Parametric instability landscape of coupled Kerr parametric oscillators

Networks of coupled Kerr parametric oscillators (KPOs) hold promise for the realization of neuromorphic and quantum computation. Yet, their rich bifurcation structure remains poorly understood. Here, we employ secular perturbation theory to map the stability regions of these networks and identify the regime where the system can be mapped to an Ising model. Starting with two coupled KPOs, we show how the bifurcations arise from the competition between the global parametric drive and linear coupling between the KPOs. We then extend this framework to larger networks with all-to-all equal coupling, deriving analytical expressions for the full cascade of bifurcation transitions. In the thermodynamic limit, we find that these transitions become uniformly spaced, leading to a highly regular structure. Our results reveal the precise bounds under which KPO networks have an Ising-like solution space, and thus provide crucial guidance for their experimental implementation.