The generator and quantum Markov semigroup for quantum walks

TL;DR

This paper develops a generator for quantum walks, extends them to continuous time, constructs the quantum Markov semigroup, and analyzes their limit distributions, providing new insights into their mathematical structure and behavior.

Contribution

It introduces a generator for quantum walks, extends discrete to continuous time, and constructs the quantum Markov semigroup with characterization and limit distribution analysis.

Findings

Derived a generator for quantum walk evolution.

Extended discrete quantum walks to continuous time.

Obtained limit distributions matching previous results.

Abstract

The quantum walks in the lattice spaces are represented as unitary evolutions. We find a generator for the evolution and apply it to further understand the walks. We first extend the discrete time quantum walks to continuous time walks. Then we construct the quantum Markov semigroup for quantum walks and characterize it in an invariant subalgebra. In the meanwhile, we obtain the limit distributions of the quantum walks in one-dimension with a proper scaling, which was obtained by Konno by a different method.