Machine Learning Approaches to Decoding Topological Quantum Codes

Changwon Lee; Tak Hur; Jeongwoo Jae; Daniel K. Park

quant-phAug 16, 2026

Machine Learning Approaches to Decoding Topological Quantum Codes

Authors: Changwon Lee, Tak Hur, Jeongwoo Jae, Daniel K. Park

Organizations: Department of Statistics and Data Science, Yonsei University, Seoul 03722, Republic of Korea · R&D center, Samsung SDS, Seoul 05510, Republic of Korea · Department of Applied Statistics, Yonsei University, Seoul 03722, Republic of Korea · Department of Quantum Information, Yonsei University, Seoul 03722, Republic of Korea

Abstract

Decoding is an essential component of quantum error correction (QEC), translating stabilizer measurement outcomes into corrective actions that suppress logical errors and preserve logical quantum information. Building fault-tolerant architectures requires increasing the code distance, which in turn places growing demands on decoding accuracy, scalability, and practical deployability. While a wide range of decoding algorithms have been proposed and demonstrated, achieving reliable, scalable, and real-time decoding remains a significant challenge. Machine-learning (ML) approaches are particularly well suited to this setting, as quantum error decoding is fundamentally a problem of processing large volumes of classical data with complex spatiotemporal correlations. This chapter surveys ML-based methods for quantum error decoding, with a focus on topological codes and an emphasis on architectural principles, practical performance, and real-time considerations. We first frame decoding as a learning problem and outline key paradigms, including discriminative, generative, and reinforcement-learning formulations. We then introduce the neural network building blocks that underpin most contemporary neural decoders and discuss how these components can be integrated to balance expressivity, scalability, and latency. Building on this architectural perspective, we review recent progress and benchmarks in neural decoding for memory experiments, and discuss real-time decoding, open challenges, and future directions toward scalable fault-tolerant quantum computing.

Figures & tables

Figure 1: Overview of the ML-based decoding pipeline for stabilizer codes. (a) A rotated surface code undergoes repeated syndrome extraction over $r$ rounds; data qubits (black dots) are arranged on a lattice with $X$ -type (white) and $Z$ -type (blue) stabilizer measurements. (b) The measured syndrome outcomes at each round are represented as binary vectors and stacked across time steps to form the decoder input. (c) A machine-learning decoder—which may be instantiated as a feedforward network, CNN, GNN, recurrent network, or transformer—processes the syndrome history. (d) Two output formulations: a logical class label , where the decoder predicts the most likely logical error class $\bar{L}\in\{\bar{I},\bar{X},\bar{Y},\bar{Z}\}$ via softmax classification (top); or a per-qubit error probability $P(E_{i}\,|\,\mathbf{s})$ for each data qubit, shown as colored nodes on the code lattice (bottom), where color intensity indicates error likelihood.

Figure 2: Visualization of the error decomposition.

Figure 3: Three canonical neural decoder architectures. (a) Two stacked LSTM layers process defects $d_{0},d_{1},\ldots,d_{N-1}$ sequentially. The auxiliary head receives the recurrent output alone and predicts $p_{\mathrm{aux}}$ ; the main head additionally ingests the final defects $d_{N}$ from data-qubit readout and predicts $p_{\mathrm{main}}$ . (b) Detector error graph spanning three measurement rounds; red nodes are $X$ -type and blue nodes are $Z$ -type detection events, with edges weighted by shared fault probability. Successive graph-convolution layers propagate local features along the space-time graph; mean pooling collapses all node embeddings into a fixed-size vector, and two evaluation heads output $\lambda_{X}$ and $\lambda_{Z}$ . (c) qecGPT causal Transformer. The joint configuration $(\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\gamma})$ of stabilizer generators, logical operators, and syndrome is first projected via an input embedding, augmented with positional encoding, then passed through $N_{l}=2$ – $3$ stacked self-attention layers; the EVAL head outputs the joint distribution $q_{\theta}(\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\gamma})$ .

Panel of Figure 4 — Figure 4: AlphaQubit family architectures. Top: AlphaQubit 1 (AQ1) [ 47 ] module pipeline. Adapted from Ref. [ 26 ] . Bottom: AlphaQubit 2 (AQ2) [ 86 ] with frame compression and alternating recurrent and transformer layers.

Panel of Figure 5 — Figure 5: Sliding-window and parallel-window decoding. Green blocks indicate committed regions, where corrections are accepted and passed forward. Red blocks indicate buffer regions, which provide additional syndrome context but are not committed at that decoding step. Top: In sliding-window decoding, a temporal window is moved forward over the syndrome stream. At each decode step, the decoder commits the correction in the green core region and uses the red buffer region to account for error chains that extend across the window boundary. The buffer is retained until enough future syndrome information is available. Bottom: In parallel-window decoding, windows are organized into decoding layers. Within a layer, several non-overlapping windows can be decoded concurrently. Boundary information from the first layer is passed to a second layer, which resolves the regions left between the committed cores of the first-layer windows. In both cases, the purpose of the buffer is to localize the decoding problem while suppressing boundary-induced decoding errors.

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Hanyan Cao, Ge Yan, Yuxuan Du, Feng Pan

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Quantum Error Correction Quantum Error Correction Decoding Error-Correcting Codes

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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.

Abstract

Quantum Error Correction Decoding

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