Four-component Schrödinger cat (4C) states are important physical resources for fault-tolerant quantum computing. However, the generation of 4C states in solid-state platforms remains challenging due to stringent nonlinearity requirements. We propose a Floquet-engineered scheme to generate magnonic four-component Schrödinger cat states in a hybrid superconducting-ferromagnetic system. Periodically driving the two superconducting qubits induces the effective conditional-displacement-type interactions between the magnon mode and the selected superconducting qubit sideband, where the coupling is mediated by virtual photon excitations in the microwave cavity. Notably, our scheme does not require higher-order nonlinearities of the magnon mode, and the relative phase symmetry of the 4C state's coherent-state constellation in phase space is fully controlled by the phase difference of the Floquet drives. Numerical simulations show that the high-fidelity 4C states can be generated even if the decoherence of the system is considered. These results provide a scalable route to multicomponent cat-state engineering in solid-state platforms and open additionally avenues for quantum computation.

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• Quantum state engineering