History Dependent Quantum Walk on the Cycle with an Unbalanced Coin

TL;DR

This paper investigates a new history-dependent quantum walk model on cycle graphs, analyzing how its memory parameter and initial conditions influence its distribution, revealing that only a few parameters produce non-uniform behavior and establishing a relationship with other memory-based quantum walks.

Contribution

It provides the first detailed analysis of the recycled coin quantum walk on cycles, highlighting the effects of the memory parameter and initial state, and connecting it to other memory-based models.

Findings

Few parameters produce non-uniform distributions.

Initial state and cycle size modulo four significantly affect behavior.

Established a relationship between recycled coin and other memory-based quantum walks.

Abstract

Recently, a new model of quantum walk, utilizing recycled coins, was introduced; however little is yet known about its properties. In this paper, we study its behavior on the cycle graph. In particular, we will consider its time averaged distribution and how it is affected by the walk's "memory parameter" - a real parameter, between zero and eight, which affects the walk's coin flip operator. Despite an infinite number of different parameters, our analysis provides evidence that only a few produce non-uniform behavior. Our analysis also shows that the initial state, and cycle size modulo four all affect the behavior of this walk. We also prove an interesting relationship between the recycled coin model and a different memory-based quantum walk recently proposed.