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arXiv · 2305.07645

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

Adam Burchardt·Frederik Hahn
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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.

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Adam Burchardt, Frederik Hahn. 2023-05-12. The Foliage Partition: An Easy-to-Compute LC-Invariant for Graph States. https://doi.org/10.22331/q-2025-04-24-1720

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