Return probability of one-dimensional discrete-time quantum walks with   final-time dependence
  We analyze final-time dependent discrete-time quantum walks in one dimension. We compute asymptotics of the return probability of the quantum walk by a path counting approach. Moreover, we discuss a relation between the quantum walk and the corresponding final-time dependent classical random walk. 
Yusuke Ide, Norio Konno, Takuya Machida, Etsuo Segawa
quant-ph