A Simple Method for Finding the Scattering Coefficients of Quantum Graphs

TL;DR

This paper introduces a straightforward method to compute the frequency-dependent scattering coefficients of quantum graphs, enabling efficient analysis of their responses and simplifying complex graph behaviors in quantum walks.

Contribution

The paper presents a novel, simple technique to determine scattering coefficients of quantum graphs using the characteristic polynomial of the time step operator.

Findings

Scattering coefficients can be expressed via the characteristic polynomial.

The method allows derivation of the impulse response for quantum graphs.

Enables rapid analysis and reduction of large quantum graphs.

Abstract

Quantum walks are roughly analogous to classical random walks, and like classical walks they have been used to find new (quantum) algorithms. When studying the behavior of large graphs or combinations of graphs it is useful to find the response of a subgraph to signals of different frequencies. In so doing we can replace an entire subgraph with a single vertex with frequency dependent scattering coefficients. In this paper a simple technique for quickly finding the scattering coefficients of any quantum graph will be presented. These scattering coefficients can be expressed entirely in terms of the characteristic polynomial of the graph's time step operator. Moreover, with these in hand we can easily derive the "impulse response" which is the key to predicting the response of a graph to any signal. This gives us a powerful set of tools for rapidly understanding the behavior of graphs or for reducing a large graph into its constituent subgraphs regardless of how they are connected.