Quantum Information Processing, Vol. 3, Nos. 1–5, October 2004 (© 2004)
One-Dimensional Continuous-Time Quantum Walks
E. M. Bollt,
and C. Tamon
Received December 1, 2003; accepted March 31, 2004
We survey the equations of continuous-time quantum walks on simple
one-dimensional lattices, which include the ﬁnite and inﬁnite lines and the ﬁnite
cycle, and compare them with the classical continuous-time Markov chains. The
focus of our expository article is on analyzing these processes using the Laplace
transform on the stochastic recurrences. The resulting time evolution equations,
classical vs. quantum, are strikingly similar in form, although dissimilar in behav-
ior. We also provide comparisons with analyses performed using spectral methods.
KEY WORDS: Quantum walks; continuous time; Laplace transform.
The theory of Markov chains on countable structures is an important area
in Mathematics and Physics. A quantum analog of continuous-time Mar-
kov chains on the inﬁnite line is well studied in Physics (for example,
it can be found in Ref. 12, Chapters 13 and 16). More recently, it was
studied by Aharonov et al.
and by Farhi and Gutmann.
work placed the problem in the context of quantum algorithms for search
problems on graphs. Here, the symmetric stochastic matrix of the graph
is viewed as a Hamiltonian of the quantum process. Using Schr
equation with this Hamiltonian, we obtain a quantum walk on the under-
lying graph, instead of a classical random walk.
Department of Physics, Clarkson University. E-mail: firstname.lastname@example.org
Department of Mathematics and Computer Science, and Department of Physics, Clarkson
University. E-mail: email@example.com
Department of Mathematics and Computer Science, and Center for Quantum Device Tech-
nology, Clarkson University. E-mail: firstname.lastname@example.org
To whom correspondence should be addressed.
1570-0755/04/1000-0295/0 © 2004 Springer Science+Business Media, Inc.