  We analyze in detail the so-called "pushing gate" for trapped ions,
introducing a time dependent harmonic approximation for the external motion. We
show how to extract the average fidelity for the gate from the resulting
semi-classical simulations. We characterize and quantify precisely all types of
errors coming from the quantum dynamics and reveal for the first time that
slight nonlinearities in the ion-pushing force can have a dramatic effect on
the adiabaticity of gate operation. By means of quantum optimal control
techniques, we show how to suppress each of the resulting gate errors in order
to reach a high fidelity compatible with scalable fault-tolerant quantum
computing.
