  The quantum walks in the lattice spaces are represented as unitary
evolutions. We find a generator for the evolution and apply it to further
understand the walks. We first extend the discrete time quantum walks to
continuous time walks. Then we construct the quantum Markov semigroup for
quantum walks and characterize it in an invariant subalgebra. In the meanwhile,
we obtain the limit distributions of the quantum walks in one-dimension with a
proper scaling, which was obtained by Konno by a different method.
