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Paper 1

Universal transversal gates with color codes - a simplified approach

Aleksander Kubica, Michael E. Beverland

Year
2014
Journal
arXiv preprint
DOI
arXiv:1410.0069
arXiv
1410.0069

We provide a simplified, yet rigorous presentation of the ideas from Bombín's paper "Gauge Color Codes" [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 $R_n=\text{diag}(1,e^{2πi/2^n})$, which deviates from the method in "Gauge Color Codes", 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 $\{H,CNOT,R_3\}$ without state-distillation.

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Paper 2

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