The foliage partition: An easy-to-compute lc-invariant for graph states

This paper introduces the foliage partition, an easy-to-compute LC-invariant for graph states, of computational complexity O(n3) 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.

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