Error-Correcting Codes

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30 papers4 papers in the last four weeks, level with the four weeks before. 0.0% of all new papers.

Oct 13, 2026

Distance Prediction for Quantum Codes

The distance of a quantum code is a fundamental metric that dictates its error-correction capability. While quantum low-density parity-check (qLDPC) families such as Bivariate Bicycle (BB) codes offer a promising pathway to low-overhead fault-tolerant quantum computing, computing their distance is challenging. Current exact solvers and even search heuristics that provide a distance upper bound suffer from severe computational bottlenecks, restricting the discovery of optimal codes. In this paper, we construct a dataset of BB codes with corresponding distance upper bounds obtained through high-budget heuristic searches. We introduce BB-Net, an attention-based neural network architecture for predicting the distance of BB codes. It achieves 96.39% accuracy on the reference distances in the test set, compared with 94.86% for state-of-the-art heuristics with a budget of up to

10^{4}

times more computational runtime. On a subset with certified exact distances, BB-Net achieves 98.94%. By relieving the distance-calculation bottleneck, BB-Net enables faster screening of large code spaces, particularly in ML-driven search pipelines, and thereby can accelerate the discovery of novel quantum error-correcting codes, an important capability for advancing scalable fault-tolerant quantum computing. Our code is publicly available: https://github.com/NadavSc/BBNet

Nadav Schneider, Eliya Nachmani

School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Israel

Quantum LDPC Codes Quantum Error Correction Error-Correcting Codes

Oct 5, 2026

Large Language Model-Guided Discovery of Weight-Five Bivariate Bicycle Codes

Building on our earlier program-evolution workflow guided by large language models (LLMs), we study weight-five bivariate bicycle (BB) and perturbed bivariate bicycle (PBB) codes. The resulting catalogue contains 1,142 distinct code proposals, including 1,081 nonbaseline proposals attributable to LLM-generated programs. Across the catalogue, we certify connected Calderbank--Shor--Steane (CSS) realizations [[96,4,10]], [[140,6,10]], and [[180,4,14]]. A post-search comparison certifies seven imported Lin--Pryadko archive constructions. For leading parameter triples also represented in that archive, we provide exact distance evidence, explicit bivariate presentations, and verified component reductions. A basis-independent connectivity analysis identifies 409 of the 1,142 catalogue entries as disconnected and shows that 73.1% of the classes with exact distance certificates contain repeated connected components. Algebraic analysis organizes the connected CSS classes into order-3, order-7, and order-15 cyclotomic-kernel strata. The strongest exact connected PBB parameter point is [[216,4,10]], attained by two distinct component classes. Among the 936 distinct CSS proposals from the LLM-guided campaign with a recorded positive distance, 816 (87.18%) are certified at

d\geq5

. For comparison, three random-search controls each sample 6,444 CSS code proposals uniformly without replacement, using the same per-lattice and encoded-dimension sample counts as the LLM-guided campaign. In these controls, 4,672--4,785 proposals (72.50--74.26%) meet the same criterion. The LLM-guided campaign has the higher certified yield, while the random controls cover more connected classes. Together, these results extend LLM-guided discovery to a more constrained code family and provide a reproducible structural and exact-distance account of its strongest candidates.

Juan Cruz-Benito

IBM Research

Quantum LDPC Codes Quantum Error Correction Large Language Model-Guided Scientific Discovery Error-Correcting Codes

Sep 30, 2026

Anchor-ECC: Local Integrity Checking for Watermarked LLM Outputs via Error-Correcting Codes

LLM watermarking has become an effective approach to distinguishing AI-generated text from human-written text by embedding detectable patterns during generation. However, a small post-generation edit may change the meaning of the text without removing its overall watermark signal, creating a risk that the modified content is still attributed to the original model. We propose Anchor-ECC, which incorporates the error-correcting code (ECC) constraints and explicit boundary anchors into the watermark structure and pairs them with a dynamic-programming decoder to detect and localize post-generation edits. Across Qwen3-8B, Mistral-7B-Instruct-v0.3, and OPT-125M, the approximate-hard setting achieves about 99.7% block-level true positive rate (TPR) with at most 7.6% false alarm rate (FAR) for edit detection under mixed insertions, deletions, and substitutions, while preserving the distinction between watermarked outputs and unwatermarked text. Additional quality experiments identify lower-perplexity configurations that retain strong edit-detection performance. Together, these results extend LLM watermarking from source identification to local integrity verification while supporting configurable trade-offs between detection reliability and generation quality.

