Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decoding techniques, while possibly having similar logical error rates, can perform differently on the same error syndrome. We use machine learning techniques to assign a given error syndrome to the decoder which is likely to decode it correctly. We apply our framework to an ensemble of Minimum-Weight Perfect Matching (MWPM) and Hard-Decision Re-normalization Group (HDRG) decoders for the surface code in the depolarizing noise model. Our simulations show an improvement of 38.4%, 14.6%, and 7.1% over the pseudo-threshold of MWPM in the instance of distance 5, 7, and 9 codes, respectively. Lastly, we discuss the advantages and limitations of our framework and applicability to other error-correcting codes. Our framework can provide a significant boost to error correction by combining the strengths of various decoders. In particular, it may allow for combining very fast decoders with moderate error-correcting capability to create a very fast ensemble decoder with high error-correcting capability.

In this paper, we study how to draw Halin-graphs, i.e., planar graphs that consist of a tree T and a cycle among the leaves of that tree. Based on tree-drawing algorithms and the pathwidth pw(T ), a well-known graph parameter, we find poly-line drawings of height at most 6pw(T ) + 3 ∈ O(log n). We also give an algorithm for straight-line drawings, and achieve height at most 12pw(T ) + 1 for Halin-graphs, and smaller if the Halin-graph is cubic. We show that the height achieved by our algorithms is optimal in the worst case (i.e. for some Halin-graphs).

In this paper, we study how to draw Halin-graphs, i.e., planar graphs that consist of a tree $T$ and a cycle among the leaves of that tree. Based on tree-drawing algorithms and the pathwidth $pw(T) $, a well-known graph parameter, we find poly-line drawings of height at most $6pw(T)+3\in O(\log n)$. We also give an algorithm for straight-line drawings, and achieve height at most $12pw(T)-1$ for Halin-graphs, and smaller if the Halin-graph is cubic. We show that the height achieved by our algorithms is optimal in the worst case (i.e. for some Halin-graphs).