Generic quantum walks with memory on regular graphs

Quantum walks with memory (QWM) are a type of modified quantum walks that record the walker's latest path. As we know, only two kinds of QWM have been presented up to now. It is desired to design more QWM for research, so that we can explore the potential of QWM. In this work, by presenting the one-to-one correspondence between QWM on a regular graph and quantum walks without memory (QWoM) on a line digraph of the regular graph, we construct a generic model of QWM on regular graphs. This construction gives a general scheme for building all possible standard QWM on regular graphs and makes it possible to study properties of different kinds of QWM. Here, by taking the simplest example, which is QWM with one memory on the line, we analyze some properties of QWM, such as variance, occupancy rate, and localization.