Limit Theorems for the Discrete-Time Quantum Walk on a Graph with Joined Half Lines

TL;DR

This paper derives limit theorems for a discrete-time quantum walk on a graph composed of joined half lines, revealing localization and asymptotic behaviors, and extends analysis to asymmetric initial states.

Contribution

It introduces an enlarged basis for quantum walks, enabling reduction to half-line walks even with asymmetric initial states, and establishes new limit theorems for these walks.

Findings

Demonstrates localization and oscillatory asymptotic behavior.

Proves weak convergence to a density function with scaling order t.

Applies results to general initial states and various coin operators.

Abstract

We consider a discrete-time quantum walk $W_{t,\kappa}$ at time $t$ on a graph with joined half lines $\mathbb{J}_\kappa$, which is composed of $\kappa$ half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with \textit{symmetric} initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that $W_{t,\kappa}$ can be reduced to the walk on a half line even if the initial state is \textit{asymmetric}. For $W_{t,\kappa}$, we obtain two types of limit theorems. The first one is an asymptotic behavior of $W_{t,\kappa}$ which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for $W_{t,\kappa}$. On each half line, $W_{t,\kappa}$ converges to a density function like the case of the one-dimensional lattice with a scaling order of $t$. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.