State-level bond-two obstruction for kagome and triangular lattices

Determine whether the graph states on kagome and triangular lattices, after a nearest-neighbor coherent perturbation $e^{-i\varepsilon X_uX_v}$ with nonzero generic $\varepsilon$, admit any PEPS representation with bond dimension at most two on the same connectivity graph, and, if not, prove a state-level exclusion analogous to the honeycomb and square-torus Schmidt-rank obstructions.

Background

The paper proves that on sufficiently large honeycomb and square tori, the perturbed state $e^{-i\varepsilon X_uX_v}G$ has no bond-two PEPS representation on the same graph for $0<|\varepsilon|<\pi/2$. These results are state-level statements and do not rely solely on the tangent-space obstruction.

For kagome and triangular lattices, the paper establishes that the nearest-neighbor $XX$ direction is absent from the tangent image at the graph-state point, but the matching-cut Schmidt argument cannot be applied because every edge lies in a triangle. Consequently, it remains unresolved whether a different bond-two representation might represent the perturbed state.