Generic Quantum Walks with Memory

TL;DR

This paper introduces a general framework for quantum walks with memory on regular graphs, enabling systematic study of their properties and potential for new quantum walk models.

Contribution

It establishes a one-to-one correspondence between quantum walks with memory on regular graphs and quantum walks without memory on line digraphs, providing a comprehensive construction scheme.

Findings

Analyzed variance, occupancy rate, and localization of QWM with 1 memory on the line.

Provided a general method to construct all standard QWM on regular graphs.

Enabled exploration of properties of various QWM types.

Abstract

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 are 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, through presenting the one-to-one correspondence between QWM on a regular graph and quantum walks without memory(QWoM) on 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 1 memory on the line, we analyze some properties of QWM, such as variance, occupancy rate and localization.