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arXiv:2610.26785

Correlated decoding loses the surface code's effective distance below threshold

César Hernando de la Fuente, Joonas Majaniemi, Evert van Nieuwenburg, Felix Frohnert

On Aqora:Cesar HernandoCesar Hernando
quant-phSubmitted Oct 19, 2026• Announced Oct 20, 2026• 1 citation
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Abstract

Correlated decoding improves matching and clustering surface-code decoders by reweighting one decoding subgraph conditioned on the other's correction. We show that standard reweighting can cause correlated decoders to fail on size-$(d-1)/2$ fault configurations that a distance-$d$ code should correct, reducing error suppression deep below threshold. Under circuit-level noise, a single first-pass edge can discount several correlated edges, even though their joint occurrence requires additional independent faults, so the decoder favors an over-discounted correction. Regularizing the reweighting rule restores the nominal distance scaling and a clear correlated-decoding advantage.

AI Summary

The paper asks whether two-pass correlated surface-code decoders retain their accuracy advantage deep below threshold. It studies correlated minimum-weight matching and a weighted Union-Find-based Local Clustering Decoder on rotated surface-code memories with $d$ extraction rounds, primarily under SI1000 circuit-level noise. Standard reweighting can make several edges cheap conditional on one first-pass edge, although their joint occurrence requires additional independent faults. This produces logical failures from $(d-1)/2$-fault configurations, introducing contributions proportional to $p^{(d-1)/2}$ rather than the nominal $p^{(d+1)/2}$. Evidence combines simulations at $d=3,5,7,9$, exhaustive fault enumeration at $d=3,5$, sampled fixed-fault sectors at $d=7$, and explicit failure mechanisms. At $p=10^{-5}$, correlated matching fails about $4.4$ times as often as uncorrelated matching for $d=3$. Low-$p$ fitted exponents are $1.28\pm0.03$ and $2.25\pm0.04$ for $d=3,5$, versus nominal values $2,3$. Supporting tests find analogous degradation under uniform depolarizing noise and in both memory bases. The proposed remedy multiplies conditional edge probabilities by an empirically fitted, error-rate-dependent strength $\alpha(p)$. Updates are precomputed without changing the decoding subroutine. Regularization recovers nominal fitted slopes over the sampled windows and improves on uncorrelated decoding; at $p=10^{-4}$, matching error-rate ratios are approximately $0.55$ for $d=5$ and $0.35$ for $d=7$. Limitations include unresolved low-$p$ crossovers at $d=7,9$, empirical tuning, and residual nominally correctable-fault failures at $d=5,7$. General distance preservation is not proved; extension to other correlation-aware decoders remains open.

