Comparison of memory thresholds for planar qudit geometries

Understand

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.