with new schedule functions At, Bt and Ct, and transverse fields ai. Physical spins are arranged on a square lattice (see blue spheres in Fig. 1), where the index i labels the entries of the matrix . The number of physical spins K equals the number of connections in the original model, which is quadratically larger than in the original problem, for example, K=N(N−1)/2 for all-to-all connected graphs. This enlarged state space contains states, where collections of physical spins encode conflicting relative orientations of the logical spins. These states can be locally identified and energetically penalized by four-body constraints H□ at each plaquette □ of the square lattice, such that at the end of the sweep plaquettes either contain all an even12, or all an odd13 number of spins in the state, thus realising an even or odd parity representation of equation (2). This ensures that the final ground state of the LHZ Hamiltonian (2) corresponds to the final ground state of the logical Hamiltonian (1), and thus to the optimal solution of the optimization problem. Importantly, the optimization problem is now encoded in local fields , corresponding to single-particle energy shifts. We show that the above model for a programmable quantum annealer can be emulated in an atomic Rydberg setup, which builds on the remarkable recent advances towards realizing complex spin models with cold atoms in lattices interacting via designed Rydberg–Rydberg interactions1415161718.