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arXiv:2610.02734

SMP: A General Hyperedge-Based Framework for Circuit-Level Quantum Error Correction

Siying Wang, Yue Yan, Zhixin Xia, Canwei Shi, Hannuo Yuan, Xiang-Bin Wang

quant-phSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

Hyperedge faults remain a major challenge in circuit-level quantum error correction. Accurately exploiting hyperedge correlations often requires substantial computational resources, which complicates real-time decoding. Matching-based decoders are fast and scalable, but their pairwise graph representation limits the use of higher-order fault information. Here, we introduce Syndrome Motif Projection (SMP), a general framework for incorporating hyperedge information into matching-based decoding. The key idea is that a hyperedge fault produces a characteristic local syndrome motif. Observing such a motif provides evidence for the corresponding fault. SMP extracts this information directly from the measured syndrome with linear-complexity preprocessing while preserving the matching backend. It can therefore serve as a lightweight front end to existing scalable and real-time matching decoders. We demonstrate SMP across surface code memories, color code memories, and logical gate circuits. On Google's Willow experimental surface code data, SMP improves both standard and correlated matching and outperforms the tested belief matching baseline. For color code memories, SMP combined with Chromobius reduces logical failure probabilities by up to 77.8\%. For logical gate circuits, SMP reduces the failure probability of a six-CNOT circuit by 35.5\% relative to logical observable matching while achieving performance comparable to iterative belief matching. These results demonstrate that SMP provides a lightweight and transferable approach to hyperedge decoding, with a clear path toward integration into scalable real-time decoding architectures for fault-tolerant quantum computation.

