Symmetry of distribution for the one-dimensional Hadamard walk
  In this paper we study a one-dimensional quantum random walk with the Hadamard transformation which is often called the Hadamard walk. We construct the Hadamard walk using a transition matrix on probability amplitude and give some results on symmetry of probability distributions for the Hadamard walk. 
Norio Konno, Takao Namiki and Takahiro Soshi