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Physica Scripta

Resgru: Syndrome decoder for heavy-hexagon quantum error correction code

Ashutosh Kumar, Majid Haghparast, Lauri Kettunen

Published Jul 8, 2026• 1 citation
Publisher

Abstract

Abstract Decoding is a central bottleneck in near-term quantum error correction (QEC), where repeated syndrome extraction must be translated into reliable logical outcomes under realistic noise and hardware constraints. We study heavy-hexagon (HH) quantum memory experiments and propose Resgru , a lightweight recurrent neural decoder that maps time-ordered detection events to a binary logical-class prediction for each shot. We benchmark Resgru against a standard minimum-weight perfect matching (MWPM) decoder instantiated on the same detector error model for HH code at code distances d ∈ { 3 , 5 , 7 } under circuit-level depolarizing noise with physical error rate p = 0.1 % . Resgru consistently achieved higher logical fidelity than MWPM decoder and yields lower per-round logical error rates, including a 28% reduction for d = 5 and a 14% reduction for d = 7 . Despite being trained only on short sequences, Resgru maintains an advantage when evaluated on longer simulated memory experiments up to 300 QEC rounds.

DOI
10.1088/1402-4896/ae87e2

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Abstract Decoding is a central bottleneck in near-term quantum error correction (QEC), where repeated syndrome extraction must be translated into reliable logical outcomes under realistic noise and hardware constraints. We study heavy-hexagon (HH) quantum memory experiments and propose Resgru , a lightweight recurrent neural decoder that maps time-ordered detection events to a binary logical-class prediction for each shot. We benchmark Resgru against a standard minimum-weight perfect matching (MWPM) decoder instantiated on the same detector error model for HH code at code distances d ∈ { 3 , 5 , 7 } under circuit-level depolarizing noise with physical error rate p = 0.1 % . Resgru consistently achieved higher logical fidelity than MWPM decoder and yields lower per-round logical error rates, including a 28% reduction for d = 5 and a 14% reduction for d = 7 . Despite being trained only on short sequences, Resgru maintains an advantage when evaluated on longer simulated memory experiments up to 300 QEC rounds.
Neural decoders for quantum error correction have achieved high accuracy on surface code memory experiments, but practical fault-tolerant computation requires decoders that transfer across devices, noise conditions, and logical operations. We introduce a modular graph-recurrent decoder designed for this setting. Built from reusable local syndrome encoders, it is pretrained in simulation and fine-tuned to hardware using a common pipeline, with only lightweight spatial aggregation adapted between experiments. The decoder performs competitively on conventional memory benchmarks, outperforming minimum-weight perfect matching (MWPM) at distances 3 and 5 on an IBM Nighthawk-generation processor and closely approaching AlphaQubit on Google's below-threshold hardware data. We then test the framework in two settings beyond memory decoding on the same IBM device. First, in an XXX-type stability experiment, we continuously tune the effective measurement error rate using a variable pre-readout rotation and train a single decoder jointly across the resulting noise range. It matches or outperforms MWPM across the sweep without requiring a separately calibrated decoder for each setting. Second, we decode a ZZZZZZ-merge lattice-surgery experiment between two distance-3 surface-code patches, improving the decoded correlation between the merge outcome and the final ZZZZZZ parity relative to MWPM on real hardware. Together, these experiments demonstrate neural decoding that is portable not only across hardware, but also across qualitatively different surface code observables and logical operations, using models trainable on single-GPU academic resources.
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Real-time decoding is a major bottleneck in scaling quantum error correction (QEC) from noisy intermediate-scale quantum (NISQ) devices to fault-tolerant quantum computing. We present an adaptive confidence-gated decoding framework for the rotated surface code that treats decoding as a two-stage inference problem. A lightweight feed-forward neural network performs fast-path decoding for the majority of syndrome measurements, while only low-confidence predictions are escalated to a minimum-weight perfect matching (MWPM) refinement stage. We benchmark the framework on rotated surface codes with distances d∈{3,5,7,9,11}d \in \{3,5,7,9,11\}d∈{3,5,7,9,11} under circuit-level depolarising noise using the Stim stabiliser simulator. The evaluation characterises logical accuracy, confidence-controlled accuracy-latency trade-offs, decoding throughput, per-shot latency, and decoding-graph resource scaling. Routing only 3.3%-6.2% of syndromes to the refinement stage improves logical accuracy from 99.21% for the neural-only baseline to 99.81% at a confidence threshold of 0.95 while incurring only a bounded increase in average decoding cost. Neural-decoder throughput saturates near 4.6×1054.6 \times 10^{5}4.6×105 samples s−1^{-1}−1 at batch size 512 on commodity CPU hardware, indicating that the neural fast path is not the dominant throughput bottleneck beyond code distance d=7d=7d=7. We release the complete benchmarking pipeline, trained models, raw benchmark data, and source code, and explicitly distinguish the experimentally validated contributions from the broader hardware-aware QEC co-design roadmap, including hardware-constrained code discovery, GPU-accelerated inference, and multi-noise optimisation, which remain directions for future work.
Realizing the full potential of quantum computation requires Quantum Error Correction (QEC). QEC reduces error rates by encoding logical information across redundant physical qubits, enabling errors to be detected and corrected. A common decoder used for this task is Minimum Weight Perfect Matching (MWPM) a graph-based algorithm that relies on edge weights to identify the most likely error chains. In this work, we propose a data-driven decoder named Neural Minimum Weight Perfect Matching (NMWPM). Our decoder utilizes a hybrid architecture that integrates Graph Neural Networks (GNNs) to extract local syndrome features and Transformers to capture long-range global dependencies, which are then used to predict dynamic edge weights for the MWPM decoder. To facilitate training through the non-differentiable MWPM algorithm, we formulate a novel proxy loss function that enables end-to-end optimization. Our findings on the toric code under depolarizing noise demonstrate thresholds of 17.9% and 10.95%, nearing the 18.9% and 11.0% maximum likelihood bounds, highlighting the advantage of hybrid decoders that combine the predictive capabilities of neural networks with the algorithmic structure of classical matching.
Decoding error syndromes for topological quantum error correcting codes, such as surface and heavy hexagonal codes, is computationally expensive. While minimum weight perfect matching (MWPM) algorithms have been commonly used for decoding, recent works have demonstrated the efficacy of machine learning (ML), particularly neural networks, in decoding syndromes for these codes. In this study, we introduce a ML-based decoder tailored to heavy hexagonal code to address asymmetric noise channels which reflect real-world scenario better than the depolarization model considered in previous works. Our proposed decoder shows∼5×\sim 5\times∼5×and∼22×\sim 22\times∼22×improvements in the threshold values for amplitude and amplitude-phase damping noise models respectively over MWPM methods. Our decoder is also robust to changes in asymmetry, with the threshold reducing by only ~ 3.6% for a10×10\times10×change in asymmetry.