Universal transversal gates with color codes - a simplified approach

TL;DR

This paper presents a simplified, rigorous construction of color codes in arbitrary dimensions, demonstrating how to implement a universal set of fault-tolerant quantum gates transversally, enhancing quantum error correction methods.

Contribution

It provides a self-contained, explicit construction of color codes and introduces a simplified method for transversal implementation of the generalized phase gate.

Findings

Explicit construction of color codes in arbitrary dimensions

Simplified proof for transversal generalized phase gate implementation

Universal fault-tolerant gate set without state-distillation in 3D

Abstract

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