Engineering the Nonlinearity of Bosonic Modes with a Multi-loop SQUID

Engineering high-order nonlinearities while suppressing lower-order terms is crucial for quantum error correction and state control in bosonic systems, yet it remains an outstanding challenge. Here, we introduce a general framework of Nonlinearity-Engineered Multi-loop SQUID (NEMS) device, enabling the realization of arbitrary nonlinearities by tuning fluxes in multiple loops within superconducting circuits. We demonstrate specific examples of NEMS devices that selectively engineer pure cubic, quartic, and quintic interactions with suppressed parasitic couplings, showing great promise for realizing Kerr-cat bias-preserving {\scshape cnot} gates and stabilizing four-leg cat qubits. By opening new avenues for tailoring nonlinear Hamiltonians of superconducting devices, this work enables sophisticated and precise manipulation of bosonic modes, with potential applications in quantum computation, simulation, and sensing.