Quantum symmetries of regular lattices

Determine whether regular lattices, including square lattices, possess genuinely quantum symmetries, specifically BQC quantum permutations that are not quantum controlled and act as quantum automorphisms of the lattice graph.

Background

The paper interprets BQC permutations as genuinely quantum graph symmetries and uses them to construct new symmetries of an Ising model on an explicitly designed graph. The authors note that almost all general graphs do not have quantum symmetries, whereas almost all trees do, making the existence of such symmetries for physically important graph families nontrivial.

Regular lattices, such as square lattices, are important models for statistical mechanics and physical field theories. Establishing whether they admit genuinely quantum symmetries would determine whether the paper's BQC-based extension of Ising-model symmetries applies to standard lattice geometries rather than only to specially constructed graphs.