  We present a new scheme for a discrete-time quantum walk on two- and
three-dimensional lattices using a two-state particle. We use different Pauli
basis as translational eigestates for different axis and show that the coin
operation, which is necessary for one-dimensional walk is not a necessary
requirement for two- and three- dimensional walk but can serve as an additional
resource. Using this scheme, the probability distribution from Grover walk
using four-state particle and other equivalent schemes on a square lattice
using coin operation is reproduced. We also present the Hamiltonian form of
evolution which can serve as a general framework to simulate, control, and
study the dynamics in different physical systems.
