Szegedy quantum walks with memory on regular graphs

Quantum walks with memory (QWM) are types of modified quantum walks that record the walker's latest path. The general model of coined QWM is presented in Li et al. (Phys Rev A 93:042323, 2016). In this paper, we present the general Szegedy QWM model and we describe its relationship with the coined QWM model. A coined QWM can be transformed into a Szegedy QWM, while a Szegedy QWM can be transformed into a coined QWM with any partition. These results may help in the analysis of the coined QWM. By transforming a coined QWM into a Szegedy QWM, the essential structure of the coined QWM is revealed. We give an example and we prove that two known QWMs are equal when they have a proper position-dependent coin operator.