Universal transversal gates with color codes: A simplified approach

Abstract

We provide a simplified yet rigorous presentation of the ideas from Bomb\'{\i}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 ${R}_{n}=\text{diag}(1,{e}^{2\ensuremath{\pi}i/{2}^{n}})$, 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.