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Hyperedge faults remain a major challenge in circuit-level quantum error correction. Accurately exploiting hyperedge correlations often requires substantial computational resources, which complicates real-time decoding. Matching-based decoders are fast and scalable, but their pairwise graph representation limits the use of higher-order fault information. Here, we introduce Syndrome Motif Projection (SMP), a general framework for incorporating hyperedge information into matching-based decoding. The key idea is that a hyperedge fault produces a characteristic local syndrome motif. Observing such a motif provides evidence for the corresponding fault. SMP extracts this information directly from the measured syndrome with linear-complexity preprocessing while preserving the matching backend. It can therefore serve as a lightweight front end to existing scalable and real-time matching decoders. We demonstrate SMP across surface code memories, color code memories, and logical gate circuits. On Google's Willow experimental surface code data, SMP improves both standard and correlated matching and outperforms the tested belief matching baseline. For color code memories, SMP combined with Chromobius reduces logical failure probabilities by up to 77.8%. For logical gate circuits, SMP reduces the failure probability of a six-CNOT circuit by 35.5% relative to logical observable matching while achieving performance comparable to iterative belief matching. These results demonstrate that SMP provides a lightweight and transferable approach to hyperedge decoding, with a clear path toward integration into scalable real-time decoding architectures for fault-tolerant quantum computation.
Quantum error correction (QEC) is essential for building fault-tolerant quantum computers, requiring decoders that are simultaneously accurate, fast, and scalable. Most state-of-the-art neural decoders achieve high accuracy but process the full dense syndrome array of size $O(d^2 R) $regardless of the actual error rate, where d is the code distance and R is the number of measurement rounds. At physically relevant error rates (p ~ 0.1%), fewer than 5% of syndrome entries contain active detection events -- yet existing decoders process the entire syndrome volume. We introduce the Sparse Mamba Decoder (SMD), a defect-centric neural decoder that processes only the k active detection events using a 13-dimensional feature representation per defect and a Mamba state-space backbone, achieving O(k)O(k)O(k) complexity. Across depolarizing, uniform circuit-level, SI1000, and Google Sycamore experimental benchmarks, SMD reduces the MWPM logical error rate by up to 49% at d≤5d \le 5d≤5 under SI1000 noise, runs 95-467x faster than the Tesseract near-MLD decoder and 232-463x faster than Belief Matching, and maintains nearly constant latency (24-57 us) across d = 3-9 under uniform circuit-level noise. On the Sycamore experimental dataset, the SMD ensemble matches or slightly surpasses the dense Mamba decoder of Varbanov et al. All results are obtained on commodity NVIDIA GPUs with 7.5-16M parameters, without specialized accelerators.
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.
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Color codes are promising alternatives to surface codes for universal fault-tolerant quantum computing due to their simpler lattice-surgery protocols and the transversal implementation of logical Clifford gates. However, their practical deployment has been limited by slower decoding algorithms and worse logical failure rates and thresholds compared to surface codes. Although AI-based logical-flip decoders have recently been proposed to address these challenges, no clear framework currently exists for implementing such decoders within the parallel block-wise decoding schemes in both space and time required for large-scale fault-tolerant computation. AI-based pre-decoders offer a scalable alternative due to their local nature. By performing spacelike corrections on physical qubits and timelike corrections on stabilizer measurements, pre-decoders are naturally compatible with parallel block-wise decoding schemes and lattice-surgery protocols. In this work, we introduce AI-based pre-decoders for triangular color codes. We present a novel neural-network architecture for their implementation and develop methods to simplify the complex training data generated by color-code syndrome-extraction circuits containing feedforward operations. Remarkably, we find that both logical failure rates (LERs) and runtimes improve relative to raw Chromobius decoding as the code distance increases. For example, at code distance d=31 and physical error rate p=0.3%p=0.3\%p=0.3%, our pre-decoder + Chromobius pipeline improves the logical failure rate by a factor of 347x while reducing runtime by 7.33x compared to raw Chromobius decoding alone. These results demonstrate that AI-based pre-decoding can substantially narrow the performance gap between color codes and surface codes, bringing color codes closer to practical large-scale fault-tolerant quantum computation.
Fast, scalable decoding architectures that operate in a block-wise parallel fashion across space and time are essential for real-time fault-tolerant quantum computing. We introduce a scalable AI-based pre-decoder for the surface code that performs local, parallel error correction with low decoding runtimes, removing the majority of physical errors before passing residual syndromes to a downstream global decoder. This modular architecture is backend-agnostic and composes with arbitrary global decoding algorithms designed for surface codes, and our implementation is completely open source. Integrated with uncorrelated PyMatching, the pipeline achieves end-to-end decoding runtimes of order O(1μs)\mathcal{O}(1 μ\text{s})O(1μs) per round at large code distances on NVIDIA GB300 GPUs while reducing logical error rates (LERs) relative to global decoding alone. In a block-wise parallel decoding scheme with access to multiple GPUs, the decoding runtime can be reduced to well below O(1μs)\mathcal{O}(1 μ\text{s})O(1μs) per round. We observe further LER improvements by training a larger model, outperforming correlated PyMatching up to distance-13. We additionally introduce a noise-learning architecture that infers decoding weights directly from experimentally accessible syndrome statistics without requiring an explicit circuit-level noise model. We show that purely data-driven graph weight estimation can nearly match uncorrelated PyMatching and exceed correlated PyMatching in certain regimes, enabling highly-optimized decoding when hardware noise models are unknown or time-varying, as well as training pre-decoders with realistic noise models. Together, these results establish a practical, modular, and high-throughput decoding framework suitable for large-distance surface-code implementations.
Fault-tolerant quantum computation critically depends on the ability to detect and correct physical qubit errors faster than they accumulate. The surface code (SC) has emerged as the most experimentally promising quantum error correction (QEC) framework due to its high threshold and local stabilizer structure, yet its practical performance is bottlenecked by decoder efficiency. Traditional decoding approaches such as minimum-weight perfect matching (MWPM) achieve near-optimal accuracy under independent noise but struggle with correlated errors, circuit-level noise, and real-time latency constraints. This paper investigates how neural network (NN)-based decoders reduce logical error rates in SC quantum systems by learning complex syndrome-to-correction mappings from data. We propose a hybrid convolutional-recurrent neural architecture that processes stabilizer measurement syndromes across both spatial and temporal dimensions, enabling the decoder to capture correlated error patterns inherent in realistic hardware. Systematic simulations under depolarizing, biased, and circuit-level noise models demonstrate that the proposed neural decoder achieves a logical error rate reduction of up to 41.3% compared to MWPM at a physical error rate of p = 0.008, while maintaining sub-millisecond decoding latency compatible with superconducting qubit cycle times. These results establish that data-driven neural decoders represent a scalable and hardware-compatible pathway toward achieving the logical error rates required for practical fault-tolerant quantum computing.