Quantum Inf Process (2012) 11:341–349
On the relation between quantum walks and zeta
Norio Konno · Iwao Sato
Received: 28 February 2011 / Accepted: 7 May 2011 / Published online: 24 May 2011
© Springer Science+Business Media, LLC 2011
Abstract We present an explicit formula for the characteristic polynomial of the
transition matrix of the discrete-time quantum walk on a graph via the second weighted
zeta function. As applications, we obtain new proofs for the results on spectra of the
transition matrix and its positive support.
Keywords Quantum walk · Transition matrix · Ihara zeta function
Mathematics Subject Classiﬁcation (2000) 05C50 · 05C60 · 81P68
As a quantum counterpart of the classical random walk, the quantum walk has
recently attracted much attention for various ﬁelds. The review and book on quan-
tum walks are Ambainis , Kempe , Kendon , Konno , for examples.
Quantum walks of graphs were applied in graph isomorphism problems. Graph iso-
morphism problems determine whether two graphs are isomorphic. Shiau et al. 
ﬁrst pointed out the deﬁciency of the simplest classical algorithm and continuous-time
one particle quantum random walks in distinguishing some non-isomorphic graphs.
N. Konno (
Department of Applied Mathematics, Faculty of Engineering, Yokohama National University,
Hodogaya, Yokohama 240-8501, Japan
Oyama National College of Technology, Oyama, Tochigi 323-0806, Japan