Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography
2026-09-03 • Data Structures and Algorithms
Data Structures and AlgorithmsMachine Learning
AI summaryⓘ
The authors use ideas from parameterised graph theory to understand how difficult it is to represent and learn tensor-network states (TNS) based on characteristics of the underlying graph. They show that two graph parameters, cutwidth and tree-cutwidth, help determine how complex MPS and TTN representations of TNS need to be. They also provide bounds on how many samples and how much computation is needed to learn these states, linking these to graph properties like degree and treewidth. Finally, they present a learning method that works even when the state does not exactly match the assumed tensor-network model, guaranteeing a close approximation with explicit complexity bounds.
parameterised graph theorytensor-network statesmatrix product statetree tensor networkcutwidthtree-cutwidthtreewidthbond dimensiontensor-network tomography
Authors
Matthias C. Caro, Natalie McHugh, Sergii Strelchuk
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.