Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography

• URL: http://arxiv.org/abs/2609.04165v1
• Date: Thu, 03 Sep 2026 17:50:38 GMT
• Title: Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography
• Abstract summary: We show that graph parameters determine whether a tensor-network state (TNS) admits a tractable matrix product state (MPS) or tree tensor network (TTN) representation, and which control the complexity of learning the state?<n>We derive graph-dependent upper bounds on the sample and computational complexity of realisable TNS tomography, with exponents that depend on cutwidth, tree-cutwidth, and a new graph parameter, learning complexity, which we bound in terms of degree and treewidth.
• Score: 0.6117371161379209
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Parameterised graph theory studies how the complexity of graph-theoretic problems depends on structural parameters of the input graph. This perspective has proved useful in analysing tensor-network simulation (Markov and Shi, 2008). Its implications for tensor-network representations and tomography are less well understood. In particular, which graph parameters determine whether a tensor-network state (TNS) admits a tractable matrix product state (MPS) or tree tensor network (TTN) representation, and which control the complexity of learning the state? We address these questions using parameterised graph theory. First, we show that cutwidth and tree-cutwidth bound the bond dimension overhead required to represent a TNS as an MPS or TTN. In the TTN case, tree-cutwidth also bounds the local dimension of the grouped subsystems. The proofs are based on entanglement rerouting, a tensor-network analogue of rerouting information in a classical network. Second, we derive graph-dependent upper bounds on the sample and computational complexity of realisable TNS tomography, with exponents that depend on cutwidth, tree-cutwidth, and a new graph parameter, learning complexity, which we bound in terms of degree and treewidth. We obtain these results by extending the disentangling MPS learner of (Cramer et al., 2010), as analysed further in (Bakshi et al., 2025; Lin et al., 2025), to TTNs and to tensor networks on arbitrary known graphs. Finally, we extend the framework beyond the realisable setting. For an arbitrary input state, our agnostic learner outputs a pure state whose fidelity is within additive error $ε$ of the optimum over tensor-network states on the given graph with a given bond dimension, with explicit graph-dependent bounds on sample and computational complexity.

Related papers

• Graph Edge Representation via Tensor Product Graph Convolutional Representation [23.021660625582854]
  This paper defines an effective convolution operator on graphs with edge features which is named as Product Graph Convolution (TPGC) It provides a complementary model to traditional graph convolutions (GCs) to address the more general graph data analysis with both node and edge features. Experimental results on several graph learning tasks demonstrate the effectiveness of the proposed TPGC.
  arXiv  Detail & Related papers  (2024-06-21T03:21:26Z)
• Tensor-view Topological Graph Neural Network [16.433092191206534]
  Graph neural networks (GNNs) have recently gained growing attention in graph learning. Existing GNNs only use local information from a very limited neighborhood around each node. We propose a novel Topological Graph Neural Network (TTG-NN), a class of simple yet effective deep learning. Real data experiments show that the proposed TTG-NN outperforms 20 state-of-the-art methods on various graph benchmarks.
  arXiv  Detail & Related papers  (2024-01-22T14:55:01Z)
• NodeFormer: A Scalable Graph Structure Learning Transformer for Node Classification [70.51126383984555]
  We introduce a novel all-pair message passing scheme for efficiently propagating node signals between arbitrary nodes. The efficient computation is enabled by a kernerlized Gumbel-Softmax operator. Experiments demonstrate the promising efficacy of the method in various tasks including node classification on graphs.
  arXiv  Detail & Related papers  (2023-06-14T09:21:15Z)
• Efficient tensor network simulation of quantum many-body physics on sparse graphs [0.0]
  We study tensor network states defined on an underlying graph which is sparsely connected. We find that message-passing inference algorithms can lead to efficient computation of local expectation values.
  arXiv  Detail & Related papers  (2022-06-09T18:00:03Z)
• Learning Graph Structure from Convolutional Mixtures [119.45320143101381]
  We propose a graph convolutional relationship between the observed and latent graphs, and formulate the graph learning task as a network inverse (deconvolution) problem. In lieu of eigendecomposition-based spectral methods, we unroll and truncate proximal gradient iterations to arrive at a parameterized neural network architecture that we call a Graph Deconvolution Network (GDN) GDNs can learn a distribution of graphs in a supervised fashion, perform link prediction or edge-weight regression tasks by adapting the loss function, and they are inherently inductive.
  arXiv  Detail & Related papers  (2022-05-19T14:08:15Z)
• Graph Kernel Neural Networks [53.91024360329517]
  We propose to use graph kernels, i.e. kernel functions that compute an inner product on graphs, to extend the standard convolution operator to the graph domain. This allows us to define an entirely structural model that does not require computing the embedding of the input graph. Our architecture allows to plug-in any type of graph kernels and has the added benefit of providing some interpretability.
  arXiv  Detail & Related papers  (2021-12-14T14:48:08Z)
• Spectral Graph Convolutional Networks With Lifting-based Adaptive Graph Wavelets [81.63035727821145]
  Spectral graph convolutional networks (SGCNs) have been attracting increasing attention in graph representation learning. We propose a novel class of spectral graph convolutional networks that implement graph convolutions with adaptive graph wavelets.
  arXiv  Detail & Related papers  (2021-08-03T17:57:53Z)
• Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting [63.04999833264299]
  "Graph Substructure Networks" (GSN) is a topologically-aware message passing scheme based on substructure encoding. We show that it is strictly more expressive than the Weisfeiler-Leman (WL) graph isomorphism test. We perform an extensive evaluation on graph classification and regression tasks and obtain state-of-the-art results in diverse real-world settings.
  arXiv  Detail & Related papers  (2020-06-16T15:30:31Z)
• Tensor Graph Convolutional Networks for Multi-relational and Robust Learning [74.05478502080658]
  This paper introduces a tensor-graph convolutional network (TGCN) for scalable semi-supervised learning (SSL) from data associated with a collection of graphs, that are represented by a tensor. The proposed architecture achieves markedly improved performance relative to standard GCNs, copes with state-of-the-art adversarial attacks, and leads to remarkable SSL performance over protein-to-protein interaction networks.
  arXiv  Detail & Related papers  (2020-03-15T02:33:21Z)
• Graph Signal Processing -- Part III: Machine Learning on Graphs, from Graph Topology to Applications [19.29066508374268]
  Part III of this monograph starts by addressing ways to learn graph topology. A particular emphasis is on graph topology definition based on the correlation and precision matrices of the observed data. For learning sparse graphs, the least absolute shrinkage and selection operator, known as LASSO is employed.
  arXiv  Detail & Related papers  (2020-01-02T13:14:27Z)

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.