An Example of the Diﬀerence Between Quantum and
Classical Random Walks
Andrew M. Childs,
and Sam Gutmann
Received March 1, 2001; accepted May 16, 2002
In this note, we discuss a general deﬁnition of quantum random walks on graphs and
illustrate with a simple graph the possibility of very diﬀerent behavior between a
classical random walkand its quantum analog. In this graph, propagation between a
particular pair of nodes is exponentially faster in the quantum case.
KEY WORDS: quantum random walk; hitting times.
Many classical algorithms are based on random walks, so it is natural to ask
whether quantum random walks might be useful for quantum computation.
Aframework for using quantum random walks to solve decision problems
was investigated in Ref. 1. There also, an exponential separation was found
between the classical and quantum times to propagate through a certain
In this note, we describe a general deﬁnition of continuous-time random
walks on graphs and give a simpler example of a graph for which the quantum
time to propagate between a particular pair of nodes is exponentially shorter
than the analogous classical propagation time. We also discuss advantages of
the continuous time formulation over discrete versions.
1570-0755/02/0400-0035/0 # 2002 Plenum Publishing Corporation
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge,
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115.
Quantum Information Processing, Vol. 1, Nos. 1/2, April 2002 (# 2002)