  A new approach to quantum walks is presented. Considering a quantum system
undergoing some unitary discrete-time evolution in a directed graph G, we think
of the vertices of G as sites that are occupied by the quantum system, whose
internal state is described by density operators. To formulate the unitary
evolution, we define reflections in the tensor product of an internal Hilbert
space and a spatial Hilbert space. We then construct unitary channels that
govern the evolution of the system in the graph. The discrete dynamics of the
system (called quantum walks) is obtained by iterating the unitary channel on
the density operator of the quantum system. It turns out that in this
framework, the action of the unitary channel on a density operator is described
by the usual matrix multiplication.
