Mapping graph state orbits under local complementation

TL;DR

This paper explores the structure of graph state orbits under local complementation up to 9 qubits, revealing complex relationships between graph properties and quantum entanglement, and providing computational tools for analysis.

Contribution

It maps the orbits generated by local complementation for small qubit systems, revealing hidden structures and correlations with entanglement measures, along with tools for visualization and analysis.

Findings

Identified rich structure in graph state orbits up to 9 qubits

Linked graph connectivity with entanglement properties

Provided tools for orbit computation and visualization

Abstract

Graph states, and the entanglement they posses, are central to modern quantum computing and communications architectures. Local complementation---the graph operation that links all local-Clifford equivalent graph states---allows us to classify all stabiliser states by their entanglement. Here, we study the structure of the orbits generated by local complementation, mapping them up to 9 qubits and revealing a rich hidden structure. We provide programs to compute these orbits, along with our data for each of the 587 orbits up to 9 qubits and a means to visualise them. We find direct links between the connectivity of certain orbits with the entanglement properties of their component graph states. Furthermore, we observe the correlations between graph-theoretical orbit properties, such as diameter and colourability, with Schmidt measure and preparation complexity and suggest potential applications. It is well known that graph theory and quantum entanglement have strong interplay---our exploration deepens this relationship, providing new tools with which to probe the nature of entanglement.