Two growth steps for weighted UF on a toric code decoder graph with p = 0.8 % and weights truncated to the nearest integer (for performance estimates, we truncate to the nearest tenth). Some edges are omitted for clarity. In this case, we have two types of edges: weight four edges (thick) in the cardinal directions, and weight five diagonal edges (thin). The orange (light) highlight indicates the growing cluster. In the top figure, a single excitation occurs in the corner of the decoder graph. In the middle, the cluster radius grows by four and it merges with clusters to the north, west, and up directions. At the bottom, the cluster radius grows by one and it merges with clusters diagonal to the original excitation.
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 match-ing 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 comparison sof timing as well as low error rate behavior. doi: 10.1103/PhysRevA.102.012419
Shilin Huang, Michael Newman, and Kenneth R. Brown
