Ising model equivalence of small networks of coupled parametric oscillators

Coupled Kerr parametric oscillators (KPOs) are often considered an ideal platform for analog simulation of Ising spin networks. Quantum-mechanical implementations of such a simulator could allow to solve general NP-hard problems more efficiently by mapping them onto an Ising model and finding its ground state. However, already two classical coupled KPOs have been found to display a variety of complex dynamics beyond the basic Ising equivalence, showing the need for a careful and detailed understanding of simple networks before moving to complicated applications.

We study the Ising equivalence of two or three coupled KPOs using classical, inductively coupled non-linear RLC circuits. Measuring the response dynamics under different driving conditions allows us to map the number of stationary oscillation solutions and their characteristics. We find that even though the solution space differs widely from the Ising model close to the parametric threshold, the equivalence recovers for higher parametric drives or weaker couplings.

Our results bring us closer to a full understanding of the requirements and restrictions for operating large, coherently coupled KPO networks as Ising simulators, opening up the possibility for further research and transfer to quantum-mechanical systems.