High-fidelity state transfer via quantum walks from delocalized states

We study the state transfer through quantum walks placed on a bounded one-dimensional path. We first consider continuous-time quantum walks from a Gaussian state. We find such a state when superposing centered on the starting and antipodal positions preserves a high fidelity for a long time and when sent on large circular graphs. Furthermore, it spreads with a null group velocity. We also explore discrete-time quantum walks to evaluate the qubit fidelity throughout the walk. In this case, the initial state is a product of states between a qubit and a Gaussian superposition of position states. Then, we add two sigma x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _x$$\end{document} gates to confine this delocalized qubit. We also find that this bounded system dynamically enables periodic recovery of the initial separable state. We outline some applications of our results in dynamic graphs and propose quantum circuits to implement them based on the available literature.