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.