Nonlinear Coupling between Motional Modes in Trapped Ion Quantum Processors

Trapped-ion crystals are a leading platform for quantum information science, but achieving the high-fidelity entangling gates required for fault-tolerant quantum computing becomes harder as system size increases. As systems scale, spectral crowding makes low-order nonlinear resonances between collective motional modes increasingly common and can limit gate performance, especially in monolithic or global-mode architectures. We develop a general model to identify and simulate nonlinear motional-mode coupling (NoMoCou) arising from third-order Coulomb terms and quantify its impact on the Molmer-Sorensen gate across linear chains and 2D crystals in rf and Penning traps. We delineate the regimes where NoMoCou dominates the error budget and provide design rules: detune operating points from low-order resonances, tune trap anisotropy to reshape spectra, and shape gate waveforms.