Limit distribution of a time-dependent quantum walk on the half line

We focus on a 2-period time-dependent quantum walk on the half line in this paper. The quantum walker launches at the edge of the half line in a localized superposition state and its time evolution is carried out with two unitary operations which are alternately cast to the quantum walk. As a result, long-time limit finding probabilities of the quantum walk turn to be determined by either one of the two operations, but not both. More interestingly, the limit finding probabilities are independent from the localized initial state. We will approach the appreciated features via a quantum walk on the line which is able to reproduce the time-dependent walk on the half line.