Self-avoiding quantum walks: realisations in subspaces and limit theorems

TL;DR

This paper investigates three types of two-dimensional self-avoiding quantum walks, analyzing their properties and limit distributions, revealing delocalization effects and providing analytical results for their evolution.

Contribution

It introduces and analyzes three novel self-avoiding quantum walk models in 2D, with analytical results on their limit distributions and delocalization behavior.

Findings

All three walks lead to delocalization from localized initial states.

Analytical limit distributions are derived for walks in coin space and combined coin-position space.

Self-avoidance in subspaces results in non-trivial, delocalized quantum dynamics.

Abstract

While completely self-avoiding quantum walks have the distinct property of leading to a trivial unidirectional transport of a quantum state, an interesting and non-trivial dynamics can be constructed by restricting the self-avoidance to a subspace of the complete Hilbert space. Here, we present a comprehensive study of three two-dimensional quantum walks, which are self-avoiding in coin space, in position space and in both, coin and position space. We discuss the properties of these walks and show that all result in delocalisation of the probability distribution for initial states which are strongly localised for a walk with a standard Grover coin operation. We also present analytical results for the evolution in the form of limit distributions for the self-avoiding walks in coin space and in both, coin and position space.