Improved HDRG decoders for qudit and non-Abelian quantum error correction

TL;DR

This paper introduces an improved HDRG decoding method for qudit and non-Abelian quantum error correction, significantly increasing error thresholds and demonstrating superior performance in complex quantum systems with efficient runtime.

Contribution

The authors develop a novel HDRG decoding algorithm that enhances error correction thresholds and applies it to both Abelian and non-Abelian quantum models, including continuous error correction with imperfect measurements.

Findings

Increases minimal errors for logical failure from Θ(L^{2/3}) to Ω(L^{1-ε})

Outperforms previous HDRG decoders in simulations of quantum double and non-Abelian models

Achieves poly(log L) runtime for perfect measurements and O(1) for Abelian systems with imperfect measurements

Abstract

Hard-decision renormalization group (HDRG) decoders are an important class of decoding algorithms for topological quantum error correction. Due to their versatility, they have been used to decode systems with fractal logical operators, color codes, qudit topological codes, and non-Abelian systems. In this work, we develop a method of performing HDRG decoding which combines strenghts of existing decoders and further improves upon them. In particular, we increase the minimal number of errors necessary for a logical error in a system of linear size $L$ from $\Theta(L^{2/3})$ to $\Omega(L^{1-\epsilon})$ for any $\epsilon>0$. We apply our algorithm to decoding $D(\mathbb{Z}_d)$ quantum double models and a non-Abelian anyon model with Fibonacci-like fusion rules, and show that it indeed significantly outperforms previous HDRG decoders. Furthermore, we provide the first study of continuous error correction with imperfect syndrome measurements for the $D(\mathbb{Z}_d)$ quantum double models. The parallelized runtime of our algorithm is $\text{poly}(\log L)$ for the perfect measurement case. In the continuous case with imperfect syndrome measurements, the averaged runtime is $O(1)$ for Abelian systems, while continuous error correction for non-Abelian anyons stays an open problem.