A quantum walk with a delocalized initial state: contribution from a coin-flip operator

TL;DR

This paper investigates the long-term behavior of a 2-state quantum walk starting from a delocalized initial state, revealing how the initial state and coin-flip operator influence the walk's limit distribution.

Contribution

It introduces a new analysis of quantum walks with delocalized initial states, extending previous studies focused on localized states, and derives their limit distributions.

Findings

Limit distributions differ from localized initial state cases.

The initial state from Fourier series expansion affects the limit density functions.

The coin-flip operator significantly influences the walk's asymptotic behavior.

Abstract

A unit evolution step of discrete-time quantum walks is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the quantum walks starting with a localized initial state have been derived. In the present paper, we compute limit distributions of a 2-state quantum walk with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the $(\alpha,\beta)$ delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.