  Recently Mc Gettrick [1] introduced and studied a discrete-time 2-state
quantum walk (QW) with a memory in one dimension. He gave an expression for the
amplitude of the QW by path counting method. Moreover he showed that the return
probability of the walk is more than 1/2 for any even time. In this paper, we
compute the stationary distribution by considering the walk as a 4-state QW
without memory. Our result is consistent with his claim. In addition, we obtain
the weak limit theorem of the rescaled QW. This behavior is striking different
from the corresponding classical random walk and the usual 2-state QW without
memory as his numerical simulations suggested.
