Unitary and Open Scattering Quantum Walks on Graphs

We consider a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. These Scattering Quantum Walks model the discrete dynamics of a system on the graph’s edges, where a scattering process at the vertices is governed by the scattering matrices assigned to each vertex. We show the Scattering Quantum Walks encompass several known Quantum Walks. We further introduce two classes of Scattering Open Quantum Walks on arbitrary graphs, also parameterized by scattering matrices: one defined on the edges, the other on the vertices of the graph. We show these walks give rise to proper quantum channels and describe their main spectral and dynamical properties, relating them to naturally associated classical Markov chains.