Localization of a multi-dimensional quantum walk with one defect

TL;DR

This paper extends quantum walk models to multiple dimensions with a defect, analyzes its spectral properties, and proves localization, linking it to a Schrödinger operator for deeper understanding.

Contribution

It introduces a multidimensional quantum walk with a defect, analyzes its spectrum, and proves localization using spectral mapping and the Feshbach map, advancing quantum walk theory.

Findings

Localization of the quantum walk is proven.

Spectral analysis reduces to a discrete Schrödinger operator.

The spectrum is characterized for the case of one defect.

Abstract

In this paper, we introduce a multidimensional generalization of Kitagawa's split-step discrete-time quantum walk, study the spectrum of its evolution operator for the case of one defect coins, and prove localization of the walk. Using a spectral mapping theorem, we can reduce the spectral analysis of the evolution operator to that of a discrete Schr\"{o}dinger operator with variable coefficients, which is analyzed using the Feshbach map.