Notes on Inhomogeneous Quantum Walks

TL;DR

This paper investigates inhomogeneous discrete-time quantum walks, analyzing their spectral symmetry properties which resemble the Hofstadter butterfly, contributing to understanding complex quantum walk behaviors.

Contribution

It establishes symmetry properties of the spectrum of inhomogeneous quantum walks, revealing a structure similar to the Hofstadter butterfly, advancing theoretical understanding.

Findings

Spectral symmetry properties identified

Resemblance to Hofstadter butterfly demonstrated

Enhanced understanding of inhomogeneous quantum walks

Abstract

We study a class of discrete-time quantum walks with inhomogeneous coins defined in [Y. Shikano and H. Katsura, Phys. Rev. E {\bf 82}, 031122 (2010)]. We establish symmetry properties of the spectrum of the evolution operator, which resembles the Hofstadter butterfly.