  Tensor networks provide succinct representations of quantum many-body states
and are an important computational tool for strongly correlated quantum
systems. Their expressive and computational power is characterized by an
underlying entanglement structure, on a lattice or more generally a
(hyper)graph, with virtual entangled pairs or multipartite entangled states
associated to (hyper)edges. Changing this underlying entanglement structure
into another can lead to both theoretical and computational benefits. We study
a natural resource theory which generalizes the notion of bond dimension to
entanglement structures using multipartite entanglement. It is a direct
extension of resource theories of tensors studied in the context of
multipartite entanglement and algebraic complexity theory, allowing for the
application of the sophisticated methods developed in these fields to tensor
networks. The resource theory of tensor networks concerns both the local
entanglement structure of a quantum many-body state and the (algebraic)
complexity of tensor network contractions using this entanglement structure. We
show that there are transformations between entanglement structures which go
beyond edge-by-edge conversions, highlighting efficiency gains of our resource
theory that mirror those obtained in the search for better matrix
multiplication algorithms. We also provide obstructions to the existence of
such transformations by extending a variety of methods originally developed in
algebraic complexity theory for obtaining complexity lower bounds. The resource
theory of tensor networks allows to compare different entanglement structures
and should lead to more efficient tensor network representations and
contraction algorithms.
