Universal transversal gates with color codes - a simplified approach

We provide a simplified, yet rigorous presentation of the ideas from Bombín's paper "Gauge Color Codes" [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\(1,e^2πi/2n\), which deviates from the method in "Gauge Color Codes", 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 CNOT, a fault-tolerant universal gate set \{H,CNOT,R₃\} without state-distillation.