Zewei Deng, Muhammad Siddeek, Liyan Xie +4

University of Minnesota · Google · Oakland University +2

Error-Correcting Codes Text Watermarking LLM Watermarking Robust Watermarking

Sep 29, 2026

Evolving Towards Better Codes: LLM-Guided Search for High-Distance Binary Linear Codes

Evolutionary program search driven by large language models (LLMs) has produced record-breaking constructions for open problems in combinatorics and beyond. We apply this approach to the longstanding problem of improving the best-known bounds for binary linear codes. Building on the EvoTune evolutionary framework and the ShinkaEvolve codebase, we introduce LinCodeEvolve, which evolves code-construction programs against an exact minimum-distance evaluator. A strategy loop combines diversity-driven search and expert supervision: when progress plateaus, new strategies are used to redirect the search. LinCodeEvolve discovers seven record-breaking codes,

[172,21,66]

,

[173,20,68]

,

[176,21,68]

,

[181,21,70]

,

[184,21,72]

,

[189,22,72]

and

[200,21,77]

, six of which have concise quasi-cyclic descriptions. With standard code modification techniques, they improve

22

entries of the tables. Every code is verified by exhaustive enumeration. These results suggest that LLM-guided search can help find improved codes and complement existing methods in coding theory.

Amal Seddas, Vladyslav Shashkov, Maryna Viazovska +1

EPFL

Automated Algorithm Discovery Error-Correcting Codes

Sep 28, 2026

ZeroCode: On-demand Error-Correcting Code Construction from the Zero Matrix via Reinforcement Learning

Error-correcting codes (ECCs) are essential across diverse applications, from wireless communications and storage to quantum computing, yet each application imposes distinct design requirements on the parity-check matrix (PCM). To address these on-demand requirements in a unified framework, we propose ZeroCode, a reinforcement learning (RL)-based approach that constructs PCMs sequentially from the all-zero matrix. ZeroCode formulates construction as a discrete sequential decision-making problem and uses proximal policy optimization with action masking to select valid edges. ZeroCode achieves a gain of approximately 1 dB over the prior RL-based construction method at a bit error rate (BER) of

10^{-4}

for the (32,16) code and outperforms existing genetic, differentiable, and classical code-design methods in our experiments. Beyond optimizing decoding performance, the masking mechanism allows on-demand structural constraints, such as a maximum degree, 4-cycle-free structure, and quasi-cyclic structure, to be flexibly incorporated. Moreover, a single policy rollout yields a library of PCMs with varying edge counts, offering trade-offs between decoding performance and complexity without retraining. Overall, ZeroCode addresses diverse code-design requirements within a unified framework, providing solutions with optimized decoding performance under given constraints.

Ju-Hyeong Lee, Yongjune Kim, Sang-Hyo Kim +2

University of Ulsan · Pohang University of Science and Technology (POSTECH) · Sungkyunkwan University

Error-Correcting Codes RL for Combinatorial Optimization

Sep 10, 2026

Reinforcement Learning for Syndrome Extraction

A key subtask of quantum error correction is to extract a syndrome that, if nontrivial, signals an error. The number of possible ways to extract a syndrome grows exponentially with the syndrome size, and these implementations vary greatly in fault tolerance, as measured by their logical error rates. This creates a natural search problem: find an implementation with a low logical error rate. Previous work solves this problem but sacrifices either solution quality or scalability. In this paper, we use reinforcement learning and importance sampling to outperform previous work at all scales. Compared with the state of the art automatic scheduling tools AlphaSyndrome and PropHunt, our tool reduces the logical error rate by 25.9% and 71.7% on average, respectively, culminating with a reduction of 97.8% for a surface code with distance 15.

John Zhuoyang Ye, Aarav Pabla, Jens Palsberg

University of California, Los Angeles Los Angeles, United States

Quantum Error Correction Error-Correcting Codes

Sep 5, 2026

A machine-checked proof of the Dong-Yang classification of optimal (n,4) binary codes for BSCs

We present a machine-checked Lean~4 formalization of Dong and Yang's classification of optimal finite-length

(n,4)

binary block codes for binary symmetric channels. The formalization was developed mainly by feeding the paper's proofs to an AI tool. To establish correctness, the authors verified the main theorem statements in Lean and the accepted axioms. This note discusses the corrections and simplifications made to the AI-generated formalization, and records discrepancies found in the paper during the formalization. The Lean code is available at https://github.com/shhyang/n4code_lean.

Shenghao Yang, Yanyan Dong

School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, P. R. China.

Error-Correcting Codes

Sep 2, 2026

WeaveMark: Robust and Scalable Multi-bit LLM Watermarking via Coded Payload Spreading

Multi-bit watermarking for large language models enables content source tracing by embedding user-identifiable messages into generated text. Existing methods face a fundamental trade-off among extraction accuracy, text quality, and payload capacity. We propose WeaveMark, a robust and scalable multi-bit LLM watermarking scheme based on coded payload spreading. WeaveMark shifts this trade-off frontier by improving payload capacity through multi-bit-per-token spreading (weaving), improving extraction accuracy through soft-decision error-correcting codes, and preserving text quality through unbiased multilayer reweighting. It further introduces dedicated zero-bit layers for reliable watermark presence detection. Extensive experiments demonstrate substantial gains in extraction performance, especially for long messages and edited text, without degrading text quality. WeaveMark achieves an 89.8% match rate for 32-bit messages at 200 tokens, compared with 20.8% for BiMark. Under 10% substitution attacks on 16-bit messages at 200 tokens, it maintains 86.0% versus 30.7%. Code is available at https://anonymous.4open.science/r/WeaveMark-ED6F.

