Spin representation of bifurcation-based quantum annealing with Kerr parametric oscillators

A scheme using parametrically driven oscillators with Kerr nonlinearity (Kerr parametric oscillators, KPOs) was proposed as a method of adiabatic quantum optimization, also called bifurcation-based quantum annealing. In this scheme a KPO with gradually increasing pump field generates the cat state, which is interpreted as superposition of up and down states of an Ising spin. In this process for interacting KPOs the interaction makes each KPO select one of the two coherent states, namely the up or down state, as the ground state at the final point of the process. The resulting state corresponds to a solution of combinatorial optimization problems. The correspondence to spin configurations is clarified at the final point, while possible spin representation is unclear in the middle of the process, where the solution is searched for by the system. We transform interacting KPOs to spin systems and discuss the scheme in terms of the spin representation. We also investigate the obtained spin systems and compare them to the original KPOs to evaluate the validity of our transformation.