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arXiv:2504.01728 · IEEE Transactions on Information Theory

Linear Time Iterative Decoders for Hypergraph-Product and Lifted-Product Codes

Asit Kumar Pradhan, Nithin Raveendran, Narayanan Rengaswamy, Bane V. Vasic

Published Apr 27, 2026• 0 citations

Abstract

Quantum low-density parity-check (QLDPC) codes with asymptotically non-zero rates are prominent candidates for achieving fault-tolerant quantum computation, primarily due to the low operational depth of their syndrome-measurement circuits. Numerous studies advocate the necessity of fast decoders to fully harness the capabilities of QLDPC codes, thus driving the focus towards designing low-complexity iterative decoders. However, empirical investigations indicate that such iterative decoders are susceptible to having a high error floor when decoding QLDPC codes. The main objective of this paper is to analyze the decoding failures of thehypergraph-product(HGP) andlifted-product(LP) codes and to design decoders that mitigate these failures, thus achieving a reduced error floor. The suboptimal performance of these codes can predominantly be ascribed to two structural phenomena: (1) stabilizer-induced trapping sets (TS), which correspond to stabilizer-induced subgraphs in the Tanner graphs, and (2) classical trapping sets (TS), which originate from the classical codes used in the construction of HGP and LP codes. The dynamics of stabilizer-induced TSs are examined, and a straightforward modification of iterative decoders is proposed to circumvent these TSs. Moreover, this work proposes a systematic methodology for designing decoders that can circumvent the classical TSs in both HGP and LP codes by deriving them from decoders capable of avoiding the TSs in the parent classical LDPC codes. When decoders that can avoid stabilizer-induced TSs are run in parallel with those that can mitigate the effect of classical TSs, the logical error rate improves significantly in the error-floor region.

DOI: 10.1109/tit.2026.3687799

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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 LP codes show a significant improvement in logical error rate relative to BP+OSD. Moreover, the improved error correction relies on low-complexity finite-state iterative decoders, targeted for fast and low-latency FPGA implementation.

Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code's graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. The purpose of this paper is to examine failure inducing sets of QLDPC codes under syndrome-based iterative decoding. As for classical LDPC codes, we show that absorbing sets play a central role in understanding decoder failures. Raveendran and Vasic initiated the study of quantum trapping sets, where beyond the classical-type trapping sets, they identified rigid symmetric structures (a.k.a symmetric stabilizers) responsible for degenerate errors. In this paper, we show that this behavior is part of a much more general phenomenon that can be described by the absorbing set framework.

Quantum low-density parity-check (QLDPC) codes with asymptotically nonzero rates are promising candidates for fault-tolerant quantum computation. Belief propagation (BP) based iterative decoding algorithms, a primary choice for classical LDPC codes, perform poorly for QLDPC codes due to stabilizer-induced trapping sets, resulting in a high error floor. Several decoding algorithms, like post-processing decoders, normalized BP decoders, and neural decoders, have been proposed to increase the performance in the error-floor region. However, this improvement comes at the expense of an increase in the execution time of the decoder. This paper proposes a general framework for error correction for a class of QLDPC codes called lifted-product codes using recurrent neural networks (RNNs). The RNN is employed to learn message-passing rules that can decode quantum-trapping sets. Then the standard message-passing rules are used with the learned rules to improve the error floor. While training the RNN, the quasi-cyclic property of the lifted product codes is exploited to reduce the size of the training set and the number of parameters in the network. This reduction in the number of parameters makes these decoders amenable to hardware implementation. Simulation results show that the proposed decoder performs better than the existing decoders in the literature.

Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code's graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. The purpose of this paper is to examine failure inducing sets of QLDPC codes under syndrome-based iterative decoding. As for classical LDPC codes, we show that absorbing sets play a central role in understanding decoder failures. Raveendran and Vasić [11] initiated the study of quantum trapping sets, where beyond the classical-type trapping sets, they identified rigid symmetric structures (a.k.a. symmetric stabilizers) responsible for degenerate errors. In this paper, we show that this behavior is part of a much more general phenomenon that can be described by the absorbing set framework.

Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code’s graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. In this paper we examine the failure inducing sets of QLDPC codes under syndrome-based iterative decoding, and their connection to absorbing sets in classical LDPC codes.

Lifted product codes are an important family of quantum low-density parity-check (QLDPC) codes, as they were the first QLDPC code family shown to be asymptotically good. Understanding the structure of their parity-check matrices

H_{\mathsf{X}}

and

H_{\mathsf{Z}}

, as well as the associated Tanner graphs, is essential for analyzing their decoding behavior and error-floor performance. In this work, we show that the Tanner graphs of

H_{\mathsf{X}}

and

H_{\mathsf{Z}}

are indeed isomorphic, and investigate their graph-theoretical structure. We establish conditions ensuring the connectivity of these graphs and provide bounds on their minimal absorbing sets, providing new insight into the combinatorial structures influencing decoding performance.