Gang-Hyun Park, Ju-Hyeong Lee, Hee-Youl Kwak +1

University of Ulsan, Ulsan, Republic of Korea

Multi-Bit Watermarking Error-Correcting Codes Watermark Removal Detection Text Watermarking LLM Watermarking Robust Watermarking

Sep 1, 2026

MaskCode: Mask Transformer for Feedback-Assisted Coding With Linear Block Codes

Feedback-based coding schemes have demonstrated substantial performance gains over today's open-loop coding schemes. Unfortunately, these gains are usually achieved in idealized settings with perfect feedback. Over the last few years, machine learning-based schemes have been shown to be promising solutions for implementing feedback-based codes, particularly when combined with short-block-length open-loop error correcting codes (ECCs) in a concatenated coding structure. However, existing ML-based feedback schemes remain agnostic to the outer code's structure, potentially misallocating feedback resources on error patterns already correctable by the outer ECC. To address this, we propose MaskCode, a Transformer-based inner feedback code for concatenated coding systems, which explicitly incorporates structural knowledge of the outer linear block code into the inner feedback encoder design via two synergistic mechanisms: 1) a soft syndrome-based input that informs the encoder about potential parity constraint violations, and 2) a code-aware attention mask derived from the Tanner graph. We further show that end-to-end training with a differentiable belief propagation (BP) decoder offers no additional gain, as MaskCode's structure-aware design already internalizes the structural knowledge of the outer code; in fact, backpropagation through the iterative BP decoder introduces gradient explosion, which degrades rather than improves performance. Extensive evaluations on BCH and LDPC outer codes demonstrate that MaskCode consistently outperforms all baselines, achieving up to 1.5 dB SNR gain.

Jonggyu Jang, Hongjae Nam, Vishrant Tripathi +2

Department of Electronics Engineering, Chung-nam National University, Daejeon 34134, Republic of Korea · Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA · Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Republic of Korea

Error-Correcting Codes

Jul 28, 2026

OmniQEC: discovering practical quantum error-correcting codes by an AI scientist

Quantum error correction (QEC) is indispensable for scalable fault-tolerant quantum computing. However, discovering QEC codes that remain effective is challenging, as logical performance depends on the interplay between code structure, hardware, syndrome extraction, and decoding, which often impose competing requirements. Here we introduce OmniQEC, an efficient AI scientist for discovering QEC codes suited to deployment on modern quantum processors. OmniQEC formulates QEC design as an iterative discovery process in which an orchestrator, implemented by advanced large language models (LLMs), coordinates code generation, code-level screening, syndrome-extraction synthesis, and decoder-based circuit evaluation. At its core, OmniQEC combines a self-evolving reasoning mechanism with a slow--fast synergistic workflow: a fast loop explores candidates using inexpensive code-level proxies, whereas a slow loop performs physically grounded circuit-level evaluation and feeds the resulting evidence back into the search. We evaluate OmniQEC across four qLDPC construction families, three LLM backends, and

14

total-physical-qubit budgets per backend. The discovered codes show steadily improving logical-error suppression with increasing physical-qubit budgets and outperform the BB codes with

[\![72,12,6]\!]

and

[\![144,12,12]\!]

under complete-implementation budgets of 98 and 240 physical qubits, respectively. The discovered codes are hardware-friendly and may be of independent interest for practical QEC implementation. These findings pave the way towards LLM-assisted QEC discovery grounded in physically informed code--circuit--decoder co-design.

Ge Yan, Shanchuan Li, Pengyue Ma +5

College of Computing and Data Science, Nanyang Technological University, Singapore · Department of Electrical Engineering and Computer Science, Tokyo University of Agriculture and Technology, Koganei, Tokyo, Japan · 3Zhejiang University of Technology, China +3

AI Agents for Scientific Discovery Quantum LDPC Codes Quantum Error Correction Large Language Model-Guided Scientific Discovery Error-Correcting Codes

Jul 21, 2026

Machine-learned syndrome post-selection for reliable quantum error correction

Quantum error correction can be enhanced by post-selecting out runs that are likely to produce a logical failure, but the most accurate measures for that require costly decoder-level information. We introduce a practical, decoder-agnostic post-selection method that learns directly from syndrome data. The method trains a supervised classifier to distinguish between syndromes from low- and high-noise regimes, and then uses the classifier's output as an abort score for new runs, without requiring logical-error labels, correction operators, or code-specific likelihood calculations. We validate the approach in three complementary settings: circuit-level simulations of the Gross bivariate-bicycle code, code-capacity simulations of the surface code, and experimental logical magic-state distillation data from the QuEra neutral-atom processor. In the Gross and surface codes, learned syndrome post-selection reduces the conditional logical error rate at a fixed acceptance rate, with performance comparable to syndrome-weight filtering. For the surface code, the learned classifier reveals a post-selection transition distinct from the conventional decoding threshold. In the experimental data, the machine-learning score outperforms syndrome-weight post-selection and, when combined with logical-gap filtering, improves the output fidelity beyond using the logical gap alone. These results show that syndrome-only learning provides a scalable and hardware-compatible route to improving the reliability of quantum error correction.

