Modifying quantum walks: A scattering theory approach

TL;DR

This paper develops a scattering theory framework for analyzing discrete-time quantum walks on directed Eulerian graphs with tails, enabling the study of graph modifications and comparisons through scattering matrices.

Contribution

It introduces a method to construct and analyze quantum walks on directed graphs using scattering matrices, including graph modifications and graph comparison techniques.

Findings

Scattering matrices relate to automorphic and reverse graph transformations.

Graph modifications like tail removal or edge addition alter the scattering matrix predictably.

Combined graphs' scattering matrices can be derived from individual components.

Abstract

We show how to construct discrete-time quantum walks on directed, Eulerian graphs. These graphs have tails on which the particle making the walk propagates freely, and this makes it possible to analyze the walks in terms of scattering theory. The probability of entering a graph from one tail and leaving from another can be found from the scattering matrix of the graph. We show how the scattering matrix of a graph that is an automorphic image of the original is related to the scattering matrix of the original graph, and we show how the scattering matrix of the reverse graph is related to that of the original graph. Modifications of graphs and the effects of these modifications are then considered. In particular we show how the scattering matrix of a graph is changed if we remove two tails and replace them with an edge or cut an edge and add two tails. This allows us to combine graphs, that is if we connect two graphs we can construct the scattering matrix of the combined graph from those of its parts. Finally, using these techniques, we show how two graphs can be compared by constructing a larger larger graph in which the two original graphs are in parallel, and performing a quantum walk on the larger graph. This is a kind of quantum walk interferometry.