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

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.

21 pages, 10 figures. Both authors contributed equally to this work

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Cited by in corpus (2)

• Many-body quantum resources of graph states
• Distinguishing Graph States by the Properties of Their Marginals