Tobias Haug, Askery Canabarro, Leandro Aolita

Quantum Research Center, Technology Innovation Institute, Abu Dhabi, UAE · Campus Arapiraca, Federal University of Alagoas, Arapiraca-AL, Brazil

Quantum Error Correction Classification Error-Correcting Codes

Jul 12, 2026

MDQEC-QAS: Meta-Decoding for Quantum Error Correction with Hardware-Aware VQC Search and Confidence-Gated Recovery

We propose a unified meta-decoding framework for quantum error correction that learns syndrome-to-recovery mappings across multiple stabilizer codes and noise settings, without requiring separate decoders for each configuration. The benchmark includes FiveQubit, Steane, Planar3x3, and Planar5x5 codes, four noise families, and five evaluation regimes: interpolation, unseen-p transfer, unseen-noise transfer, few-shot unseen-code adaptation, and few-shot held-out-size adaptation. We compare a classical Meta-MLP teacher-trained baseline with variational quantum circuit (VQC) meta-decoders selected through hardware-aware quantum architecture search over qubit count, circuit depth, and entangling topology. The Meta-MLP achieves teacher-label accuracies of 0.9993, 0.9118, 0.9342, 0.6304, and 0.7548 across the five regimes, while the hardware-aware VQC achieves 0.9400, 0.8495, 0.8415, 0.5678, and 0.7143. However, logical-level evaluation shows that high teacher-label accuracy alone is insufficient in the most challenging Planar5x5 setting. During interpolation, the raw logical-failure ratios relative to the teacher are 12.08 and 25.91 for the Meta-MLP and VQC, respectively, whereas confidence-gated fallback reduces them to 1.71 and 1.11. These results support confidence-aware selective recovery rather than unconditional teacher replacement.

Prashant Kumar Choudhary, Nouhaila Innan, Muhammad Shafique +1

Department of Physics, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, India · eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD), Abu Dhabi, UAE · Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE

Neural Decoding Selective Prediction Meta-Learning Quantum Architecture Search Quantum Error Correction Variational Quantum Circuits+1

Jul 7, 2026

Latency-Constrained Hardware-Aware Quantum Error Correction Co-Design with Adaptive Confidence-Gated Neural Decoding for the Rotated Surface Code

Real-time decoding is a major bottleneck in scaling quantum error correction (QEC) from noisy intermediate-scale quantum (NISQ) devices to fault-tolerant quantum computing. We present an adaptive confidence-gated decoding framework for the rotated surface code that treats decoding as a two-stage inference problem. A lightweight feed-forward neural network performs fast-path decoding for the majority of syndrome measurements, while only low-confidence predictions are escalated to a minimum-weight perfect matching (MWPM) refinement stage. We benchmark the framework on rotated surface codes with distances

d \in \{3,5,7,9,11\}

under circuit-level depolarising noise using the Stim stabiliser simulator. The evaluation characterises logical accuracy, confidence-controlled accuracy-latency trade-offs, decoding throughput, per-shot latency, and decoding-graph resource scaling. Routing only 3.3%-6.2% of syndromes to the refinement stage improves logical accuracy from 99.21% for the neural-only baseline to 99.81% at a confidence threshold of 0.95 while incurring only a bounded increase in average decoding cost. Neural-decoder throughput saturates near

4.6 \times 10^{5}

samples s

^{-1}

at batch size 512 on commodity CPU hardware, indicating that the neural fast path is not the dominant throughput bottleneck beyond code distance

d=7

. We release the complete benchmarking pipeline, trained models, raw benchmark data, and source code, and explicitly distinguish the experimentally validated contributions from the broader hardware-aware QEC co-design roadmap, including hardware-constrained code discovery, GPU-accelerated inference, and multi-noise optimisation, which remain directions for future work.

Sumit Chongder

Department of Physics, Quantum Information and Computation, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan 342037, India

Neural Decoding Quantum Error Correction Error-Correcting Codes

Jul 7, 2026

Low-Overhead Error-Corrected QCNNs Using Bivariate Bicycle Codes

Quantum convolutional neural networks (QCNNs) combine the power of quantum computing and classical CNN for computational speedup in classification tasks. However, noise levels on state-of-the-art quantum devices remain too high for practical QCNN execution. In addition, despite the reliable surface code providing a method for error rates below a threshold value, they have a prohibitively large qubit cost. Recently introduced bivariate bicycle (BB) codes are of particular interest for their high error threshold, constant encoding rate, and linear code distance. Through simulation with realistic hardware noise sources, we demonstrate that a 4-qubit unprotected QCNN fails to converge and exhibits a worse learning rate compared to numerical simulations. Addressing both limitations, we propose a distance-4 BB quantum error-correction (QEC) technique for QCNNs. In doing so, we validate that our low-overhead QEC technique for QCNNS represents a step toward practical QCNNs.

