We provide a simplified, yet rigorous presentation of the ideas fromBomb\'{i}n's paper "Gauge Color Codes" [arXiv:1311.0879v3]. Our presentation isself-contained, and assumes only basic concepts from quantum error correction.We provide an explicit construction of a family of color codes in arbitrarydimensions and describe some of their crucial properties. Within thisframework, we explicitly show how to transversally implement the generalizedphase gate $R_n=\text{diag}(1,e^{2\pi i/2^n})$, which deviates from the methodin "Gauge Color Codes", allowing an arguably simpler proof. We describe how toimplement the Hadamard gate $H$ fault-tolerantly using code switching. In threedimensions, this yields, together with the transversal $CNOT$, a fault-tolerantuniversal gate set $\{H,CNOT,R_3\}$ without state-distillation.
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机译：我们提供了Bomb {{}}论文“量规颜色代码” [arXiv：1311.0879v3]中的思想的简化但严格的介绍。我们的演示文稿是独立的，仅假设了量子误差校正的基本概念。我们提供了任意维度的颜色代码系列的显式构造，并描述了它们的一些关键特性。在此框架内，我们明确显示了如何横向实现广义相位门$ R_n = \ text {diag}（1，e ^ {2 \ pi i / 2 ^ n}）$，这与“仪表颜色代码”中的方法有所不同，可以说是更简单的证明。我们描述了如何使用代码切换来容错地实现Hadamard门$ H $。在三个维度上，这与横向$ CNOT $一起产生了容错通用门集$ \ {H，CNOT，R_3 \} $，而无需进行状态蒸馏。
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