We provide a simpli ed, yet rigorous presentation of the ideas from Bomb n’s paper Gauge Color
\nCodes [1]. Our presentation is self-contained, and assumes only basic concepts from quantum error
\ncorrection. We provide an explicit construction of a family of color codes in arbitrary dimensions
\nand describe some of their crucial properties. Within this framework, we explicitly show how to
\ntransversally implement the generalized phase gate Rn = diag(1; e2 i=2n
\n), which deviates from the
\nmethod in Ref. [1], allowing an arguably simpler proof. We describe how to implement the Hadamard
\ngate H fault-tolerantly using code switching. In three dimensions, this yields, together with the
\ntransversal CNOT, a fault-tolerant universal gate set fH;CNOT;R3g without state-distillation.<\/p>\n","protected":false},"excerpt":{"rendered":"
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. 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