Graphical algorithms and threshold error rates for the 2d color code

Recent work on fault-tolerant quantum computation making use of topological errorcorrection shows great potential, with the 2d surface code possessing a threshold errorrate approaching 1% [1, 2]. However, the 2d surface code requires the use of a complexstate distillation procedure to achieve universal quantum computation. The color codeof [3] is a related scheme partially solving the problem, providing a means to performall Clifford group gates transversally. We review the color code and its error correctingmethodology, discussing one approximate technique based on graph matching. We derivean analytic lower bound to the threshold error rate of 6.25% under error-free syndromeextraction, while numerical simulations indicate it may be as high as 13.3%. Inclusion offaulty syndrome extraction circuits drops the threshold to approximately 0.10 ± 0.01%.