Quantum Random Walks in One Dimension
Received August 16, 2002; accepted December 3, 2002
This letter treats the quantum random walk on the line determined by a 2 2 unitary
matrix U. A combinatorial expression for the mth moment of the quantum random
walk is presented by using 4 matrices, P; Q; R and S given by U. The dependence of
the mth moment on U and initial qubit state ’ is clariﬁed. A new type of limit
theorems for the quantum walk is given. Furthermore necessary and suﬃcient con-
ditions for symmetry of distribution for the quantum walk is presented. Our results
show that the behavior of quantum random walk is striking diﬀerent from that of the
classical ramdom walk.
KEY WORDS: Quantum random walk; the Hadamard walk; limit theorems.
PACS: 03.67.Lx; 05.40.Fb; 02.50.Cw.
Recently various problems for quantum random walks have been
widely investigated by a number of groups in connection with quantum
computing. Examples include Aharonov et al.,
Ambainis et al.,
Childs, Farhi and Gutmann,
r et al.,
Konno, Namiki and Soshi,
Konno et al.,
Mackay et al.,
Travaglione and Milburn,
Yamasaki, Kobayashi and Imai.
For a more general setting including quantum cellular automata, see
A more mathematical point of view of quantum computing can
be found in Brylinsky and Chen.
In Ambainis et al.,
they gave two general ideas for analyzing
quantum random walks. One is the path integral approach, the other is the
Schr€odinger approach. In this paper, we take the path integral approach,
1570-0755/02/1000-0345/0 # 2003 Plenum Publishing Corporation
Department of Applied Mathematics, Faculty of Engineering, Yokohama National
University, 79-5 Tokiwadai, Hodogaya, Yokohama 240-8501, Japan.
To whom correspondence should be addressed. E-mail: firstname.lastname@example.org
Quantum Information Processing, Vol. 1, No. 5, October 2002 (# 2003)