We describe a decoder for the rotated surface code that runs as parallel tempering on probabilistic bits (p-bits) built from single-flux-quantum (SFQ) cells at 4.2 K. The X graph is decoded first, and its result switches the weights of the Z graph to account for X–Z correlations from Y errors. The p-bits, the edge toggle cells with merger trees for up to five writers and the timing of the colour clock are simulated at circuit level in JoSIM, and the measured properties enter a GPU model of the full decoder. Over d rounds of circuit-level noise at p = 0.006, paired shot by shot, the decoder has 14 to 18 % fewer logical failures than minimum-weight perfect matching (MWPM) at d = 5 and 27 to 36 % fewer at d = 7 (two independent seeds) and, with both seeds pooled, 2.2 % (d = 5) and 4.7 % (d = 7) more failures than correlated matching. Without the weight switching its logical outcome agrees with MWPM in almost every shot, so the gain comes from the X–Z correlation step (97 to 104 % of it); a deterministic two-stage control with the same switching rule comes within 2 % of correlated matching, so most of the remaining gap is due to the sampler. At p = 0.004 and 0.008 the decoder has 15 to 48 % fewer failures than MWPM (one seed). A block of d rounds is decoded in 0.53 µs (d = 5) and 1.67 µs (d = 7). Counted from the simulated cells, the core of the d = 7 decoder (p-bits, toggle cells, splitters, mergers; control logic and wiring excluded) needs 6.82 million Josephson junctions and 11 to 22 mW under energy-efficient RSFQ (ERSFQ) biasing. The decoder is a simulation study; it has not been built. Files. P5_manuscript.pdf: the manuscript. sfq_pbit_decoder_p5.tar.gz: simulation and analysis scripts (JoSIM test benches, GPU decoder model and its patches, evaluation scripts), JoSIM raw results, per-shot logical outcomes of all decoder runs, the receipts from which every number and figure is generated, the automated number checks and a self-test that reruns all evaluations and compares them field by field with the receipts. See README.md in the archive. Licences. Text and data CC BY 4.0, code MIT. Use of AI. Large language models (Claude, Anthropic) wrote most of the simulation and analysis code, ran the simulations, searched and read the literature, produced the figures and drafted the text under the author's direction; AI agents were also used as adversarial reviewers. Every number is generated by scripts from the raw simulation output and checked automatically.
We describe a decoder for the rotated surface code that runs as parallel tempering on probabilistic bits (p-bits) built from single-flux-quantum (SFQ) cells at 4.2 K. The X graph is decoded first, and its result switches the weights of the Z graph to account for X–Z correlations from Y errors. The p-bits, the edge toggle cells with merger trees for up to five writers and the timing of the colour clock are simulated at circuit level in JoSIM, and the measured properties enter a GPU model of the full decoder. Over d rounds of circuit-level noise at p = 0.006, paired shot by shot, the decoder has 14 to 18 % fewer logical failures than minimum-weight perfect matching (MWPM) at d = 5 and 27 to 36 % fewer at d = 7 (two independent seeds) and, with both seeds pooled, 2.2 % (d = 5) and 4.7 % (d = 7) more failures than correlated matching. Without the weight switching its logical outcome agrees with MWPM in almost every shot, so the gain comes from the X–Z correlation step (97 to 104 % of it); a deterministic two-stage control with the same switching rule comes within 2 % of correlated matching, so most of the remaining gap is due to the sampler. At p = 0.004 and 0.008 the decoder has 15 to 48 % fewer failures than MWPM (one seed). A block of d rounds is decoded in 0.53 µs (d = 5) and 1.67 µs (d = 7). Counted from the simulated cells, the core of the d = 7 decoder (p-bits, toggle cells, splitters, mergers; control logic and wiring excluded) needs 6.82 million Josephson junctions and 11 to 22 mW under energy-efficient RSFQ (ERSFQ) biasing. The decoder is a simulation study; it has not been built. Files. P5_manuscript.pdf: the manuscript. sfq_pbit_decoder_p5.tar.gz: simulation and analysis scripts (JoSIM test benches, GPU decoder model and its patches, evaluation scripts), JoSIM raw results, per-shot logical outcomes of all decoder runs, the receipts from which every number and figure is generated, the automated number checks and a self-test that reruns all evaluations and compares them field by field with the receipts. See README.md in the archive. Licences. Text and data CC BY 4.0, code MIT. Use of AI. Large language models (Claude, Anthropic) wrote most of the simulation and analysis code, ran the simulations, searched and read the literature, produced the figures and drafted the text under the author's direction; AI agents were also used as adversarial reviewers. Every number is generated by scripts from the raw simulation output and checked automatically.
