  From the unitary operator used for implementing two-state discrete-time
quantum walk on one-, two- and three- dimensional lattice we obtain a
two-component Dirac-like Hamiltonian. In particular, using different pairs of
Pauli basis as position translation states we obtain three different form of
Hamiltonians for evolution on one-dimensional lattice. We extend this to two-
and three-dimensional lattices using different Pauli basis states as position
translation states for each dimension and show that the external coin
operation, which is necessary for one-dimensional walk is not a necessary
requirement for a walk on higher dimensions but can serve as an additional
resource to control the dynamics. The two-component Hamiltonian we present here
for quantum walk on different lattices can serve as a general framework to
simulate, control, and study the dynamics of quantum systems governed by
Dirac-like Hamiltonian.