Alejandro Rosales, Animesh Yadav

School of Electrical Engineering and Computer Science Ohio University, Athens, OH, USA

Quantum Convolutional Neural Networks Quantum Error Correction Quantum Machine Learning Quantum Neural Networks Error-Correcting Codes

Jun 27, 2026

Arbitrary Reduction of Validation Error for AI Decision Tests using Homomorphic AI and Repetition Codes

This paper presents new results and breakthrough obtained with the HbHAI techniques (Hash-based Homomorphic Artificial Intelligence) proposed in \cite{filiol0,sepp}. HbHAI is based on a novel class of key-dependent hash functions that naturally preserve most similarity properties, most AI algorithms rely on. It enables to analyse and process data in its cryptographically secure form while using existing native AI algorithms without modification, with unprecedented performances compared to existing homomorphic encryption schemes and most notably compared to the same processing on corresponding plaintext data. Two major results have been obtained further. First we enable to reduce the compression rate up to a factor of 10 thus allowing to process massive datasets while reducing the computation time and the energy footprint in the same order. Second, we show how it is possible to arbitrarily reduce the final validation error of AI-based decision tests by using repetition error-correcting codes.

Eric Filiol, Jaagup Sepp

Independent Researcher - Retired Professor · Hope4Sec, Tallinn, Estonia

Homomorphic Encryption Privacy-Preserving ML Error-Correcting Codes

Jun 23, 2026

Large-Language-Model Discovery of Quantum LDPC Codes through Structured Concept Evolution

Quantum computers could outperform classical machines on important problems, but only if the errors that pervade quantum hardware can be corrected at scale. Quantum low-density parity-check (qLDPC) codes offer a promising route to this goal by combining sparse parity checks with finite encoding rate and growing distance, but their construction remains a challenging discrete design problem. Here we introduce structured concept evolution (SCE), a search framework that pairs a large language model with a structured algebraic mutation grammar to discover lifted-product code families, a class of CSS qLDPC codes. Instead of asking the LLM to design codes from first principles, SCE evolves structured concepts consisting of algebraic specifications paired with executable programs that realize them, using hierarchical mutations that modify the group algebra, protograph geometry, or base space. Running SCE, we discover a diverse set of competitive code families, ranging from abelian constructions to families over non-abelian groups beyond those underlying standard designs such as bivariate-bicycle codes, and characterize them under code-capacity depolarizing noise with BP+OSD decoding. These results are obtained with lightweight models (GPT-5.4-mini and GPT-5.4-nano).

Zidu Liu, Florian Marquardt

Max Planck Institute for the Science of Light, 91058 Erlangen, Germany · Department of Physics, Friedrich-Alexander University Erlangen-Nürnberg, 91058 Erlangen, Germany

Quantum LDPC Codes Quantum Error Correction Large Language Model-Guided Scientific Discovery Error-Correcting Codes

Jun 3, 2026

Toward a Generalized Defense Across Sparse, Continuous, and Structured Parameter Attacks

Deep neural networks are increasingly deployed across heterogeneous and partially untrusted environments, where models are distributed through cloud storage, CI/CD pipelines, containerized services, and edge execution platforms. This broad deployment landscape exposes model parameters to various integrity risks. Unlike input-space adversarial attacks, parameter attacks directly tamper with the model's internal parameters and persist across all subsequent inferences. Existing defenses either require retraining, incur significant accuracy degradation, or are limited to specific attack classes. However, in real-world deployment scenarios, the forms of parameter attacks are often unpredictable. To address this challenge, we present ParDef, a generalized defense for deep neural networks against diverse types of parameter attacks. ParDef integrates keyed channel reparameterization, which obscures sensitive parameter directions, QC-LDPC quantization, which embeds redundancy and supports error correction, and adaptive robust inference, which stabilizes predictions under uncertainty. Our evaluation on CIFAR-10, CIFAR-100, and Tiny-ImageNet using ResNet and VGG models demonstrates that ParDef consistently reduces attack success rates across different parameter attacks while maintaining high model performance and incurring only moderate deployment overhead. These results highlight that ParDef is a practical and generalized defense for DNN deployments.

Bin Duan, Zeyu Bai, Guowei Yang

School of Electrical Engineering and Computer Science, The University of Queensland, Australia

Adversarial Attacks Adversarial Robustness Error-Correcting Codes

Jun 1, 2026

Evolutionary Discovery of Bivariate Bicycle Codes with LLM-Guided Search

Quantum LDPC code discovery requires searching large algebraic design spaces while reliably certifying the parameters and equivalence classes of any candidates found. We introduce an LLM-guided evolutionary workflow in which language models mutate Python programs that generate bivariate-bicycle and perturbed bivariate-bicycle code ansätze. Across five campaigns, the system performed approximately 1{,}650 evolutionary iterations, screened about

2 \times 10^5

candidate codes, and required

{\sim}140

hours of computation and

{\sim}

US$400 in LLM inference cost. Candidate codes are evaluated through a staged validation pipeline combining

\mathrm{GF}(2)

rank computation, distance estimation and certification, mixed-integer linear programming, BLISS Tanner-graph deduplication, decomposability analysis, and local-Clifford equivalence checks. At block length

n \leq 360

, the workflow identifies 465 distinct candidate codes: 97 CSS bivariate-bicycle codes and 368 non-CSS perturbed variants. The CSS search recovers known high-performing codes and finds new finite-length representatives, including an indecomposable [[288,16,12]] code and higher-weight codes with up to

k = 50

at distance

d = 8

. The non-CSS search produces perturbed codes matching the gross-code figure of merit at [[144,12,12]], along with additional high-distance candidates reported as certified values or upper bounds according to MILP status. Overall, these results show that LLM-guided program evolution can serve as a practical tool for structured quantum-code discovery when paired with independent evaluation.

