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