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Quantum annealing may provide advantages over simulated annealing on solvingsome problems such as Kth order binary optimization problem. No feasiblearchitecture exists to implement the high-order optimization problem (K > 2) oncurrent quantum annealing hardware. We propose a two-dimensional quantumannealing architecture to solve the 4th order binary optimization problem byencoding four-qubit interactions within the coupled local fields acting on aset of physical qubits. All possible four-body coupling terms for an N-qubitsystem can be implemented through this architecture and are readily realizablewith the existing superconducting circuit technologies. The overhead of thephysical qubits is O(N4), which is the same as previously proposedarchitectures in four-dimensional space. The equivalence between theoptimization problem Hamiltonian and the executable Hamiltonian is ensured by agauge invariant subspace of the experimental system. A scheme to realize localgauge constraint by single ancillary qubit is proposed.

2D implementation of quantum annealing algorisms for fourth order binary optimization problems