Limit theorems for quantum walks with memory

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.