Sparse Mamba Decoder for Quantum Error Correction: Efficient Defect-Centric Processing of Surface Code Syndromes

Abstract: 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 $O(k)$ complexity. Across depolarizing, uniform circuit-level, SI1000, and Google Sycamore experimental benchmarks, SMD reduces the MWPM logical error rate by up to 49% at $d \le 5$ 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.

• The paper introduces the Sparse Mamba Decoder, a defect-centric neural architecture that exploits the sparsity of syndrome data for efficient quantum error correction.
• It leverages a 13-dimensional feature representation and state-space modeling to reduce computational complexity from O(d²R) to O(k), achieving notable improvements in accuracy and latency.
• Empirical results on synthetic and experimental benchmarks, including depolarizing noise and Sycamore data, demonstrate superior performance over traditional and dense neural decoders.

Sparse Mamba Decoder for Quantum Error Correction: Defect-Centric Neural Decoding of Surface Code Syndromes

Motivation and Architectural Overview

Quantum error correction (QEC) is a necessary mechanism for achieving fault-tolerant quantum computation, with the surface code being the dominant practical class due to its high threshold and local structure. The classical decoders used for surface codes must provide optimal tradeoffs among accuracy, computational speed, and scalability, especially at the code distances required for fault tolerance. Traditional approaches, including minimum-weight perfect matching (MWPM) and recent neural methods based on recurrent transformers and state-space models, typically operate on the full syndrome volume (of size $(d^2 \times R)$ for code distance $d$ and $R$ measurement rounds), regardless of the error density. At relevant physical error rates ($p \sim 10^{-3}$), syndrome volumes are highly sparse, with less than 5% of entries containing detection events. This inefficiency motivates a defect-centric processing paradigm, leading to the Sparse Mamba Decoder (SMD) architecture.

SMD fundamentally alters the input representation: it operates only on the $k$ active detection events, each encoded as a 13-dimensional feature vector encompassing spatial coordinates, stabilizer type, spatial and temporal neighborhood flags, normalized boundary distances, and a cumulative XOR-based reconstructed measurement. These features are processed by a Mamba state-space backbone, facilitating sequence modeling with $O(k)$ complexity. This contrasts with conventional dense neural decoders ($O(d^2 R)$) and attention-based decoders ($O(d^4 R)$), and eliminates explicit spatial convolution by embedding all relevant geometry in the feature representation.

Benchmarking and Numerical Evaluation

SMD is evaluated across multiple benchmarks: synthetic depolarizing noise, uniform circuit-level noise, the physically motivated SI1000 model, and experimental data from Google's Sycamore quantum processor. All models are trained and evaluated on commodity NVIDIA hardware, typically in the 7.5–16M parameter range, without requiring specialized accelerators.

Depolarizing Noise

Under depolarizing noise with perfect stabilizer measurements—a purely spatial setting—SMD's dual-head variant achieves up to 91% lower logical error rate (LER) than MWPM, Tesseract, and Belief Matching at $d = 11$, with improvement monotonically increasing with code distance. This superiority is attributed to SMD's architectural capacity for joint $(A_z, A_x)$ decoding: it inherently captures Y-error-induced correlations across X- and Z-type stabilizers, while matching-based decoders factorize over bases.

Uniform Circuit-Level Noise

For uniform circuit-level noise with $d$0 measurement rounds, SMD surpasses MWPM in LER across all tested distances and error rates, with up to 44% LER reduction at $d$1 ($d$2). SMD achieves this with nearly constant latency (24–57 μs across $d$3–9), significantly outperforming belief matching and graph-neural-network decoders in computational efficiency.

SI1000 Benchmark

The SI1000 model, reflecting experimentally relevant superconducting qubit hardware, serves as the standard for rigorous decoder benchmarking. At $d$4, SMD reduces MWPM LER by up to 49% at $d$5 and by 16% at $d$6 via model ensembling, while running 95–467x faster than Tesseract and 232–463x faster than Belief Matching at equivalent or lower error rates. Notably, ensemble-based MWPM (Libra-style) provides negligible accuracy improvement, indicating that SMD's gains arise from learned correlations beyond matching-based factorization.

Sycamore Experimental Dataset

On the Sycamore dataset, SMD achieves a mean LER of $d$7 at $d$8 and $d$9 at $R$0, matching or marginally surpassing the dense Mamba decoder ($R$1 at $R$2, $R$3 at $R$4). SMD thus demonstrates that defect-centric sparse processing is sufficient for high-accuracy decoding in practical settings, challenging the necessity of full 2D convolutional spatial processing.

Theoretical Implications and Feature Design

SMD exploits the intrinsic sparsity in syndrome volumes at low physical error rates. The feature engineering directly encodes spatial and temporal geometry, stabilizer type, local neighborhood connectivity, boundary distances, and measurement state, enabling the state-space model to capture the geometric and topological structure without explicit convolutions. This facilitates the modeling of correlated error patterns, particularly those inaccessible to matching-based decoders, such as Y-error cross-basis correlations.

The computational complexity of SMD scales linearly with the number of active defects, not the code size, suggesting substantial asymptotic advantages as quantum processors scale to larger code distances. The model is hardware-agnostic, requiring only commodity GPUs, further enhancing practical feasibility compared to architectures necessitating specialized TPU hardware.

Scaling, Practical Constraints, and Extensions

SMD's efficiency is pronounced as code distances increase and syndrome densities remain sparse. Decoding latency is essentially invariant with respect to code distance, contrasting with the sharp scaling seen in belief matching approaches. However, real-time operation on superconducting hardware would require streaming rather than batch processing; adaptations combining SMD's $R$5 scaling with streaming architectures (e.g., AQ2-RT on Trillium TPUs) offer a promising path forward.

Future research includes:

• Extending SMD’s defect-centric paradigm to color codes and quantum LDPC codes, where syndrome sparsity is even greater
• Integrating soft information such as I/Q readout data for enhanced accuracy
• Addressing ultra-low latency, streaming decoding for physical quantum hardware
• Direct application to upcoming large-distance experimental platforms (e.g., Google's Willow distance-7 chip)

Conclusion

The Sparse Mamba Decoder provides a defect-centric, sparsity-exploiting neural decoding architecture for surface code quantum error correction. It achieves competitive or superior logical error rates compared to dense neural and classical decoding approaches, with orders-of-magnitude speedup and nearly constant latency across code distances, all on commodity GPU hardware. Its architectural and empirical results suggest that the fundamental sparsity of quantum error syndromes should be considered a structural prior in decoder design. As quantum processors expand in scale and complexity, the cost and accuracy advantages of sparse, defect-centric processing will become increasingly compelling. The SMD approach lays the foundation for practical, scalable, and efficient neural quantum error correction, with broad applicability to future quantum hardware and code families.

Reference: "Sparse Mamba Decoder for Quantum Error Correction: Efficient Defect-Centric Processing of Surface Code Syndromes" (2605.17156)