In this paper, we propose a novel message-passing decoding approach that leverages the degeneracy of quantum low-density parity-check codes to enhance decoding performance, eliminating the need for serial scheduling or post-processing. Our focus is on two-block Calderbank-Shor-Steane (CSS) codes, which are composed of symmetric stabilizers that hinder the performance of conventional iterative decoders with uniform update rules. Specifically, our analysis shows that, under the isolation assumption, the min-sum decoder fails to converge when constant-weight errors are applied to symmetric stabilizers, as variable-to-check messages oscillate in every iteration. To address this, we introduce a decoding technique that exploits this oscillatory property by applying distinct update rules: variable nodes in one block utilize messages from previous iterations, while those in the other block are updated conventionally. Logical error-rate results demonstrate that the proposed decoder significantly outperforms the normalized min-sum decoder and achieves competitive performance with belief propagation enhanced by order-zero ordered statistics decoding, all while maintaining linear complexity in the code's block length.
In this paper, we propose a novel message-passing decoding approach that leverages the degeneracy of quantum low-density parity-check codes to enhance decoding performance, eliminating the need for serial scheduling or post-processing. Our focus is on two-block Calderbank-Shor-Steane (CSS) codes, which are composed of symmetric stabilizers that hinder the performance of conventional iterative decoders with uniform update rules. Specifically, our analysis shows that, under the isolation assumption, the min-sum decoder fails to converge when constant-weight errors are applied to symmetric stabilizers, as variable-to-check messages oscillate in every iteration. To address this, we introduce a decoding technique that exploits this oscillatory property by applying distinct update rules: variable nodes in one block utilize messages from previous iterations, while those in the other block are updated conventionally. Logical error-rate results demonstrate that the proposed decoder significantly outperforms the normalized min-sum decoder and achieves competitive performance with belief propagation enhanced by order-zero ordered statistics decoding, all while maintaining linear complexity in the code's block length.
In this paper, we propose a novel message-passing decoding approach that leverages the degeneracy of quantum low-density parity-check codes to enhance decoding performance, eliminating the need for serial scheduling or post-processing. Our focus is on two-block Calderbank-Shor-Steane (CSS) codes, which are composed of symmetric stabilizers that hinder the performance of conventional iterative decoders with uniform update rules. Specifically, our analysis shows that, under the isolation assumption, the min-sum decoder fails to converge when constant-weight errors are applied to symmetric stabilizers, as variable-to-check messages oscillate in every iteration. To address this, we introduce a decoding technique that exploits this oscillatory property by applying distinct update rules: variable nodes in one block utilize messages from previous iterations, while those in the other block are updated conventionally. Logical error-rate results demonstrate that the proposed de-coder significantly outperforms the normalized min-sum decoder and achieves competitive performance with belief propagation enhanced by order-zero ordered statistics decoding, all while maintaining linear complexity in the code's block length.
We introduce a low-complexity message-passing quantum error correction algorithm for decoding Quantum Low-Density Parity-Check (QLDPC) stabilizer codes. The proposed decoder operates on the quaternary stabilizer graph but only exchanges binary messages. This leads to a significantly reduced complexity compared to other quaternary belief propagation (BP) algorithms that pass floating-point messages. The efficacy of the proposed decoder is evaluated by providing decoding examples, performance metrics using Monte-Carlo simulations, and complexity analysis. Despite its reduced complexity, the performance loss of the proposed decoder is modest compared to floating-point parallel quaternary decoders for a Calderbank-Shor-Steane (CSS) code family. In particular, experiments obtained over the [[1054, 140, 20]] lifted product (LP) Tanner code demonstrated that for low error rates (< 0.01), the proposed quaternary-binary message-passing decoder approaches the performance of quaternary BP by converging in almost the same number of iterations while requiring less complex operations. Additionally, for non-CSS codes, our decoder performs similarly as quaternary floating-point decoders despite its lower complexity.