Juan Cruz-Benito, Andrew W. Cross, David Kremer +1

IBM Research, IBM T. J. Watson Research Center, NY, USA

Automated Algorithm Discovery Quantum LDPC Codes Quantum Error Correction Large Language Model-Guided Scientific Discovery Error-Correcting Codes

May 27, 2026

Score Based Error Correcting Code Decoder

Error-correcting codes enable reliable communication, yet practical soft decoding remains challenging across code families and block lengths. We propose SB-ECC, a score-based decoder that casts decoding as continuous-time denoising. A neural denoiser defines a probability-flow ordinary differential equation (ODE) that iteratively updates the noisy channel observation toward a valid codeword, guided by parity constraints. The model is trained across noise levels without time/SNR conditioning, enabling inference without SNR estimation and supporting a direct latency accuracy trade off controlled by the ODE solver budget. We use the raw signed channel observation as input for learning a continuous denoising field. Across 42 code/SNR settings, SB-ECC achieves the best BER in 39/42 entries, with an average SNR gain of 0.17dB and a maximum gain of 0.46dB over the strongest competing baseline, we showed that swapping the solver from Euler to DPM preserves -ln(BER) while reducing end-to-end decoding time by 8.86% on average (up to 12.82%).

Alon Helvits, Eliya Nachmani

School of Electrical and Computer Engineering (ECE), Ben Gurion University, Beer Sheva, Israel.

Error-Correcting Codes

May 27, 2026

Learning Logical Operations for Arbitrary Quantum Error Correction Codes

Logical operations are essential for quantum computation within quantum error-correcting codes. However, discovering their physical realizations is challenging, especially for non-additive codes that lack a stabilizer description. We present a general learning-based framework that, given only an encoding circuit, constructs physical implementations of logical operations while enforcing structural properties such as transversality or shallow depth. Our approach is validated by rediscovering known logical operations of standard stabilizer codes. We then extend it to a co-design procedure, dubbed variational early fault-tolerant quantum computing (VarEFTQC), which tailors non-additive encodings to a given noise model and enforces desired logical gate sets, such as transversal IQP-type families or low-depth universal sets. A software library implements the complete learning pipeline, including loss-function variants, ansatz families, and optimization routines. Together, these results position VarEFTQC as a proof-of-concept framework for discovering hardware-adapted logical gadgets for early fault-tolerant quantum computing.

Nico Meyer, Christopher Mutschler, Dominik Seuß +2

Fraunhofer IIS, Fraunhofer Institute for Integrated Circuits IIS, Nuremberg, Germany · Pattern Recognition Lab, Friedrich-Alexander-University Erlangen-Nuremberg, Erlangen, Germany · University of Technology Nuremberg (UTN), Nuremberg, Germany +1

Quantum Error Correction Variational Quantum Circuits Error-Correcting Codes

May 17, 2026

Variational Diffusion Channel Decoder

Neural channel decoder, as a data-driven channel decoding strategy, has shown very promising improvement on error-correcting capability over the classical methods. However, the success of those deep learning-based decoder comes at the cost of drastically increased model storage and computational complexity, hindering their practical adoptions in real-world time-sensitive resource-sensitive communication and storage systems. To address this challenge, we propose an efficient variational diffusion model-based channel decoder, which effectively integrates the domain-specific belief propagation process to the modern diffusion model. By reaping the low-cost benefits of belief propagation and strong learning capability of diffusion model, our proposed neural decoder simultaneously achieves very low cost and high error-correcting performance. Experimental results show that, compared with the state-of-the-art neural channel decoders, our model provides a feasible solution for practical deployment via achieving the best decoding performance with significantly reduced computational cost and model size.

Chengwei Zhang, Yifan Du, Siyu Liao

The School of Integrated Circuits Sun Yat-sen University Shenzhen, China

Neural Decoding Error-Correcting Codes Diffusion Models

May 17, 2026

Maximum Likelihood Decoding of Quantum Error Correction Codes

Quantum error correction (QEC) is indispensable for realizing fault-tolerant quantum computation, yet its effectiveness hinges critically on the classical decoding algorithm that interprets noisy syndrome measurements. Among all possible decoding strategies, maximum likelihood decoding (MLD) is provably optimal, since it identifies the logical group with largest likelihood by summing over all possible errors within logical class consistent with the observed syndrome. Despite its optimality, MLD is computationally intractable in general (#P-hard), motivating a rich landscape of exact and approximate algorithms. In this topical review, we provide a unified perspective on MLD by surveying recent advances through three complementary lenses: statistical mechanics, tensor networks, and artificial intelligence. From the statistical mechanics viewpoint, the MLD problem maps onto evaluating partition functions of disordered spin models, enabling exact solutions for certain codes and noise models as well as threshold estimation via phase-transition analysis. From the tensor network perspective, approximate contraction of tensor networks on the code's factor graph yields decoders that closely approach MLD accuracy with polynomial computational cost. From the artificial intelligence perspective, neural-network-based decoders, including autoregressive generative models and recurrent transformers, learn to approximate the MLD distribution from data, achieving high accuracy with the parallelism afforded by modern hardware accelerators. We discuss the connections among these three approaches, review their application to both simulated and experimental quantum hardware, and outline open challenges including real-time decoding, scalability to large code distances, and generalization to high-rate quantum low-density parity-check codes.