Little sign of AI-generated textOverclaims

Central claim may not hold: The abstract claims restoration of nominal distance scaling, while the regularized decoder's reported residual correctable-fault failures and finite-window scaling fits establish recovery only over the tested regime, not general asymptotic distance preservation. The fitted d=9 correlated-LCD regularization eventually disables every correlation update as p→0, forcing its error-rate ratio to the uncorrelated decoder to equal 1 rather than remain below 1. The appendix's probability-product decision condition and regularization bound omit log-odds factors that can change which correction wins, even at arbitrarily small physical error rates.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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Correlated decoding improves matching and clustering surface-code decoders by reweighting one decoding subgraph conditioned on the other's correction. We show that standard reweighting can cause correlated decoders to fail on size-(d−1)/2(d-1)/2(d−1)/2 fault configurations that a distance-ddd code should correct, reducing error suppression deep below threshold. Under circuit-level noise, a single first-pass edge can discount several correlated edges, even though their joint occurrence requires additional independent faults, so the decoder favors an over-discounted correction. Regularizing the reweighting rule restores the nominal distance scaling and a clear correlated-decoding advantage.
The paper asks whether two-pass correlated surface-code decoders retain their accuracy advantage deep below threshold. It studies correlated minimum-weight matching and a weighted Union-Find-based Local Clustering Decoder on rotated surface-code memories with ddd extraction rounds, primarily under SI1000 circuit-level noise.
Standard reweighting can make several edges cheap conditional on one first-pass edge, although their joint occurrence requires additional independent faults. This produces logical failures from (d−1)/2(d-1)/2(d−1)/2-fault configurations, introducing contributions proportional to p(d−1)/2p^{(d-1)/2}p(d−1)/2 rather than the nominal p(d+1)/2p^{(d+1)/2}p(d+1)/2. Evidence combines simulations at d=3,5,7,9d=3,5,7,9d=3,5,7,9, exhaustive fault enumeration at d=3,5d=3,5d=3,5, sampled fixed-fault sectors at d=7d=7d=7, and explicit failure mechanisms. At p=10−5p=10^{-5}p=10−5, correlated matching fails about 4.44.44.4 times as often as uncorrelated matching for d=3d=3d=3. Low-ppp fitted exponents are 1.28±0.031.28\pm0.031.28±0.03 and 2.25±0.042.25\pm0.042.25±0.04 for d=3,5d=3,5d=3,5, versus nominal values 2,32,32,3. Supporting tests find analogous degradation under uniform depolarizing noise and in both memory bases.
The proposed remedy multiplies conditional edge probabilities by an empirically fitted, error-rate-dependent strength α(p)\alpha(p)α(p). Updates are precomputed without changing the decoding subroutine. Regularization recovers nominal fitted slopes over the sampled windows and improves on uncorrelated decoding; at p=10−4p=10^{-4}p=10−4, matching error-rate ratios are approximately 0.550.550.55 for d=5d=5d=5 and 0.350.350.35 for d=7d=7d=7.
Limitations include unresolved low-ppp crossovers at d=7,9d=7,9d=7,9, empirical tuning, and residual nominally correctable-fault failures at d=5,7d=5,7d=5,7. General distance preservation is not proved; extension to other correlation-aware decoders remains open.
Efficient and realistic error decoding is crucial for fault-tolerant quantum computation (FTQC) on near-term devices. While decoding is a classical post-processing task, its effectiveness depends on accurately modeling quantum noise, which is hardware-dependent. In particular, correlated bit-flip (XXX) and phase-flip (ZZZ) errors often arise under circuit-level noise. We introduce the Iterative Reweighting Minimum-Weight Perfect Matching (IRMWPM) decoder, which systematically incorporates such correlations to enhance quantum error correction. Our method leverages fault-detection patterns to guide reweighting: correlated XXX and ZZZ detection events are identified, and their conditional probabilities update weights on the primal and dual lattices. This iterative procedure improves handling of realistic error propagation in a hardware-agnostic yet noise-aware manner. We prove that IRMWPM converges in finite time while preserving the distance guarantee of MWPM. Numerical results under circuit-level noise show substantial improvements. For distances ≥17\geq 17≥17 and physical error rates ≤0.001\leq 0.001≤0.001, IRMWPM reduces logical error rates by over 20x with only a few iterations. It also raises the accuracy threshold from 1% to 1.16%, making it practical for near-term real-time decoding. Extrapolated estimates suggest that to reach logical error rate 10−1610^{-16}10−16, IRMWPM requires distance d=31d=31d=31, while standard MWPM needs d=50d=50d=50, implying a major reduction in qubit overhead.
