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Surface Code Decoding: Which Algorithm Achieves Lowest Logical Error Rate Under Circuit-Level Noise?

Which surface code decoder (minimum weight perfect matching, union-find, or neural-network-based) achieves the lowest logical error rate under circuit-level noise for code distances 3 to 13?

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10/10 claims verified · 7 sources

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Direct answer

No single decoder achieves the lowest logical error rate across all code distances and noise models. Under circuit-level noise, neural-network decoders can outperform minimum-weight perfect matching (MWPM) by approximately 25% on experimental data, and the Sparse Mamba Decoder (SMD) reduces MWPM error rates by up to 49% at d ≤ 5. Weighted matching on the toric code achieves a higher threshold (0.72%) than weighted union-find (0.62%), but union-find with weighting improves its threshold from 0.38% to 0.62%. The best choice depends on code distance, noise characteristics, and latency constraints.

Decoder landscape and why it matters

Surface code decoders turn syndrome measurements into a correction operator; the logical error rate decides whether fault-tolerant quantum computing is feasible. Three decoder families are compared: minimum-weight perfect matching (MWPM), union-find (UF), and neural-network (NN) decoders. Under circuit-level noise, the threshold — the physical error rate below which logical errors decrease with code distance — is a key metric. Weighted matching on the toric code reaches a threshold of 0.72%, while weighted union-find reaches 0.62% under the same noise [4]. The Sparse Mamba Decoder (SMD) reduces MWPM logical error rates by up to 49% at d ≤ 5 under SI1000 noise [6].

Mechanism: how decoders handle correlated errors

Neural-network decoders outperform matching decoders because they handle errors that produce multiple correlated syndrome defects, such as Y errors, better [5]. Weighting improves both UF and matching decoders: under circuit-level depolarizing noise on the toric code, weighting raises the UF threshold from 0.38% to 0.62%, and the matching threshold from 0.65% to 0.72% [4]. With quantum non-demolition measurements, weighted UF reaches a threshold of 0.76% versus 0.90% for weighted matching [4]. The SMD keeps almost constant latency (24–57 μs) across d=3–9 under uniform circuit-level noise, which matters for real-time decoding [6].

Measured performance: logical error rates and thresholds

On experimental data from a transmon-qubit processor, the neural network decoder achieves logical error rates about 25% lower than MWPM [5]. Adding soft information from analog readout lowers the NN decoder's logical error rate by a further 10% [5]. The SMD reduces MWPM logical error rates by up to 49% at d ≤ 5 under SI1000 noise [6]. Under circuit-level depolarizing noise on the toric code, weighted matching (threshold 0.72%) outperforms weighted union-find (threshold 0.62%) [4].

Limits and open questions

The claims do not give a head-to-head comparison of all three decoders under identical noise models and code distances. Neural-network decoders are tested on experimental data (transmon processor) and show a 25% improvement over MWPM, but no comparison to union-find is given [5]. The SMD results are for SI1000 noise, not circuit-level depolarizing noise, and only up to d=5 [6]. Weighted union-find and weighted matching are compared only on the toric code, not the planar surface code [4]. Latency data for NN decoders is missing, and training overhead is not quantified. The best decoder for a given system may depend on hardware constraints, noise bias, and the availability of soft information.

Practical

How to build it, or how to use it

  1. Choose a decoder family: For low latency (d=3–9), implement the Sparse Mamba Decoder (SMD) using the architecture described in [6]; for higher thresholds, implement weighted matching from [4].
  2. Syndrome extraction: Use a circuit-level depolarizing noise model with quantum non-demolition measurements for the toric code (or planar surface code).
  3. Weight assignment: For weighted union-find or weighted matching, assign edge weights based on the probability of error mechanisms (e.g., depolarizing noise) as in [4].
  4. Decoder implementation: For SMD, build a sparse Mamba state-space model that processes defect clusters; for NN, train a feedforward or convolutional network on simulated syndromes with soft information from analog readout [5].
  5. Baseline measurement: Measure logical error rate for code distances d=3,5,7,9,11,13 under circuit-level noise, comparing against MWPM (e.g., using PyMatching).
  6. Evaluation: Compute the threshold by fitting logical error rate against physical error rate; for SMD, also measure latency per round.

Our take

What we would build

We would build a benchmark suite that compares SMD, weighted union-find, and a neural-network decoder on the same circuit-level depolarizing noise model for the planar surface code at d=3,5,7,9. The suite would measure logical error rate, threshold, and decoding latency. Success would be a decoder that achieves a logical error rate at least 30% lower than MWPM at d=7 and a latency under 1 μs per round. This would show which decoder family is most practical for near-term quantum processors and give a reproducible baseline for the community.

Claim record

What this note is based on

  1. factsupported

    Neural-network decoders can achieve a lower logical error rate compared to minimum-weight perfect matching when decoding the surface code.

    [5] Neural network decoder for near-term surface-code experiments — abstract arXiv:2307.03280v2
    “Neural-network decoders can achieve a lower logical error rate compared to conventional decoders, like minimum-weight perfect matching, when decoding the surface code. Furthermore, these decoders require no prior information about the physical error rates, making them highly adap…”
  2. resultsupported

    On experimental data from a transmon-qubit processor, the neural network decoder achieves logical error rates approximately 25% lower than minimum-weight perfect matching.

    [5] Neural network decoder for near-term surface-code experiments — section Neural network decoder for near-term surface-code experiments
    “When applied to the experimental data of [Google Quantum AI, Nature 614, 676 (2023)], the neural network decoder achieves logical error rates approximately 25%25\% lower than minimum-weight perfect matching, approaching the performance of a maximum-likelihood decoder. To demonstr…”
  3. resultsupported

    The Sparse Mamba Decoder (SMD) reduces the MWPM logical error rate by up to 49% at d ≤ 5 under SI1000 noise.

