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 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 Quantum error-correction decoders must convert noisy syndrome records into logical decisions while operating under strict latency and reliability constraints. Minimum-weight perfect matching (MWPM) remains a strong baseline for surface-code decoding because it is fast, interpretable, and exact for its graphlike optimization problem [1–8]. However, the graphlike independent-error approximation can become mismatched when syndrome records contain temporal drift, measurement corruption, crosstalk, bursts, or mixtures of mechanisms [9–18]. We introduce SAGE-QEC, a failure-aware selective decoder that retains nominal PyMatching as its default decision and learns when an observable syndrome structure warrants an alternative candidate. SAGE-QEC uses train-frozen structure thresholds, a candidate ranker, validation-only policy selection, and a paired blind evaluation. Across 45 simulator conditions, SAGE-QEC reduced the aggregate logical error rate from 27.1285% for PyMatching to 24.8320%, corresponding to an 8.47% relative reduction. The campaign used rotated surface-code memory circuits at distances 3, 5, and 7, nominal circuit parameters 0.002, 0.003, and 0.005, five seeds, and four test regimes withheld from training: Burst-3, Temporal Drift, Crosstalk, and Mixed Noise. The test set contained 81,000 paired shots. All twelve distance-by-regime aggregate comparisons favored SAGE-QEC in the executed campaign, with paired bootstrap intervals below zero for SAGE-QEC minus PyMatching. The result is evidence of a reproducible simulator-level improvement, not a claim of universal decoder superiority. SAGE-QEC currently uses logical-output proxy candidates rather than a complete physical correction search, and its intervention rate remains high under distribution shift. It has not yet been compared with a fully calibrated correlated-noise space-time PyMatching baseline or hardware calibration data. We therefore characterize SAGE-QEC as a strong qualification-stage prototype and release a complete audit protocol for the next industrial validation stage.
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.
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.
We consider the rotated surface code of odd distance d with its standard syndrome-extraction circuit, under circuit-level depolarizing noise in which every gate, idle step, preparation and measurement fails independently with probability p, decoded by minimum-weight perfect matching with the log-likelihood weights of single faults. We prove that a memory experiment with T rounds fails with probability at most 0.061 (T+1)(d+1) λd/(1−λ), where λ = 0.999 (p/p**)9/25, for every p ≤ p** = 2×10−3. Hence the logical error rate per round decays exponentially in d for all p ≤ p**. Earlier rigorous thresholds for matching on surface-code circuits are 1.7×10−4, due to Dennis, Kitaev, Landahl and Preskill, and 7.4×10−4, due to Fowler. We keep their reduction of a failure to a weighted self-avoiding path in the decoding graph, but bound these paths by a local criterion for non-backtracking walks, which gives the threshold 1.04×10−3 in closed form. To reach 2×10−3 we use the minimality of the correction along the path, for a canonical choice among the minimum-weight corrections: labellings of short stretches of the path into error and correction edges that contradict this minimality are excluded, and the remaining labelled walks are counted with a potential function that is checked in exact arithmetic. For a leaner schedule with fused measure-and-reset steps the two arguments give 1.53×10−3 and 3×10−3, respectively. With error rates that depend on the type of operation, they certify exponential decay for error models built from the published component error rates of Quantinuum's Helios and H2 and of IBM's Heron r3 processor ibm_boston, mapped onto this circuit. All theorems are also proved in the proof assistant Lean 4. Any argument that bounds the failure probability by the probability of an increasing event, ours included, is limited by the smallest such event that contains all failures, and Monte Carlo estimates place the threshold of this event near 4×10−3 for this circuit. Changes from version 1: a third exclusion rule, which uses a canonical choice among the minimum-weight corrections, raises the certified thresholds to 2×10−3 for the standard schedule and 3×10−3 for the lean one and strengthens the certificates of the device error models. Lean 4 proofs of all theorems are new in this version. Programs, certificates, formal proofs and the outputs of all checks: doi:10.5281/zenodo.23485205. Mathematics Subject Classification (2020): 81P73; 94B35, 05C38, 68V05.
Surface codes exploit topological protection to increase error resilience in quantum computing devices and can in principle be implemented in existing hardware. They are one of the most promising candidates for active error correction, not least due to a polynomial-time decoding algorithm which admits one of the highest predicted error thresholds. We consider the dependence of this threshold on underlying assumptions including different noise models, and analyze the performance of a minimum weight perfect matching (MWPM) decoding compared to a mathematically optimal maximum likelihood (ML) decoding. Our ML algorithm tracks the success probabilities for all possible corrections over time and accounts for individual gate failure probabilities and error propagation due to the syndrome measurement circuit. We present the very first evidence for the true error threshold of an optimal circuit level decoder, allowing us to draw conclusions about what kind of improvements are possible over standard MWPM.
The minimum-weight perfect matching (MWPM) decoder is a standard decoding strategy for surface codes, but its performance can degrade considerably under biased noise. In this work, a modified family of surface codes, termed the XYZ planar code, is introduced, and MWPM is extended to posterior MWPM (pMWPM), which exploits the information provided by mixed stabilizers with almost no increase in decoding complexity. The XYZ--pMWPM and XZZX code--decoder combinations are compared under homogeneous independent and identically distributed (IID) Pauli noise and processor-derived independent nonidentically distributed (INID) noise. Under homogeneous biased Pauli noise, pMWPM substantially increases the threshold of the XYZ planar code relative to conventional MWPM. At finite code distances, particularly for small codes under strong bias, XYZ--pMWPM achieves logical error rates comparable to or lower than those of XZZX. Under INID noise, the pseudothreshold ordering observed under the corresponding homogeneous IID references is not always preserved; in some hardware--distance settings, XYZ--pMWPM achieves pseudothresholds comparable to or higher than those of XZZX. Within XZZX, qubit-specific weighting consistently yields higher INID pseudothresholds than average-rate weighting for the hardware instances considered. It is further found that the IID--INID pseudothreshold shift depends strongly on the physical-error scalarization and the construction of the homogeneous IID reference. These results show that pMWPM efficiently exploits the mixed-stabilizer structure of the XYZ planar code and provides competitive finite-distance decoding under biased and spatially nonuniform noise.