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

Adam Burchardt; Frederik Hahn

doi:10.22331/q-2025-04-24-1720

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

¹QuSoft, CWI and University of Amsterdam, Science Park 123, 1098 XG Amsterdam, the Netherlands
²Electrical Engineering and Computer Science Department, Technische Universität Berlin, 10587 Berlin, Germany
³Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Published:	2025-04-24, volume 9, page 1720
Editor:	Daniel Grier
Eprint:	arXiv:2305.07645v2
Doi:	https://doi.org/10.22331/q-2025-04-24-1720
Citation:	Quantum 9, 1720 (2025).

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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.

Featured image: An example of the foliage partition for a graph with 25 vertices. Here, the foliage partition partitions these into five sets of equal size.

Popular summary

This paper introduces an elegant new way to analyze and classify quantum graph states through a concept called the foliage partition. Graph states are fundamental quantum resources that can be visualized as graphs, where vertices represent qubits and edges represent quantum operations between them.

While graph states are essential for quantum computing, determining when two different-looking graph states are essentially the same (LC-equivalent) has been computationally challenging. Previously, verifying this equivalence required analyzing an exponentially large set of properties, making it impractical for systems with many qubits.

We address this problem by introducing the foliage partition – a much more efficient way to group vertices in a graph based on how they connect to other vertices. This partition creates a fingerprint of the graph that remains unchanged under important quantum operations (local Clifford operations).

The foliage partition identifies three types of special structures within graphs:
1. Star-like formations where peripheral vertices connect only through a central point
2. Fully connected clusters where all vertices share identical neighborhoods
3. Completely disconnected sets where, again, all vertices share identical neighborhoods

What makes this approach powerful is its efficiency – the foliage partition can be computed in O(n³) time, vastly outperforming previous methods. We prove that this invariant works not just for qubit-based quantum systems but also for higher-dimensional qudit systems.

Beyond classification, the foliage partition reveals interesting connections to quantum entanglement. We show that a graph state has a trivial foliage partition (where each vertex forms its own group) if and only if its 2-body entanglement is maximally mixed – a property called 2-uniformity.

We further demonstrate how the foliage partition can be used to create more compact representations of graphs through “foliage graphs,” provide lower bounds on the number of non-equivalent graph states, and illuminate the symmetry properties of graph states.

► BibTeX data

@article{Burchardt2025foliagepartition,
  doi = {10.22331/q-2025-04-24-1720},
  url = {https://doi.org/10.22331/q-2025-04-24-1720},
  title = {The {F}oliage {P}artition: {A}n {E}asy-to-{C}ompute {LC}-{I}nvariant for {G}raph {S}tates},
  author = {Burchardt, Adam and Hahn, Frederik},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1720},
  month = apr,
  year = {2025}
}

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[1] Lina Vandré, Jarn de Jong, Frederik Hahn, Adam Burchardt, Otfried Gühne, and Anna Pappa, "Distinguishing graph states by the properties of their marginals", Physical Review A 111 5, 052449 (2025).

[2] Marcin Płodzień, Maciej Lewenstein, and Jan Chwedeńczuk, "Many-body quantum resources of graph states", Reports on Progress in Physics 88 7, 077601 (2025).

[3] Adam Burchardt, Jarn de Jong, and Lina Vandré, "Algorithm to Verify Local Equivalence of Stabilizer States", arXiv:2410.03961, (2024).

[4] Emma Hughes, William Munizzi, and Prineha Narang, "A Compact Framework for Analyzing Asynchronous Entanglement Distribution in Quantum Networks", arXiv:2507.22992, (2025).

[5] Derek Zhang, "Bell pair extraction using graph foliage techniques", Journal of Mathematical Physics 66 2, 022204 (2025).

[6] Nathan Claudet and Simon Perdrix, "Covering a Graph with Minimal Local Sets", arXiv:2402.10678, (2024).

[7] Konstantinos-Rafail Revis, Hrachya Zakaryan, and Zahra Raissi, "Orbit classification and analysis of qutrit graph states under local complementation and local scaling", arXiv:2506.05478, (2025).

[8] Romain Bourneuf, Nathan Claudet, Sang Yoon Kim, Rose McCarty, Blair D. Sullivan, and Stéphan Thomassé, "A combinatorial framework for clustering graph states: Algorithms and hardness for rank-integrity", arXiv:2607.09469, (2026).

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