Hanyan Cao, Ge Yan, Yuxuan Du +1

1Science, Mathematics and Technology Cluster, Singapore University of Technology and Design, 8 Somapah Road, 487372, Singapore · College of Computing and Data Science, Nanyang Technological University, Singapore, Singapore · School of Physical and Mathematical Sciences, Nanyang Technological University,2026 Singapore, Singapore

Quantum Error Correction Error-Correcting Codes

May 16, 2026

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

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.

Samira Sayedsalehi, Nader Bagherzadeh, Maxim Shcherbakov +1

Department of Electrical Engineering and Computer Science University of California, Irvine, CA 92697, USA

Neural Decoding Quantum Error Correction Error-Correcting Codes

May 11, 2026

Scalable Mamba-Based Message-Passing Neural Decoder for Error-Correcting Codes

Forward error correction is essential for reliable communication over noisy channels. Attention-based model-free neural decoders have shown strong performance for short codes, but their scalability to longer codes is limited by the quadratic memory and computational cost of attention. In this paper, we introduce the Mamba message-passing decoder (MMPD), an attention-free syndrome-based neural decoder for binary linear codes. MMPD retains the Tanner-graph structure of a message-passing decoder by performing local pairwise aggregation along variable-check edges. To enable efficient long-range information propagation, these local updates are combined with bidirectional Mamba state-space blocks. By avoiding dense attention matrices, MMPD scales more favorably for long codes in both memory and computation. Experiments on the (1056, 880) LDPC code show that MMPD achieves a 0.45 dB gain over the state-of-the-art CrossMPT decoder at a specified target bit error rate, while reducing memory consumption by a factor of 1.5. This reduction factor increases substantially for longer codes, demonstrating the applicability of MMPD to scalable neural decoding of practical long codes.

Rostislav Gusev, Nikita Aleksandrov, Artem Solomkin +1

Skolkovo Institute of Science and Technology, Moscow, Russia

Neural Decoding Message Passing Neural Networks Error-Correcting Codes

May 5, 2026

Leveraging Code Automorphisms for Improved Syndrome-Based Neural Decoding

Syndrome-based neural decoding (SBND) has emerged as a promising deep learning approach for soft-decision decoding of high-rate, short-length codes. However, this approach still has substantial room for improvement. In this paper, we show how to leverage code automorphisms to enhance the ability of existing SBND models to learn and generalize through data augmentation during training and inference. As a result, for the short high-rate codes considered, we obtain models that closely approach MLD performance using small datasets and proper training. Our findings also suggest that many prior results for SBND models in the literature underestimate their true correction capability due to undertraining. Code to reproduce all results is available at: https://github.com/lebidan/sbnd.

Raphaël Le Bidan, Ahmad Ismail, Elsa Dupraz +1

IMT Atlantique, Lab-STICC UMR CNRS 6285, Brest, France

Neural Decoding Error-Correcting Codes

Apr 24, 2026

Semantic Error Correction and Decoding for Short Block Codes

This paper presents a semantic-enhanced receiver framework for transmitting natural language sentences over noisy wireless channels using multiple short block codes. After ASCII encoding, the sentence is divided into segments, each independently encoded with a short block code and transmitted over an AWGN channel. At the receiver, segments are decoded in parallel, followed by a semantic error correction (SEC) model, which reconstructs corrupted segments using language model context. We further propose the semantic list decoding (SLD), which generates multiple candidate reconstructions and selects the best one via weighted Hamming distance, and a semantic confidence-guided HARQ (SHARQ) mechanism that replaces CRC-based error detection with a confidence score, enabling selective segment retransmission without CRC overhead. All modules are designed and trained using bidirectional and auto-regressive transformers (BART). Simulation results demonstrate that the proposed scheme significantly outperforms conventional capacity-approaching short codes and long codes at the same rate. Specifically, SEC provides approximately 0.4 dB BLER gain over plain short-code transmission, while SLD extends this to 0.8 dB. Compared to transmitting the entire sentence as a single long 5G LDPC codeword, our approach significantly improves semantic fidelity and reduces decoding latency by up to 90%. SHARQ further provides an additional 1.5 dB gain over conventional HARQ.

