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We propose a scheme to implement high-fidelity conditional phase gates onpair of trapped ions immersed in a two-dimensional Coulomb crystal, usinginteraction mediated by all axial modes without side-band addressing. We showthrough numerical calculations that only local modes can be excited to achieveentangling gates through shaping the laser beams, so that the complexity of thequantum gate does not increase with the size of the system. These resultssuggest a promising approach for realization of large scale fault-tolerantquantum computation in two dimensional traps architecture.

Implementation of local and high-fidelity quantum conditional phase gates in a scalable two-dimensional ion trap