Simulating the 3-State Potts Hamiltonian with a Coherent Network of Photonic Oscillators

Recently, there has been much interest in physical simulation of spin models. This is largely driven by the fact that many important combinatorial optimization problems that are considered computationally hard problems can be formulated in terms of lattice spin models. These optimization problems can thus be efficiently solved by finding the ground state of the corresponding physical system. Here, we propose direct optical implementation of a three-state Potts spin model by using networks of coupled three-photon parametric down-conversion oscillators. The tristable phase of such a parametric oscillator acts like a three-state classical spin degree of freedom. We show that a dissipatively coupled network of such oscillators emulates the three-state classical Potts model and can be used as an all-optical solver for optimization problems such as MAX-3-CUT and graph 3-COL. The results can be further generalized to implement the n-state planar Potts model.