Theoretical study of the Spectroscopic measurements of Kerr non-linear resonators with four-body interaction

Quantum annealing provides a promising way to solve combinational optimization problems where the solutions correspond to the ground state of the Ising Hamiltonian. We can implement quantum annealing using the Kerr non-linear resonators, with bifurcation phenomena emerging when subjected to a parametric drive. These bifurcated states can function as bases of qubits. Moreover, integrating four-body interactions between physical qubits enables the establishment of effective all-to-all long-range interactions between logical qubits, which is essential for practical quantum annealing. While theoretical proposals exist for creating four-body interactions within Kerr non-linear resonators, there has not been experimental verification through their spectroscopic signatures. In this paper, we theoretically investigate the spectroscopic measurements of Kerr non-linear resonators featuring four-body interaction. We identify six distinct frequencies exhibiting population changes by employing resonant driving on one resonator and weak driving on another. Analytical and numerical calculations validate these findings. Our study demonstrates the potential of spectroscopy in characterizing systems with four-body interactions, offering insights for realizing quantum annealing with Kerr parametric oscillators.