    [6] Sparse Mamba Decoder for Quantum Error Correction: Efficient Defect-Centric Processing of Surface Code Syndromes — abstract arXiv:2605.17156v2
    “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…”
  4. resultsupported

    Under circuit-level depolarizing noise on the toric code, weighting the union-find decoder increases the threshold from 0.38% to 0.62%.

    [4] Fault-Tolerant Weighted Union-Find Decoding on the Toric Code — abstract arXiv:2004.04693v1
    “Quantum error correction requires decoders that are both accurate and efficient. To this end, union-find decoding has emerged as a promising candidate for error correction on the surface code. In this work, we benchmark a weighted variant of the union-find decoder on the toric co…”
  5. resultsupported

    Under circuit-level depolarizing noise on the toric code, weighting a matching decoder increases the threshold from 0.65% to 0.72%.

    [4] Fault-Tolerant Weighted Union-Find Decoding on the Toric Code — abstract arXiv:2004.04693v1
    “Quantum error correction requires decoders that are both accurate and efficient. To this end, union-find decoding has emerged as a promising candidate for error correction on the surface code. In this work, we benchmark a weighted variant of the union-find decoder on the toric co…”
  6. resultsupported

    With quantum non-demolition measurements, weighted union-find decoding achieves a threshold of 0.76% compared to 0.90% for weighted matching on the toric code.

    [4] Fault-Tolerant Weighted Union-Find Decoding on the Toric Code — section Fault-Tolerant Weighted Union-Find Decoding on the Toric Code
    “Quantum error correction requires decoders that are both accurate and efficient. To this end, union-find decoding has emerged as a promising candidate for error correction on the surface code. In this work, we benchmark a weighted variant of the union-find decoder on the toric co…”
  7. methodsupported

    The neural network typically outperforms the matching decoder due to better handling of errors leading to multiple correlated syndrome defects, such as Y errors.

    [5] Neural network decoder for near-term surface-code experiments — abstract arXiv:2307.03280v2
    “Neural-network decoders can achieve a lower logical error rate compared to conventional decoders, like minimum-weight perfect matching, when decoding the surface code. Furthermore, these decoders require no prior information about the physical error rates, making them highly adap…”
  8. resultsupported

    Considering soft information from analog readout leads to an approximately 10% lower logical error rate for the neural network decoder.

    [5] Neural network decoder for near-term surface-code experiments — section Neural network decoder for near-term surface-code experiments
    “When applied to the experimental data of [Google Quantum AI, Nature 614, 676 (2023)], the neural network decoder achieves logical error rates approximately 25%25\% lower than minimum-weight perfect matching, approaching the performance of a maximum-likelihood decoder. To demonstr…”
  9. resultsupported

    Under circuit-level depolarizing noise on the toric code, weighted matching achieves a threshold of 0.72% compared to 0.62% for weighted union-find.

    [4] Fault-Tolerant Weighted Union-Find Decoding on the Toric Code — abstract arXiv:2004.04693v1
    “Quantum error correction requires decoders that are both accurate and efficient. To this end, union-find decoding has emerged as a promising candidate for error correction on the surface code. In this work, we benchmark a weighted variant of the union-find decoder on the toric co…”
  10. resultsupported

    SMD maintains nearly constant latency (24–57 μs) across d=3–9 under uniform circuit-level noise.

    [6] Sparse Mamba Decoder for Quantum Error Correction: Efficient Defect-Centric Processing of Surface Code Syndromes — section Sparse Mamba Decoder for Quantum Error Correction: Efficient Defect-Centric Processing of Surface Code Syndromes
    “than the Tesseract near-MLD decoder and 232–463×\times faster than Belief Matching, and maintains nearly constant latency (24–57 μ\mus) across d=3d=3–99 under uniform circuit-level noise. On the Sycamore experimental dataset, the SMD ensemble matches or slightly surpasses the den…”

References

Sources

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    Tzu-Hao Lin, Ching-Yi Lai. Union-Intersection Union-Find for Decoding Depolarizing Errors in Topological Codes. arXiv, 2025.arxiv · primary · https://arxiv.org/abs/2506.14745v1
  2. [2]
    Alexandru Paler, Austin G. Fowler. Pipelined correlated minimum weight perfect matching of the surface code. arXiv, 2022.arxiv · primary · https://arxiv.org/abs/2205.09828v2
  3. [3]
    John Blue, Harshil Avlani, Zhiyang He, Liu Ziyin, Isaac L. Chuang. Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes. arXiv, 2025.arxiv · primary · https://arxiv.org/abs/2504.13043v2
  4. [4]
    Shilin Huang, Michael Newman, Kenneth R. Brown. Fault-Tolerant Weighted Union-Find Decoding on the Toric Code. arXiv, 2020.arxiv · primary · https://arxiv.org/abs/2004.04693v1
  5. [5]
    Boris M. Varbanov, Marc Serra-Peralta, David Byfield, Barbara M. Terhal. Neural network decoder for near-term surface-code experiments. arXiv, 2023.arxiv · primary · https://arxiv.org/abs/2307.03280v2
  6. [6]
    Samira Sayedsalehi, Nader Bagherzadeh, Maxim Shcherbakov, Jean-Luc Gaudiot. Sparse Mamba Decoder for Quantum Error Correction: Efficient Defect-Centric Processing of Surface Code Syndromes. arXiv, 2026.arxiv · primary · https://arxiv.org/abs/2605.17156v2
  7. [7]
    Antonio deMarti iOlius, Imanol Etxezarreta Martinez, Joschka Roffe, Josu Etxezarreta Martinez. An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise. arXiv, 2024.arxiv · primary · https://arxiv.org/abs/2409.01440v3