Universal transversal gates with color codes: A simplified approach

We provide a simpli ed, yet rigorous presentation of the ideas from Bomb n’s paper Gauge Color
Codes [1]. 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; e2 i=2n
), which deviates from the
method in Ref. [1], 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 fH;CNOT;R3g without state-distillation.