Percolated quantum walks with a general shift operator: From trapping to transport

We present an alternative definition of discrete-time coined quantum walks convenient for capturing a rather broad spectrum of a walker's behavior on arbitrary graphs. It includes and covers both the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find that for these walks with cyclic shift operators, the existence of an edge-3 coloring of the graph allows for nonstationary asymptotic behavior of the walk. For different shift operators, the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory, we treat a single-excitation transport on a percolated cube.