We provide a simplified yet rigorous presentation of the ideas from Bombín's paper (arXiv:1311.0879v3). Our presentation is self-contained, and assumes only basic concepts from quantum error correction. We provide an explicit construction of a family of color codes in arbitrary dimensions and describe some of their crucial properties. Within this framework, we explicitly show how to transversally implement the generalized phase gate R_n =diag(1,e^(2πi/2^n)), which deviates from the method in the aforementioned paper, allowing an arguably simpler proof. We describe how to implement the Hadamard gate H fault tolerantly using code switching. In three dimensions, this yields, together with the transversal controlled-NOT (CNOT), a fault-tolerant universal gate set {H,cnot,R_3} without state distillation.
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