Reduction of branching graphs supporting continuous time return quantum walks

TL;DR

This paper shows that quantum walks on complex branching graphs can often be simplified to walks on linear chains, providing a new way to analyze and understand quantum walk dynamics on intricate structures.

Contribution

It introduces a method to reduce quantum walks on certain branching graphs to simpler linear chain graphs and discusses cases where this reduction is not possible.

Findings

Quantum walks on some branching graphs are equivalent to walks on linear chains.

A general approach for analyzing complex branching graphs is proposed.

An example with a cube graph demonstrates adjustable quantum walk solutions.

Abstract

We demonstrate that continuous time quantum walks on several types of branching graphs, including graphs with loops, are identical to quantum walks on simpler linear chain graphs. We also show graph types for which such equivalence does not exist. Several instructive examples are discussed, and a general approach to analyze more complex branching graphs is formulated. It is further illustrated with a return quantum walk solution for a cube graph with adjustable complex hopping amplitudes.