An explicit formula of the Hamiltonians generating one-dimensional
discrete-time quantum walks is given. The formula is deduced by using the
algebraic structure introduced previously. The square of the Hamiltonian turns
out to be an operator without, essentially, the `coin register', and hence it
can be compared with the one-dimensional continuous-time quantum walk. It is
shown that, under a limit with respect to a parameter, which expresses the
magnitude of the diagonal components of the unitary matrix defining the
discrete-time quantum walks, the one-dimensional continuous-time quantum walk
is obtained from operators defined through the Hamiltonians of the
one-dimensional discrete-time quantum walks. Thus, this result can be regarded,
in one-dimension, as a partial answer to a problem proposed by Ambainis.Comment: 9 page
