Fault-tolerant weighted union-find decoding on the toric code

Quantum error correction requires decoders that are both accurate and efficient. To this end, union-find decoding has emerged as a promising candidate for error correction on the surface code. In this work, we benchmark a weighted variant of the union-find decoder on the toric code under circuit-level depolarizing noise. This variant preserves the almost-linear time complexity of the original while significantly increasing the performance in the fault-tolerance setting. In this noise model, weighting the union-find decoder increases the threshold from $0.38%$ to $0.62%$, compared to an increase from $0.65%$ to $0.72%$ when weighting a matching decoder. Further assuming quantum nondemolition measurements, weighted union-find decoding achieves a threshold of $0.76%$ compared to the $0.90%$ threshold when matching. We additionally provide comparisons of timing as well as low error rate behavior.