Universal transversal gates with color codes — a simplified approach

We present a simplified yet rigorous explanation of the ideas from Bomb́ın’s paper Gauge Color Codes [1]. Our discussion 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 Rm = diag(1, e 2iπ/2m), which deviates from the method in Ref. [1], allowing an arguably simpler verification. We describe how to implement the Hadamard gate H fault-tolerantly using code switching technique. In three dimensions, this yields, together with the transversal CNOT , a fault-tolerant universal gate set {H,CNOT,R3} without state-distillation.