Jiafu Hao, Chentao Yue, Wanchun Liu +2

School of Electrical and Computer Engineering, The University of Sydney, Sydney, NSW 2006, Australia

Semantic Communication Error-Correcting Codes

Apr 16, 2026

Learning to Concatenate Quantum Codes

Concatenating quantum error correction codes scales error correction capability by driving logical error rates down double-exponentially across levels. However, the noise structure shifts under concatenation, making it hard to choose an optimal code sequence. We automate this choice by estimating the effective noise channel after each level and selecting the next code accordingly. In particular, we use learning-based methods to tailor small, non-additive encoders when the noise exhibits sufficient structure, then switch to standard codes once the noise is nearly uniform. In simulations, this level-wise adaptation achieves a target logical error rate with far fewer qubits than concatenating stabilizer codes alone--reducing qubit counts by up to two orders of magnitude for strongly structured noise. Therefore, this hybrid, learning-based strategy offers a promising tool for early fault-tolerant quantum computing.

Nico Meyer, Christopher Mutschler, Dominik Seuß +2

Fraunhofer IIS, Fraunhofer Institute for Integrated Circuits IIS, Nuremberg, Germany · Machine Learning and Positioning Systems Lab, University of Technology Nuremberg (UTN), Nuremberg, Germany · Center for Artificial Intelligence (CAIRO), Technical University of Applied Sciences Würzburg-Schweinfurt, Germany +1

Quantum Error Correction Error-Correcting Codes

Apr 9, 2026

Scalable Neural Decoders for Practical Fault-Tolerant Quantum Computation

Quantum error correction (QEC) is essential for scalable quantum computing. However, it requires classical decoders that are fast and accurate enough to keep pace with quantum hardware. While quantum low-density parity-check codes have recently emerged as a promising route to efficient fault tolerance, current decoding algorithms do not allow one to realize the full potential of these codes in practical settings. Here, we introduce a convolutional neural network decoder that exploits the geometric structure of QEC codes, and use it to probe a novel "waterfall" regime of error suppression, demonstrating that the logical error rates required for large-scale fault-tolerant algorithms are attainable with modest code sizes at current physical error rates, and with latencies within the real-time budgets of several leading hardware platforms. For example, for the

[144, 12, 12]

Gross code, the decoder achieves logical error rates up to

\sim 17

x below existing decoders - reaching logical error rates

\sim 10^{-10}

at physical error

p=0.1\%

- with 3-5 orders of magnitude higher throughput. This decoder also produces well-calibrated confidence estimates that can significantly reduce the time overhead of repeat-until-success protocols. Taken together, these results suggest that the space-time costs associated with fault-tolerant quantum computation may be significantly lower than previously anticipated.

Andi Gu, J. Pablo Bonilla Ataides, Mikhail D. Lukin +1

Department of Physics, Harvard University, Cambridge, MA 02138, USA · Harvard Quantum Initiative, Harvard University, Cambridge, MA 02138, USA

Quantum LDPC Codes Quantum Error Correction Neural Decoding Error-Correcting Codes Convolutional Neural Networks

Mar 27, 2026

Gaussian Shannon: High-Precision Diffusion Model Watermarking Based on Communication

Diffusion models generate high-quality images but pose serious risks like copyright violation and disinformation. Watermarking is a key defense for tracing and authenticating AI-generated content. However, existing methods rely on threshold-based detection, which only supports fuzzy matching and cannot recover structured watermark data bit-exactly, making them unsuitable for offline verification or applications requiring lossless metadata (e.g., licensing instructions). To address this problem, in this paper, we propose Gaussian Shannon, a watermarking framework that treats the diffusion process as a noisy communication channel and enables both robust tracing and exact bit recovery. Our method embeds watermarks in the initial Gaussian noise without fine-tuning or quality loss. We identify two types of channel interference, namely local bit flips and global stochastic distortions, and design a cascaded defense combining error-correcting codes and majority voting. This ensures reliable end-to-end transmission of semantic payloads. Experiments across three Stable Diffusion variants and seven perturbation types show that Gaussian Shannon achieves state-of-the-art bit-level accuracy while maintaining a high true positive rate, enabling trustworthy rights attribution in real-world deployment. The source code have been made available at: https://github.com/Rambo-Yi/Gaussian-Shannon

Yi Zhang, Hongbo Huang, Liang-Jie Zhang

Center for AI Services Computing, College of Computer Science and Software Engineering, Shenzhen University

Error-Correcting Codes Diffusion Model Watermarking AI-Generated Image Watermarking Robust Watermarking

Feb 26, 2026

Automated Discovery of Improved Constant Weight Binary Codes

A constant weight binary code consists of

n

-bit binary codewords, each with exactly

w

bits equal to 1, such that any two codewords are at least Hamming distance

d

apart.

A(n,d,w)

is the maximum size of a constant weight binary code with parameters

n,d,w

. We establish improved lower bounds on

A(n,d,w)

by constructing new larger codes, for 24 values of

(n,d,w)

with

6 \leq d \leq 18

and

18 \leq n \leq 35

. The improved lower bounds come from two strategies. The first is a tabu search that operates at the level of bit swaps. The second is a novel greedy heuristic that repeatedly chooses the candidate codeword that maximizes a randomly-scored histogram of distances to previously-added codewords. These strategies were proposed by CPro1, an automated protocol that generates, implements, and tests diverse strategies for combinatorial constructions.

Christopher D. Rosin

Error-Correcting Codes