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 non-demolition 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.

中文翻译：

---

复曲面代码上的容错加权联合查找解码

量子纠错要求解码器既准确又有效。为此，联合查找解码已经成为表面码纠错的有希望的候选者。在这项工作中，我们在电路级去极化噪声下，在复曲面代码上对联合查找解码器的加权变体进行基准测试。该变体保留了原件几乎线性的时间复杂度，同时显着提高了容错设置中的性能。在此噪声模型中，对联合查找解码器加权会增加$0.38％$ 至 $0.62％$，相比于 $0.65％$ 至 $0.72％$对匹配的解码器加权时。进一步假设量子不爆破测量，加权联合查找解码达到了阈值$0.76％$ 与 $0.90％$匹配时的阈值。我们还提供时序比较和低错误率行为。