Reduction of branching graphs supporting continuous time return quantum walks

We demonstrate that continuous time quantum walks on several types of branching graphs, including graphs with loops, are identical to quantum walks on simpler linear chain graphs. We also show graph types for which such equivalence does not exist. Several instructive examples are discussed, and a general approach to analyze more complex branching graphs is formulated. It is further illustrated with a return quantum walk solution for a cube graph with adjustable complex hopping amplitudes.