Comparison of fault-tolerant thresholds for planar qudit geometries. (arXiv:1701.02335v1 [quant-ph])

We introduce and analyze a new type of decoding algorithm called General
Color Clustering (GCC), based on renormalization group methods, to be used in
qudit color codes. The performance of this decoder is analyzed under code
capacity depolarizing noise, and is used to obtain the first fault-tolerant
threshold estimates for qudit 6-6-6 color codes. The proposed decoder is
compared with similar decoding schemes for qudit surface codes as well as the
current leading qubit decoders for both sets of codes. We find that, as with
surface codes, clustering performs sub-optimally for qubit color codes, giving
a threshold of $9.75\%$ compared to the $11.4\%$ obtained through surface
projection decoding methods. However, the threshold rate increases by up to
$75\%$ for large qudit dimensions, plateauing around $17.1\%$. All the analysis
is performed using QTop, a new open-source software for simulating and
visualizing topological quantum error correcting codes.

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