  We describe a pipeline approach to decoding the surface code using minimum
weight perfect matching, including taking into account correlations between
detection events. An independent no-communication parallelizable processing
stage reweights the graph according to likely correlations, followed by another
no-communication parallelizable stage for high confidence matching. A later
general stage finishes the matching. This is a simplification of previous
correlated matching techniques which required a complex interaction between
general matching and re-weighting the graph. Despite this simplification, which
gives correlated matching a better chance of achieving real-time processing, we
find the logical error rate practically unchanged. We validate the new
algorithm on the fully fault-tolerant toric, unrotated, and rotated surface
codes, all with standard depolarizing noise. We expect these techniques to be
applicable to a wide range of other decoders.