Abstract We study decoder-representation sensitivity in simulated rotated surface-code Z-memory circuits with local spatially correlated XX pair events. At distance 3, ordinary minimum-weight perfect matching (MWPM) produced 4,610 versus 1,260 logical failures at p_corr = 10^-4 and 5,437 versus 2,239 at p_corr = 2 × 10^-4 on the same 500,000-shot samples using original-order versus canonical-sorted detector error models (DEMs). Across 50 additional seeded permutations per condition, failure counts varied widely. Canonical sorting tied for the minimum in this finite tested set; it is not claimed to be universally optimal. The ordering experiment reorders complete error instructions in a flattened DEM, preserving their probabilities and absolute detector and logical targets. PyMatching documents retaining the first-added logical-label set when parallel graph edges are merged; we therefore interpret the findings as a workflow-specific reproducibility result, not a newly discovered general software defect. At distances 5 and 7, correlation-aware MWPM had lower observed failure rates in all eight tested conditions, with six paired contrasts remaining significant after Bonferroni correction. An exact maximum-a-posteriori calculation gave a 0.2367% Bayes risk for the flattened-DEM distribution at p_corr = 10^-4; this is optimal for that modeled distribution only. This bounded classical simulation study proposes no new decoder, estimates no threshold, and provides no hardware evidence. Some historical source-to-result provenance remains incomplete, as disclosed in the manuscript. Code and evidence: https://github.com/tgiwn/QEC-surface-code-correlated-noise/tree/106d32e815c5f6009941afc36d3f65be7ecf35a3 Status: This is a preprint and has not undergone peer review.
Abstract One of the critical requirements for fault-tolerant quantum computation is fast and reliable decoding,Minimum-weight perfect matching (MWPM) is a reliable surface-code decoder,but it will face long decoding time when decoding instances grow. Neural-network decoders was used to solve this problem, but large models often require costly retraining when the code distance or noise model changes.In this work, we propose a two-stage decoder that combines pretrained graph neural network (GNN) residual pre-decoding with global MWPM cleanup.The GNN receives a detector graph generated by the established decoding backend and assigns confidence scores to local correction candidates.Two staggered layers of parallel local windows process confidence corrections.The residual syndrome is passed to the NVIDIA-accelerated MWPM backend.On matched batches of 32,768 samples, the complete method reduced the logical failure rate by 6.25% to 18.01% relative to Pure MWPM for code distances d=3,5,7 and 9 at physical error rates p = 0.007 to 0.013. At d=3,it decoded 3.48x – 4.38x faster over p=0.003 to 0.021. We also trained separate GNN residual pre-decoders for four circuit-level noise models over 160 code-distance/error-rate configurations and obtained 2.25x to 5.41x speedups in every case.The logical-failure benefit depended on the noise model,but the pace is all improving. Our work can accelerate surface-code decoding pace and extend to others noise models.
SUPERSEDED DUPLICATE — do not cite this record. Cite 10.5281/zenodo.21698294 instead. Both records were deposited on 2026-07-30 and contain the same paper. This one carries an earlier, superseded NOTE.md whose resource-scaling section still claimed L≈6–7 (d=36–49); the surviving record carries the corrected text (measured L=4, [[1536,6,16]]) and the toric4d_lowp.py stratified estimator behind it. The independent cross-validation scripts that were unique to this record (magic_state4-7.py) have been added to the surviving record. Flagged 2026-07-31. We simulate a 15-to-1 magic-state factory under full circuit-level depolarizing noise and ask whether a 4D toric code — which corrects errors single-shot, without repeated syndrome rounds — relieves the factory's dominant cost. It does not. Distillation output error is confirmed at 35p³ by exact enumeration and by stim, but the factory's circuit-level error floor is first order in the gate error rate and schedule-dependent, measured at 12–15·p_gate for the real [[15,1,3]] quantum Reed–Muller code with all 14 stabilizers. This is 3.5× larger than a previously reported 4.18·p_gate, which we trace to a circuit that omitted the code's 10 Z-type checks; that figure is withdrawn here. The floor originates in the final syndrome sweep: faults deposited on data qubits by the last round of gates are never checked by anything after them. Whichever check type is measured last, that sector's faults escape, so rescheduling shifts the floor between sectors and interleaving shaves roughly 20%, but cannot remove it. Reordering, redundant checks and flag qubits leave it first order; decoding weight-1 errors instead of discarding them makes it 5× worse. We construct the 4D toric code on an L⁴ torus from its cubical chain complex and verify [[486,6,9]] at L=3 (k=6 