  A two-dimensional discrete-time quantum walk (DTQW) can be realized by
alternating a two-state DTQW in one spatial dimension followed by an evolution
in the other dimension. This was shown to reproduce a probability distribution
for a certain configuration of a four-state DTQW on a two-dimensional lattice.
In this work we present a three-state alternate DTQW with a parameterized
coin-flip operator and show that it can produce localization that is also
observed for a certain other configuration of the four-state DTQW and
non-reproducible using the two-state alternate DTQW. We will present two limit
theorems for the three-state alternate DTQW. One of the limit theorems
describes a long-time limit of a return probability, and the other presents a
convergence in distribution for the position of the walker on a rescaled space
by time. We will also outline the relevance of these walks in physical systems.

Quantum Physics;Probability