# Comparison of memory thresholds for planar qudit geometries

## TL;DR

This paper introduces a new decoding algorithm for qudit color codes, analyzes its performance under noise, and estimates fault-tolerance thresholds, revealing significant improvements with larger qudit dimensions.

## Contribution

The paper presents the first fault-tolerant threshold estimates for qudit 6-6-6 color codes using a novel renormalization group-based decoder.

## Key findings

- Clustering performs sub-optimally for qubit color codes with a 9.75% threshold.
- Threshold increases by up to 75% for large qudit dimensions, reaching around 17.1%.
- The analysis utilizes the new open-source software QTop.

## Abstract

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.