  We present a robust quantum optimal control framework for implementing fast
entangling gates on ion-trap quantum processors. The framework leverages
tailored laser pulses to drive the multiple vibrational sidebands of the ions
to create phonon-mediated entangling gates and, unlike the state of the art,
requires neither weak-coupling Lamb-Dicke approximation nor perturbation
treatment. With the application of gradient-based optimal control, it enables
finding amplitude- and phase-modulated laser control protocols that work beyond
the Lamb-Dicke regime, promising gate speed at the order of microseconds
comparable to the characteristic trap frequencies. Also, robustness
requirements on the temperature of the ions and initial optical phase can be
conveniently included to pursue high-quality fast gates against experimental
imperfections. Our approach represents a step in speeding up quantum gates to
achieve larger quantum circuits for quantum computation and simulation, and
thus can find applications in near-future experiments.
