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

The paper applies parameterised graph theory to tensor networks, showing that cutwidth and tree‑cutwidth bound the bond‑dimension overhead needed to represent a tensor‑network state as a matrix product state or tree tensor network. It derives graph‑dependent upper bounds on the sample and computational complexity of tensor‑network tomography, introducing a new graph parameter called learning complexity. Finally, it extends the framework to an agnostic learner that approximates any state with a tensor‑network state of given bond dimension, providing explicit graph‑dependent complexity bounds.

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