  We propose a scheme to implement high-fidelity conditional phase gates on
pair of trapped ions immersed in a two-dimensional Coulomb crystal, using
interaction mediated by all axial modes without side-band addressing. We show
through numerical calculations that only local modes can be excited to achieve
entangling gates through shaping the laser beams, so that the complexity of the
quantum gate does not increase with the size of the system. These results
suggest a promising approach for realization of large scale fault-tolerant
quantum computation in two dimensional traps architecture.
