Fast Fault-Tolerant Decoders for Hypergraph Product and Lifted-Product Codes

Abstract:We design low-complexity, fault-tolerant decoders for quantum low-density parity-check (QLDPC) codes with the goal of reducing decoding latency. We target two major bottlenecks of decoding under the \emph{circuit-level} noise model: (i) post-processing via order-statistics decoding (OSD), and (ii) the large number of auxiliary variable nodes commonly introduced to represent CNOT-induced correlations during syndrome extraction. Our key observation is that propagating CNOT faults (\emph{hook errors}) create \emph{stabilizer-induced} trapping sets (TSs) that are intrinsic to hypergraph-product (HGP) and lifted-product (LP) constructions. Therefore, instead of modeling each such fault with an explicit correlation node and relying on OSD to clean up the resulting failures, we design message-passing decoders that resolve the corresponding \emph{stabilizer-induced} TSs directly. We obtain these decoders by deriving QLDPC decoders from decoders for the parent classical LDPC codes and using them collectively to correct broad families of \emph{stabilizer-induced} TSs. For CNOT faults that manifest primarily as syndrome errors, we show that their effect is equivalent to a data error together with syndrome-bit measurement errors. Consequently, given repeated measurements and a decoding graph that already includes nodes representing syndrome-bit errors, no distinct variable node is needed for each CNOT fault. Using a \emph{phenomenological} Tanner graph with nodes representing only data errors and syndrome-bit errors, simulations on the LP codes show a reduction in, or comparable, logical error rates relative to BP+OSD, at substantially lower decoding complexity.

Information Theory (cs.IT); Quantum Physics (quant-ph)

arXiv:2608.31040 [cs.IT]

(or arXiv:2608.31040v1 [cs.IT] for this version)

https://doi.org/10.48550/arXiv.2608.31040 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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