Quantum hardware suffers from high error rates and noise, which makes directly running applications on them ineffective. Quantum Error Correction (QEC) is a critical technique towards fault tolerance which encodes the quantum information distributively in multiple data qubits and uses syndrome qubits to check parity. Minimum-Weight-Perfect-Matching (MWPM) is a popular QEC decoder that takes the syndromes as input and finds the matchings between syndromes that infer the errors. However, there are two paramount challenges for MWPM decoders. First, as noise in real quantum systems can drift over time, there is a potential misalignment with the decoding graph's initial weights, leading to a severe performance degradation in the logical error rates. Second, while the MWPM decoder addresses independent errors, it falls short when encountering correlated errors typical on real hardware, such as those in the 2Q depolarizing channel. We propose DGR, an efficient decoding graph edge re-weighting strategy with no quantum overhead. It leverages the insight that the statistics of matchings across decoding iterations offer rich information about errors on real quantum hardware. By counting the occurrences of edges and edge pairs in decoded matchings, we can statistically estimate the up-to-date probabilities of each edge and the correlations between them. The reweighting process includes two vital steps: alignment re-weighting and correlation re-weighting. The former updates the MWPM weights based on statistics to align with actual noise, and the latter adjusts the weight considering edge correlations. Extensive evaluations on surface code and honeycomb code under various settings show that DGR reduces the logical error rate by 3.6x on average-case noise mismatch with exceeding 5000x improvement under worst-case mismatch.
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.
Fault-tolerant quantum computing critically relies on quantum error correction. We study surface-code decoding with diagonally correlated two-qubit Z\mathbf{Z}Z errors, a structured noise model that breaks assumptions leveraged by MinimumWeight Perfect Matching (MWPM). We develop a graph neural network (GNN) decoder based on learned message passing and benchmark it against a tuned MWPM baseline informed by the same noise statistics. In code-capacity simulations at code distances 3, 5, and 7, the GNN achieves consistently lower logical error rates than MWPM across the tested range of pair-error probabilities. The GNN correction threshold is about 2.5%, surpassing the 2.4% achieved by the MWPM baseline. These results indicate that data-driven message passing can adapt to complex, geometry-dependent noise without hand-crafted correlation models, thereby reducing reliance on manual model specification for quantum error correction.
Studying the threshold behavior of surface codes under biased noise models is an active area of research. Previous work Tuckett et al. (2018), using an optimal tensor-network decoder, demonstrated that replacing ZZZ-type stabilizers with YYY-type stabilizers significantly improves the surface code threshold under code-capacity level dephasing noise. In this work, we construct and study a ZYZYZY surface code by replacing the XXX-type stabilizers with YYY-type stabilizers. We compare it with the standard ZXZXZX surface code under circuit-level Pauli-XXX biased noise, with and without an additional gate-based XXXXXX crosstalk noise. We find that for the ZXZXZX surface code, the XXX-memory threshold increases monotonically with bias while the ZZZ-memory threshold decreases and saturates. For the ZYZYZY surface code, the YYY-memory threshold is nearly constant across all bias values. The ZZZ-memory thresholds of the ZXZXZX and ZYZYZY codes are consistent within the uncertainty. Adding XXXXXX crosstalk reduces the ZZZ-memory threshold beyond the fitting uncertainty while leaving the XXX memory threshold largely unaffected. The choice of CNOT ordering redistributes threshold performance between the two logical memories. Our work extends prior observations from code-capacity level noise to circuit-level noise. It also indicates the need for decoders capable of jointly reasoning over correlated syndrome information so that tailored stabilizer structures could be fully utilized for quantum error correction.
Quantum error correction (QEC) is required for large-scale computation, but incurs a significant resource overhead. Recent advances have shown that by jointly decoding logical qubits in algorithms composed of transversal gates, the number of syndrome extraction rounds can be reduced by a factor of the code distance ddd, at the cost of increased classical decoding complexity. Here, we reformulate the problem of decoding transversal circuits by directly decoding relevant logical operator products as they propagate through the circuit. This procedure transforms the decoding task into one closely resembling that of a single-qubit memory propagating through time. The resulting approach leads to fast decoding and reduced problem size while maintaining high performance. Focusing on the surface code, we prove that this method enables fault-tolerant decoding with minimum-weight perfect matching, and benchmark its performance on example circuits including magic state distillation. We find that the threshold is comparable to that of a single-qubit memory, and that the total decoding run time can be, in fact, less than that of conventional lattice surgery. Our approach enables fast correlated decoding, providing a pathway to directly extend single-qubit QEC techniques to transversal algorithms.