Absorption problems for quantum walks in one dimension
  This paper treats absorption problems for the one-dimensional quantum walk determined by a 2 times 2 unitary matrix U on a state space {0,1,...,N} where N is finite or infinite by using a new path integral approach based on an orthonormal basis P, Q, R and S of the vector space of complex 2 times 2 matrices. Our method studied here is a natural extension of the approach in the classical random walk. 
Norio Konno, Takao Namiki, Takahiro Soshi and Aidan Sudbury