Parametric Stability Analysis for Circuit Quantum Electrodynamical Hardwares

The transmon qubit, essential to quantum computation, exhibits disordered dynamics under strong parametric drives critical to its control. We present a combined theoretical and numerical study of stability regions in circuit QED using Floquet theory, focusing on the appearance of Arnold tongues that distinguish stable from unstable regimes. Starting from simple Josephson circuits and progressing to full multimode qubit-cavity systems, we show how time-dependent modulation maps the dynamics to Mathieu-type equations, revealing thresholds for parametric resonances. Perturbative corrections capture effects like higher harmonics and weak nonlinearities. Simulations validate these predictions and expose sensitivity to fabrication parameters. These findings inform thresholds for readout fidelity, amplifier gain, and multi-qubit gate stability.