  We define the discrete-time quantum walk on complex networks and utilize it
for community detection. We numerically show that the quantum walk with the
Fourier coin is localized in a community to which the initial node belongs.
Meanwhile, the quantum walk with the Grover coin tends to be localized around
the initial node, not over a community. The probability of the classical random
walk on the same network converges to the uniform distribution with a
relaxation time generally a priori. We thus claim that the time average of the
probability of the Fourier-coin quantum walk on complex networks reveals the
community structure more explicitly than that of the Grover-coin quantum walk
and a snapshot of the classical random walk. We first demonstrate our method of
community detection for a prototypical three-community network, producing the
correct grouping. We then apply our method to two real-world networks, namely
Zachary's karate club and the US Airport network. We successfully reveals the
community structure, the two communities of the instructor and the
administrator in the former and major airline companies in the latter.
