Testing membership in general tensor-network-state classes

Construct a two-sided-error tester for membership in \(\mathcal S_d(G,\chi)\) whose copy complexity is polynomial in \(n\) and either \(\chi^{\tcw(G)}\) or \(\chi^{\cw(G)}\).

Background

The paper discusses existing property testers for MPS and TTN classes and explains why the paper’s rerouting reductions do not directly yield sound testers for a general graph class: grouping vertices into tree-cut bags can hide internal edge structure.

The unresolved problem is to test exact membership in the original general-graph tensor-network class, rather than membership in a larger class obtained after grouping or rerouting, with complexity controlled by structural graph parameters such as tree-cutwidth or cutwidth.

References

Is there a tester for membership in $\mathcal S_d(G,\chi)$, with two-sided error and copy complexity polynomial in $n$ and $\chi{\tcw(G)}$, or in $n$ and $\chi{\cw(G)}$?

— Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography  (2609.04165 - Caro et al., 3 Sep 2026) in Question “Testing tensor network structure,” subsection “TNS testing”