  Ion trap is one of the most promising candidates for quantum computing.
Current schemes mainly focus on a linear chain of up to about one hundred ions
in a Paul trap. To further scale up the qubit number, one possible direction is
to use 2D or 3D ion crystals (Wigner crystals). In these systems, ions are
generally subjected to large micromotion due to the strong fast-oscillating
electric field, which can significantly influence the performance of entangling
gates. In this work, we develop an efficient numerical method to design
high-fidelity entangling gates in a general 3D ion crystal. We present
numerical algorithms to solve the equilibrium configuration of the ions and
their collective normal modes. We then give a mathematical description of the
micromotion and use it to generalize the gate scheme for linear ion chains into
a general 3D crystal. The involved time integral of highly oscillatory
functions is expanded into a fast-converging series for accurate and efficient
evaluation and optimization. As a numerical example, we show a high-fidelity
entangling gate design between two ions in a 100-ion crystal, with a
theoretical fidelity of 99.9\%.
