Abstract:
Following Konno [1], it is natural to ask: What is the Ito’s formula for the discrete time quantum walk on a graph different than Z, the set of integers? In this paper we answer the question for the discrete time quantum walk on Z2, the square lattice.

Abstract:
We review (not exhaustively) the quantum random walk on the line in various settings, and propose some questions that we believe have not been tackled in the literature. In a sense, this article invites the readers (beginner, intermediate, or advanced), to explore the beautiful area of quantum random walks.

Abstract:
Using the technique of path counting we show non-existence of sojourn times in the Grover walk which is related to the Grover's algorithm in computer science.

Abstract:
We study the motion of M particles performing a quantum walk on the line. Under various conditions on the initial coin states for quantum walkers controlled by the Hadamard operator, we give theoretical criterion to observe the quantum walkers at an initial location with high probability.

Abstract:
We investigate a generalized Hadamard walk in two dimensions with five inner states. The particle governed by a five-state quantum walk (5QW) moves, in superposition, either leftward, rightward, upward, or downward according to the inner state. In addition to the four degrees of freedom, it is allowed to stay at the same position. We calculate rigorously the wave function of the particle starting from the origin in the plane for any initial state, and give the spatial distribution of probability of finding the particle. We also investigate the localization problem for the two-dimensional five-state quantum walk: Does the probability of finding a particle anywhere on the plane converge to zero even after infinite time steps except initial states?

Abstract:
We consider the Grover walk as a 4-state quantum walk without memory in one dimension. The walker in our 4-state quantum walk moves to the left or right. We compute the stationary distribution of the walk, in addition, we obtain the weak limit theorem

Abstract:
We study the discrete-time quantum walk in one-dimension governed by the Fibonacci transformation .We show localization does not occur for the Fibonacci quantum walk by investigating the stationary distribution of the walk, in addition, we obtain the weak limit theorem.