Local Equivalences of Graph States

• URL: http://arxiv.org/abs/2511.22271v1
• Date: Thu, 27 Nov 2025 09:49:57 GMT
• Title: Local Equivalences of Graph States
• Authors: Nathan Claudet,
• Abstract summary: Graph states form a large family of quantum states that are in one-to-one correspondence with mathematical graphs.<n>It is crucial to understand when two such states have the same entanglement, i.e. when they can be transformed into each other using only local operations.<n>We prove the existence of an infinite strict hierarchy of local equivalences between LC- and LU-equivalence.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Graph states form a large family of quantum states that are in one-to-one correspondence with mathematical graphs. Graph states are used in many applications, such as measurement-based quantum computation, as multipartite entangled resources. It is thus crucial to understand when two such states have the same entanglement, i.e. when they can be transformed into each other using only local operations. In this case, we say that the graph states are LU-equivalent (local unitary). If the local operations are restricted to the so-called Clifford group, we say that the graph states are LC-equivalent (local Clifford). Interestingly, a simple graph rule called local complementation fully captures LC-equivalence, in the sense that two graph states are LC-equivalent if and only if the underlying graphs are related by a sequence of local complementations. While it was once conjectured that two LU-equivalent graph states are always LC-equivalent, counterexamples do exist and local complementation fails to fully capture the entanglement of graph states. We introduce in this thesis a generalization of local complementation that does fully capture LU-equivalence. Using this characterization, we prove the existence of an infinite strict hierarchy of local equivalences between LC- and LU-equivalence. This also leads to the design of a quasi-polynomial algorithm for deciding whether two graph states are LU-equivalent, and to a proof that two LU-equivalent graph states are LC-equivalent if they are defined on at most 19 qubits. Furthermore, we study graph states that are universal in the sense that any smaller graph state, defined on any small enough set of qubits, can be induced using only local operations. We provide bounds and an optimal, probabilistic construction.

Related papers

• Deciding Local Unitary Equivalence of Graph States in Quasi-Polynomial Time [0.0]
  We describe an algorithm with quasi-polynomial runtime $nlog_2(n)+O(1)$ for deciding local unitary (LU) equivalence of graph states.<n>We show that LU-equivalence reduces to solving a system of quasi-polynomially many linear equations, avoiding an exponential blow-up.
  arXiv  Detail & Related papers  (2025-02-10T15:34:41Z)
• Algorithm to Verify Local Equivalence of Stabilizer States [0.0]
  We present an algorithm for verifying the local unitary equivalence of graph and stabilizer states.<n>Our approach reduces the problem to solving a system of linear equations in modular arithmetic.
  arXiv  Detail & Related papers  (2024-10-04T22:51:11Z)
• Local equivalence of stabilizer states: a graphical characterisation [0.0]
  A fundamental property of graph states is that applying a local complementation results in a graph that represents the same entanglement as the original.<n>This property served as the cornerstone for capturing non-trivial quantum properties in a simple graphical manner.<n>We introduce a generalisation of local complementation which graphically characterises the LU-equivalence of graph states.
  arXiv  Detail & Related papers  (2024-09-30T10:51:15Z)
• Distinguishing Graph States by the Properties of Their Marginals [0.0]
  We study equivalence relations between graph states under local unitaries (LU)<n>We show that these invariants uniquely identify the entanglement classes of every graph state up to 8 qubits.<n>We generalize tools to test for local Clifford (LC) equivalence of graph states that work by condensing large graphs into smaller graphs.
  arXiv  Detail & Related papers  (2024-06-14T12:03:10Z)
• The Foliage Partition: An Easy-to-Compute LC-Invariant for Graph States [0.0]
  This paper introduces the foliage partition, an easy-to-compute LC-invariant for graph states.<n>Inspired by the foliage of a graph, our invariant has a natural graphical representation in terms of leaves, axils, and twins.
  arXiv  Detail & Related papers  (2023-05-12T17:55:45Z)
• One-step Bipartite Graph Cut: A Normalized Formulation and Its Application to Scalable Subspace Clustering [56.81492360414741]
  We show how to enforce a one-step normalized cut for bipartite graphs, especially with linear-time complexity. In this paper, we first characterize a novel one-step bipartite graph cut criterion with normalized constraints, and theoretically prove its equivalence to a trace problem. We extend this cut criterion to a scalable subspace clustering approach, where adaptive anchor learning, bipartite graph learning, and one-step normalized bipartite graph partitioning are simultaneously modeled.
  arXiv  Detail & Related papers  (2023-05-12T11:27:20Z)
• Graphon Pooling for Reducing Dimensionality of Signals and Convolutional Operators on Graphs [131.53471236405628]
  We present three methods that exploit the induced graphon representation of graphs and graph signals on partitions of [0, 1]2 in the graphon space. We prove that those low dimensional representations constitute a convergent sequence of graphs and graph signals. We observe that graphon pooling performs significantly better than other approaches proposed in the literature when dimensionality reduction ratios between layers are large.
  arXiv  Detail & Related papers  (2022-12-15T22:11:34Z)
• Compilation of algorithm-specific graph states for quantum circuits [55.90903601048249]
  We present a quantum circuit compiler that prepares an algorithm-specific graph state from quantum circuits described in high level languages. The computation can then be implemented using a series of non-Pauli measurements on this graph state.
  arXiv  Detail & Related papers  (2022-09-15T14:52:31Z)
• Graph Spectral Embedding using the Geodesic Betweeness Centrality [76.27138343125985]
  We introduce the Graph Sylvester Embedding (GSE), an unsupervised graph representation of local similarity, connectivity, and global structure. GSE uses the solution of the Sylvester equation to capture both network structure and neighborhood proximity in a single representation.
  arXiv  Detail & Related papers  (2022-05-07T04:11:23Z)
• Finite-Function-Encoding Quantum States [52.77024349608834]
  We introduce finite-function-encoding (FFE) states which encode arbitrary $d$-valued logic functions. We investigate some of their structural properties.
  arXiv  Detail & Related papers  (2020-12-01T13:53:23Z)
• Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs [62.997667081978825]
  We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. We show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.
  arXiv  Detail & Related papers  (2020-08-11T17:58:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.