The Foliage Partition: An Easy-to-Compute LC-Invariant for Graph States

TL;DR

The paper presents the foliage partition, a computationally efficient LC-invariant for graph states that captures structural and marginal properties, with applications to orbit size and automorphism bounds.

Contribution

Introduces the foliage partition, a new LC-invariant for graph states that is easy to compute and applicable to weighted and qudit graphs.

Findings

Computational complexity is $ ext{O}(n^3)$ for $n$ qubits.

Foliage partition captures graph structure and 2-body marginals.

Bounds LC-orbit sizes and automorphisms.

Abstract

This paper introduces the foliage partition, an easy-to-compute LC-invariant for graph states, of computational complexity $\mathcal{O}(n^3)$ in the number of qubits. Inspired by the foliage of a graph, our invariant has a natural graphical representation in terms of leaves, axils, and twins. It captures both, the connection structure of a graph and the $2$-body marginal properties of the associated graph state. We relate the foliage partition to the size of LC-orbits and use it to bound the number of LC-automorphisms of graphs. We also show the invariance of the foliage partition when generalized to weighted graphs and qudit graph states.