Many recently proposed Calderbank-Shor-Steane (CSS) quantum low-density parity-check (QLDPC) codes have sparse decoding graphs, enabling syndrome-based belief propagation (BP) decoding at low complexity. Their construction, however, often results in properties that impair BP performance, such as short cycles and degeneracy. In this work, we propose a general framework for introducing auxiliary variable nodes (AVNs) and auxiliary check nodes (ACNs) into the decoding graph of CSS codes, compatible with the standard stabilizer measurement framework. This provides an additional degree of freedom in the design of the decoding graph itself and can be used to tackle the aforementioned shortcomings. We show that recently proposed techniques, 4-cycle removal and subcode ensemble decoding, can be interpreted as instances of this framework. For 4-cycle removal, we find that the gains depend strongly on the BP iteration count and check-node message scaling. Building on this framework, we further propose a graph-derived subcode ensemble decoder and demonstrate under circuit-level noise that it substantially reduces the per-round logical error rate compared with BP on the corresponding 4-cycle-free decoding graph.
Practical quantum computations can require logical error rates of 10−12 or lower. Quantum low-density parity-check (QLDPC) codes are promising candidates for reaching such rates with low qubit overhead, provided that their syndromes can be decoded efficiently. Repeated noisy syndrome measurements introduce temporal degeneracy into the space-time decoding graph. We characterize the temporal kernel, whose vectors have zero total data parity, and show that it is generated by elementary temporal codewords of weight dv+2, where dv is the variable-node degree. For (dv,dc)-regular parity-check matrices with fixed degrees dv,dc>2 and girth greater than four, we derive counting bounds and construct degenerate error patterns of weight (dv+2)/2 for even dv and dv+1 for odd dv, with multiplicity Θ(nr) in the blocklength n and number of measurement rounds r. Under the isolation assumption, these patterns cause local failures of deterministic isotropic message-passing decoders under a flooding schedule. These failures depend on the node degrees rather than the minimum distance; for the bivariate bicycle (BB) codes considered, their weight is equal to four. Monte carlo simulations and fixed-weight importance sampling quantify harmful low-weight degenerate error patterns for the min-sum (MS) decoder beyond the isolation assumption. Motivated by the identified symmetries, we propose space-time anisotropic min-sum decoding with syndrome-guided realizations. Its complexity is linear in the number of Tanner graph edges for fixed maximum number of iterations and number of decoders. Numerical results on BB codes show threshold-like behavior acrossthe tested code sizes and improved failure probabilities relative to MS decoding with ordered statistics decoding post-processing.
In this letter, we propose a novel iterative decoding algorithm that exploits the degenerate nature of three different families of quantum low-density parity-check codes, i.e., surface, toric, and row-degree-4 bicycle codes. Such families of codes share harmful trapping sets that constitute symmetric stabilizers, making it impossible for any parallel-scheduled iterative message-passing decoder to converge even for error patterns of weight as low as two. By biasing subsets of nodes in the symmetric stabilizers, the decoder is able to converge to a valid error pattern. Furthermore, the proposed decoder has low decoding complexity - linear in the code’s blocklength - and a fully parallel schedule, making it suitable for low-latency efficient implementation.
Real-time quantum error correction requires classical decoders to process growing syndrome workloads with low and predictable latency. For quantum low-density parity-check (qLDPC) codes, iterative belief propagation (BP) repeatedly updates messages over sparse Tanner graphs, creating substantial memory-access and data-movement demands. We map normalized Min-Sum BP decoding of the [[144,12,12]] Bivariate Bicycle qLDPC code onto a DPU-based Processing-in-Memory (PIM) architecture. Within each DPU, 11 tasklets cooperatively decode one syndrome, while multiple DPUs process independent syndrome instances in parallel. Using uPIMulator and a data-qubit Pauli error model with ideal syndrome measurements, we compare throughput, per-syndrome processing time, logical error rate (LER), and single-syndrome tail latency against a 16-logical-CPU baseline. At a component-wise physical error probability of p=0.001 and one BP iteration, the projected aggregate kernel throughput of 2,560 DPUs reaches 1.071 x 10^7 decodes/s, compared with 1.22 x 10^6 decodes/s for the CPU, an 8.8x improvement. From two iterations onward, the measured LER remains below the physical error probability for every evaluated value of p. For one to five iterations, the maximum sampled serialized X+Z DPU compute latency remains below the 1 ms decoder-side reference for trapped-ion QEC, reaching approximately 0.873 ms at five iterations. These results show that near-memory processing can provide high aggregate throughput and sub-millisecond compute latency for qLDPC BP decoding under the evaluated conditions.