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We present a generalized definition of discrete-time quantum walks convenientfor capturing a rather broad spectrum of walker's behavior on arbitrary graphs.It includes and covers both: the geometry of possible walker's positions withinterconnecting links and the prescribed rule in which directions the walkerwill move at each vertex. While the former allows for the analysis ofinhomogeneous quantum walks on graphs with vertices of varying degree, thelatter offers us to choose, investigate, and compare quantum walks withdifferent shift operators. The synthesis of both key ingredients constitutes awell-suited playground for analyzing percolated quantum walks on a quitegeneral class of graphs. Analytical treatment of the asymptotic behavior ofpercolated quantum walks is presented and worked out in details for the Groverwalk on graphs with maximal degree 3. We find, that for these walks with cyclicshift operators the existence of an edge-3-coloring of the graph allows fornon-stationary asymptotic behavior of the walk. For different shift operatorsthe general structure of localized attractors is investigated, which determinesthe overall efficiency of a source-to-sink quantum transport across adynamically changing medium. As a simple nontrivial example of the theory wetreat a single excitation transport on a percolated cube.

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