Abstract
We show that belief propagation combined with ordered statistics post-processing is a general decoder for quantum low density parity check codes constructed from the hypergraph product. To this end, we run numerical simulations of the decoder applied to three families of hypergraph product code: topological codes, fixed-rate random codes and a new class of codes that we call semi-topological codes. Our new code families share properties of both topological and random hypergraph product codes, with a construction that allows for a finely-controlled trade-off between code threshold and stabilizer locality. Our results indicate thresholds across all three families of hypergraph product code, and provide evidence of exponential suppression in the low error regime. For the Toric code, we observe a threshold in the range $9.9\pm0.2\%$. This result improves upon previous quantum decoders based on belief propagation, and approaches the performance of the minimum weight perfect matching algorithm. We expect semi-topological codes to have the same threshold as Toric codes, as they are identical in the bulk, and we present numerical evidence supporting this observation.
Highlights
Any scalable computer architecture must be robust against hardware imperfections
In contrast to the toric and semitopological codes, the belief propagation (BP) decoder alone shows a crossing, pointing to a threshold in the range 6.5% ± 0.1%. The existence of this threshold for the BP decoder can be attributed to the fact that random Quantum LDPC (QLDPC) codes are less structured than toric and semitopological codes; the repeating patterns present in stabilizer checks of topological codes lead to high densities of degenerate errors that cause BP to fail
The practicality of QLDPC codes has been hindered by the lack of a general purpose decoder: designing a new family of QLDPC codes would necessitate the development of a special-purpose decoding strategy [19,20]
Summary
Any scalable computer architecture must be robust against hardware imperfections. In quantum computing, where qubits are realized as fragile quantum two-level systems, fault tolerance necessitates active error correction [1,2,3,4,5]. We expand on the results of Panteleev and Kalachev to provide further evidence that the BP + OSD decoder is a general decoder for all QLDPC codes that can be constructed from the hypergraph product. To this end, we first propose a new class of semitopological codes which share properties of both topological and random QLDPC codes. We show that in addition to random QLDPC codes, BP + OSD enables high-performance decoding of both topological QLDPC codes and our new class of semitopological codes.
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