Constructing non-Abelian quantum spin liquids using combinatorial gauge symmetry

We construct Hamiltonians with only 1- and 2-body interactions that exhibit an exact non-Abelian gauge symmetry (specifically, combinatiorial symmetry). Our spin Hamiltonian realizes the quantum double associated to group of quaternions. It contains ferromagnetic anti-ferromagnetic ZZ ZZ interactions, plus longitudinal transverse fields, therefore is explicit example a no sign problem topological phase. In addition model, we propose superconducting circuit version same symmetry.