  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.
