A superconducting circuit realization of combinatorial gauge symmetry

We propose a superconducting wire array that realizes a family of quantum Hamiltonians that possess combinatorial gauge symmetry — a local symmetry where monomial transformations play a central role. This physical system exhibits a rich structure. In the classical limit its ground state consists of two superimposed spin liquids; one is a crystal of small loops containing disordered U(1) degrees of freedom, and the other is a soup of loops of all sizes associated to Z2 topological order. We show that the classical results carry over to the quantum case when fluctuations are gradually tuned via the wire capacitances, yielding Z2 quantum topological order. In an extreme quantum limit where the capacitances are all small, we arrive at an effective quantum spin Hamiltonian that we conjecture would sustain Z2 quantum topological order with a gap of the order of the Josephson coupling in the array. The principles behind the construction for superconducting arrays extends to other bosonic and fermionic systems, and offers a promising path towards topological qubits and the study of other many-body systems.