An algebraic structure for one-dimensional quantum walks and a new proof of the weak limit theorem

TL;DR

This paper introduces an algebraic framework for one-dimensional quantum walks, enabling a new proof of the weak limit theorem by deriving an effective characteristic function formula and utilizing Chebyshev polynomial properties.

Contribution

It presents a novel algebraic structure that characterizes quantum walks and provides a simplified proof of the weak limit theorem.

Findings

Derived an effective formula for the characteristic function of transition probabilities

Provided a new proof of the weak limit theorem using algebraic and polynomial properties

Enhanced understanding of quantum walk behavior through algebraic characterization

Abstract

An algebraic structure for one-dimensional quantum walks is introduced. This structure characterizes, in some sense, one-dimensional quantum walks. A natural computation using this algebraic structure leads us to obtain an effective formula for the characteristic function of the transition probability. Then, the weak limit theorem for the transition probability of quantum walks is deduced by using simple properties of the Chebyshev polynomials.