  We present an efficient approach to optimising pulse sequences for
implementing fast entangling two-qubit gates on trapped ion quantum information
processors. We employ a two-phase procedure for optimising gate fidelity, which
we demonstrate for multi-ion systems in linear Paul trap and microtrap
architectures. The first phase involves a global optimisation over a
computationally inexpensive cost function constructed under strong
approximations of the gate dynamics. The second phase involves local
optimisations that utilise a more precise ODE description of the gate dynamics,
which captures the non-linearity of the Coulomb interaction and the effects of
finite laser repetition rate. We propose two novel gate schemes that are
compatible with this approach, and we demonstrate that they outperform existing
schemes in terms of achievable gate speed and fidelity for feasible laser
repetition rates. In optimising sub-microsecond gates in microtrap
architectures, the proposed schemes achieve orders of magnitude higher
fidelities than previous proposals. Finally, we investigate the impact of pulse
imperfections on gate fidelity and evaluate error bounds for a range of gate
speeds.
