Quantum Graph Theory by Example

TL;DR

This paper introduces a collection of non-trivial quantum graphs with a diagrammatic formalism, enabling analytical computation of key graph parameters and revealing classical-quantum structural decompositions.

Contribution

It presents a parametric family of quantum graphs with explicit formulas for important parameters, bridging classical and quantum graph theory insights.

Findings

Explicit formulas for quantum graph parameters like chromatic number and independence number.

Decomposition of quantum graphs into classical and quantum components.

First large parametric family of quantum graphs with analytically computable parameters.

Abstract

Quantum graphs have been introduced by Duan, Severini, and Winter to describe the zero-error behaviour of quantum channels. Since then, quantum graph theory has become a field of study in its own right. A substantial source of difficulty in working with quantum graphs compared to classical graphs stems from the fact that they are no longer discrete objects. This makes it generally difficult to construct insightful, non-trivial examples. We present a collection of non-trivial quantum graphs that can be thought of in discrete terms, and that can be expressed in the diagrammatic formalism introduced by Musto, Reutter, and Verdon. The examples arise as the quantum graphs acted on by increasingly smaller classical matrix groups, and are parametrised by triples of matrices $(A, B, C)$. The parametrisation reveals a clean decomposition of quantum graph structure into classical and genuinely quantum components: $A$ and $C$ are described by a classical weighted graph called the strange graph, while $B$ provides a purely quantum contribution with no classical analogue. Based on this model, we give exact formulas or establish bounds for quantum graph parameters, such as the number of connected components, the chromatic number, the independence number, and the clique number. Our results provide the first large, parametric families of quantum graphs for which standard graph parameters can be computed analytically.