Short answer: Belief propagation with ordered statistics decoding, BP+OSD, is a two-stage decoder for quantum low-density parity-check codes. Belief propagation runs first and ordered statistics post-processing handles cases where it fails. A 2020 Physical Review Research paper simulates it on hypergraph product codes and reports a toric-code threshold of 9.9 plus or minus 0.2 percent, ahead of earlier belief-propagation decoders. These are numerical simulation results.
The decoder
A 2020 paper shows that belief propagation combined with ordered statistics post-processing is a general decoder for quantum LDPC codes built from the hypergraph product. The two-stage idea was first proposed by Panteleev and Kalachev, who showed that for many random quantum LDPC codes the combination improves decoding performance substantially over belief propagation alone. The post-processing technique, ordered statistics decoding, comes from classical coding work of Fossorier and others in the 1990s and 2000s.
The documentation for the LDPC software package describes BP+OSD as a modified belief propagation that works well for quantum LDPC codes, and notes an advantage: it always returns a decoding that satisfies the syndrome equation. Its implementation includes a fast-solve variant, OSD_0, that stops Gaussian elimination as soon as the syndrome becomes linearly dependent on the columns eliminated so far, and a local-cluster variant. It also describes an alternative, belief propagation with a union-find decoder as post-processing.
What the 2020 simulations found
The authors run numerical simulations on three families of hypergraph product codes: topological codes, fixed-rate random codes, and a new class they call semitopological codes, which share properties of both and let the code threshold and stabilizer locality be traded off in a finely controlled way. Their results indicate thresholds across all three families and provide evidence of exponential suppression of the logical error rate in the low-error regime. For the toric code they observe a threshold of 9.9 plus or minus 0.2 percent, which they say improves on previous quantum belief-propagation decoders and approaches the performance of minimum-weight perfect matching. Because semitopological codes are identical to toric codes in the bulk, they expect the same threshold and present numerical evidence for this.
What the software documentation adds
The LDPC package documentation says the post-processing step is where the decoder returns a syndrome-consistent output, and that the fast-solve OSD_0 variant ends elimination early once the syndrome depends linearly on the columns eliminated so far. It also describes a clustered variant in which reprocessing is done locally for each cluster, and the belief-find alternative, which runs a union-find decoder if belief propagation fails and uses the belief propagation output to guide cluster growth. A decoding that satisfies the syndrome equation is not necessarily the correct one, which is why the threshold and logical error rate results from the 2020 simulations matter.
What this does not tell you
The threshold is for a particular noise model and decoder in simulation. Runtime on real hardware is not addressed, and the Gaussian elimination at the heart of OSD is a cost the software documentation tries to reduce with its fast-solve variant.
FAQ
What is BP+OSD?
A two-stage decoder: belief propagation first, then ordered statistics post-processing.
Why add the second stage?
Panteleev and Kalachev showed it improves decoding for many random quantum LDPC codes.
What threshold was found for the toric code?
9.9 plus or minus 0.2 percent in the 2020 simulations.
What is a semitopological code?
A new hypergraph product family from that paper that trades off threshold against stabilizer locality.
Is it ready for hardware?
These sources are simulations and software documentation, so they do not show that.
All field notes