Generalize tensor cross-interpolation to looped tensor networks

Generalize tensor cross-interpolation (TCI) from tree tensor networks to tensor networks with looped topologies, such as projected entangled-pair states or multiscale entanglement-renormalization ansätze.

Background

The paper develops TCI for tree tensor networks (TTNs), whose acyclic structure permits efficient contractions and an interpolative gauge. The authors note that this construction relies substantially on the absence of loops in the underlying network graph.

Extending TCI to looped tensor networks would broaden the class of multivariate functions that can be represented and prepared, but looped geometries introduce contraction and gauge-structure difficulties absent from TTNs. The paper identifies this generalization as unresolved without providing a method or theorem addressing it.

References

These are suitable for our purposes due to their loop-free structure; it remains unclear how TCI could be generalized to tensor networks with looped topology.

— Multivariate quantum state preparation with optimized tensor networks  (2609.09304 - Sims-Goh et al., 8 Sep 2026) in Section 2.3, subsection “Tree tensor networks”