition, we can obtain a measure of how fast the quantum walk spreads or how
conned the quantum walk stays in a small neighborhood. We give denitions of mixing time, lling 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.
1 Introduction
Markov chains or random walks on graphs have proved to be a fundamental tool, with broad applications
in various elds of mathematics, computer science and the natural sciences, such as mathematical modeling of
physical systems, simulated annealing, and the Markov Chain Monte Carlo method. In the physical sciences they
provide a fundamental model for the emergence of global properties from local interactions. In the algorithmic
context, they provide a general paradigm for sampling and exploring an exponentially large set of combinatorial
structures (such as matchings in a graph), by using a sequence of simple, local transitions.
In this paper, we initiate a study of quantum walks on graphs  the motivation, as in the case of Markov
