Universal transversal gates with color codes: A simplified approach

We provide a simplified yet rigorous presentation of the ideas from Bomb\'{}n'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 Rₙ=diag(1,e^2πi/2ⁿ), 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,R₃ without state distillation.