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arXiv:2608.27901

High-Throughput Normalized Min-Sum Belief Propagation Decoding for Quantum LDPC Codes with Near-Memory Processing

Jeonggeun Seo, Youngsun Han, Leanghok Hour, Dongmin Kim

quant-phSubmitted Aug 28, 2026• Announced Aug 31, 2026• 0 citations
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Abstract

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.

Comments
22 pages, 8 figures, 1 table. Submitted to Physica Scripta

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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.
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Quantum low-density parity-check (QLDPC) codes represent a promising approach for error correction in quantum computing. The recently proposed Tesseract decoder uses the A* search algorithm that guarantees finding the most likely error pattern. However, practical implementations of Tesseract often involve an extremely large graph, and the inherently sequential nature of the search results in high computational overhead and long runtime. To improve decoding efficiency, we propose a two-stage decoding framework. First, a belief propagation (BP) decoder efficiently processes the syndrome. This step generates hard decisions (a binary error vector) and soft information (per-qubit confidence levels). In non-convergent BP cases, a gating mechanism examines the BP decoder's output to identify and filter out qubits with oscillating confidence values. In these cases, the refined output serves as input to Tesseract, which then attempts to identify and correct any residual errors. This hybrid approach leverages the high speed of BP decoding and delegates the more challenging decoding instances to Tesseract. Numerical results demonstrate a substantial reduction in overall decoding complexity while maintaining the logical error rate (LER) of the stand-alone Tesseract. Across all tested physical error rates, the proposed method achieves at least 5x reduction in the number of expanded nodes in the Tesseract, with a peak reduction of nearly 15x at a physical error rate of p = 0.05 for the [[126, 12, d < 11]] T1 code, and approximately 8.8x at p = 0.05 for the [[72, 12, 6]] bicycle bivariate (BB) code.
Abstract In practical quantum error correction implementations, the measurement of syndrome information is an unreliable step—typically modeled as a binary measurement outcome flipped with some probability. However, the measured syndrome is in fact a discretized value of the continuous voltage or current values obtained in the physical implementation of the syndrome extraction. In this paper, we use this “soft” or analog information to benefit iterative decoders for decoding quantum low-density parity-check (QLDPC) codes. Syndrome-based iterative belief propagation decoders are modified to utilize the soft syndrome to correct both data and syndrome errors simultaneously. We demonstrate the advantages of the proposed scheme not only in terms of comparison of thresholds and logical error rates for quasi-cyclic lifted-product QLDPC code families but also with faster convergence of iterative decoders. Additionally, we derive hardware (FPGA) architectures of these soft syndrome decoders and obtain similar performance in terms of error correction to the ideal models even with reduced precision in the soft information. The total latency of the hardware architectures is about 600 ns (for the QLDPC codes considered) in a 20 nm CMOS process FPGA device, and the area overhead is almost constant—less than 50% compared to min-sum decoders with noisy syndromes.
Accurate decoding of quantum low-density parity-check (qLDPC) codes often relies on expensive post-processing search, although decoding difficulty varies strongly between syndromes. We find that the benefit of deeper post-processing search is highly concentrated in a small subset of decoding instances, and that these instances can be identified directly from the decoder itself. To this end, we introduce an adaptive decoder based on belief propagation (BP) and ordered-statistics decoding (OSD), using the disagreement between the BP hard decision and the syndrome-consistent order-zero OSD solution as an internal risk signal to determine where deeper search is required. On a [[144,12,12]][[144,12,12]][[144,12,12]] bivariate bicycle code under circuit-level depolarising noise, encoding 121212 logical qubits in 144144144 data qubits with distance 121212, escalating only the highest-risk 20%20\%20% of instances recovers 85%85\%85% to 92%92\%92% of the improvement in logical error rate available from a full one-free-variable sweep, while reducing the mean serial decoding cost by a factor of 3.63.63.6 relative to applying the same truncated search to every instance. At matched budget, disagreement-guided routing also outperforms routing based on the BP syndrome residual, syndrome weight and random selection. The same behaviour reappears on a structurally distinct radial qLDPC code obtained from a lifted-product construction, where escalating 20%20\%20% of instances recovers 87%87\%87% to 94%94\%94% of the available gain. A ZZZ-memory experiment on the Quantinuum H2 trapped-ion processor, implementing both XXX-type and ZZZ-type checks, further shows that the signal remains predictive under device noise. These results show that decoder-internal disagreement can expose where additional decoding effort is valuable, allowing classical computation to be concentrated on the instances most likely to benefit from it.
We present a real-time FPGA decoder for a high-rate [[512,174,8]][[512,174,8]][[512,174,8]] 3D product qLDPC code. We construct a circuit-level detector-error model for a compact syndrome-extraction schedule and propose a small parallel ensemble of row-wise serial min-sum decoders with cumulative memory. The decoder matches BP+OSD-CS10\text{BP+OSD-CS}_\text{10}BP+OSD-CS10​ accuracy with one core and Relay-BP accuracy with three cores. The FPGA implementation meets the latency budget of neutral-atom and trapped-ion systems. An ASIC implementation would improve the latency, enabling real time decoding even for superconducting systems. At a physical error rate of p=10−3p=10^{-3}p=10−3, three FPGA devices decode 174 logical qubits with a logical error rate of 5×10−75\times10^{-7}5×10−7 and effective distance 6. For p=10−4p=10^{-4}p=10−4, and assuming the number of BP iterations is capped to 30, the decoder's average latency is 5.1 5.1\,5.1\textmu s and its worst-case latency is 97.9 97.9\,97.9\textmu s.
In this paper, we propose a belief-propagation (BP)-based decoder, termed the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for quantum low-density parity-check (QLDPC) codes. The key idea is to generate \emph{structured decoding diversity} by constructing multiple redundant parity-check representations via cycle-free subtree decompositions of the Tanner graph, and running BP decoding in parallel across these representations. This extends the classical Multiple-Bases Belief-Propagation (MBBP) framework to the quantum setting while preserving the linear-time complexity and efficiency of standard BP decoding, and avoids the need for super-linear post-processing. Simulation results demonstrate that MBBP-LD improves upon existing BP-based decoders, including BP with ordered statistics decoding (BP-OSD) and belief propagation with guided decimation (BPGD) across several QLDPC codes, while requiring substantially fewer total BP iterations. For bivariate bicycle codes [[144,12,12]][[144,12,12]][[144,12,12]] and [[288,12,18]][[288,12,18]][[288,12,18]], MBBP-LD achieves up to 20%20\%20% reduction in error rate compared to BPGD and up to 30%30\%30% compared to BP-OSD in the low- and moderate-error regimes. For the larger B1 code [[882,24,18≤d≤24]][[882, 24, 18 \leq d \leq 24]][[882,24,18≤d≤24]], MBBP-LD attains comparable or improved performance relative to BPGD while maintaining BP-like decoding latency under parallel implementation.