exactly, explicit weight-9 logicals in both sectors, d=4 exact at L=2 by exhaustive search), and measure a code-capacity threshold between 5% and 7% under BP+OSD. Single-shot repair using the 80 independent metachecks recovers near-perfect-syndrome performance, but only while syndrome noise stays roughly 10–30× below the data error rate: it improves logical error by 195× at q=10⁻³ and by only 4.5× at q=10⁻². Crucially, metachecks constrain the syndrome, not the data, and cannot see faults deposited directly on data qubits by the final sweep — which is precisely where the factory floor originates. Single-shot therefore buys latency, not factory fidelity. Because the factory's Cliffords act on encoded blocks, p_gate is the logical error rate of the hosting code. The correction moves the surface-code requirement from d=15 to d=17 (578 physical qubits per patch); on the 4D toric side, the smallest sufficient block is L=4, [[1536,6,16]], with [[486,6,9]] missing the fidelity demand by four to five orders of magnitude. We situate the result against Xu et al. (arXiv:2407.18490), whose parallel distillation across qLDPC patches reduces space–time cost but leaves the per-factory floor untouched. A corrections log records eight claims withdrawn or corrected during this work, including our own earlier 4.18·p_gate figure and a d=15 requirement that rested on a single error event. Section 6 states what is not established: these are code-capacity numbers for the 4D code, the distance requirement is quoted at p=3–5% rather than 10⁻³, d=L² is verified exactly only at L=2, the Λ extrapolations are long, and no end-to-end injection simulation was run. Correction, 2026-07-31. The files deposited with this record on 2026-07-30 predate a low-p re-measurement of the resource-scaling section. The originally deposited text stated that hosting the factory requires L≈6–7 (d=36–49, n≈7800–14400); that figure came from fitting a single suppression factor Λ to near-threshold points at p=3–5%. The single-Λ ansatz does not hold for this code: the measured suppression per two units of distance runs from 10.9 (d=4→9 at p=0.02) to 793 (d=9→16 at p=0.01). Re-measuring at the factory's actual operating regime with a weight-stratified estimator gives d ≥ 14.5 at p=0.02 and d ≥ 9.7 at p=0.01, so the smallest sufficient 4D block is L=4, [[1536,6,16]], not L=6–7. The L≈6–7 figure is withdrawn. These are point estimates whose upper confidence bands remain wide at the smallest p, and should be read as order-of-magnitude. The corrected text and the estimator script (toric4d_lowp.py) are not in this record's file set; replacing files requires a new version, which will carry the corrected note. Nothing in Sections 1–4 — the 35p³ distillation result, the 12–15·p_gate factory floor, the last-sweep mechanism, or the d=17 surface-code requirement — is affected.
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) 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) 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.
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 (X) and phase-flip (Z) 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 X and Z 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 and physical error rates ≤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 10−16, IRMWPM requires distance d=31, while standard MWPM needs d=50, implying a major reduction in qubit overhead.
Abstract. Suppressing qubit errors is one of the main challenges in building fault-tolerant quantum computers, requiring quantum error correction to protect fragile quantum information during computation cycles. The surface code is a leading approach because of its high error tolerance and compatibility with realistic hardware, but it requires a decoder that can accurately and quickly use the syndrome measurements to discern whether a logical error has occurred. This study investigates whether a feedforward neural-network can serve as a practical decoder for the surface code under circuit-level noise. Using the Stim software package, this study generates surface code circuits at multiple code distances and physical error rates and compiles them into detector samplers, then trains a PyTorch neural-network to predict whether a logical observable flip occurred from the syndrome data. The neural-network decoder is compared against Minimum-Weight Perfect Matching (MWPM), focusing on metrics including logical error rate and inference speed. The results show that the neural-network decoder performs competitively with MWPM for code distances 3 and 5 and provides inference time advantages at higher physical error rates as its inference time remains relatively constant over increasing physical error rates. However, MWPM outperforms the neural-network at distance 7, suggesting that scalability remains a major limitation for neural-network decoders. These results show that neural-networks are a promising approach for tackling quantum error correction, although future work is still needed to improve these models for real systems at larger code distances.