Quantum Low Density Parity Check Codes Decoding Achieves Faster Inference with Diffusion Models

Correcting errors in quantum information represents a major challenge in building practical quantum computers, and researchers are actively exploring new decoding strategies to address this. Zejun Liu, Anqi Gong, and Bryan K. Clark, working at the University of Illinois at Urbana-Champaign and ETH Zurich, now demonstrate a powerful new approach using diffusion models to interpret error signals, known as syndromes, in low-density parity-check codes. Their work establishes that these diffusion-based decoders not only achieve greater accuracy than existing methods, including established algorithms and other neural network approaches, but also offer significantly improved speed, particularly in challenging scenarios. Importantly, the team reveals that the decoder learns the underlying structure of the codes without explicit instruction, and scales more effectively than alternative diffusion techniques, representing a substantial advance in the field of quantum error correction.

These models learn to infer logical errors from syndrome measurements, offering a flexible alternative to traditional decoding methods. The diffusion model establishes a probabilistic mapping from syndromes to error probabilities, enabling accurate error inference even with limited training data and demonstrating strong performance on codes with up to 100 qubits. This provides a novel pathway towards practical quantum error correction by offering a robust and data-efficient decoding strategy.

When tested on the bicycle code, incorporating realistic circuit-level noise, a masked diffusion decoder achieved greater accuracy, faster average speeds, and consistently outperformed other state-of-the-art decoders in worst-case scenarios. Reducing the number of diffusion steps during inference significantly increases speed with only a minimal reduction in accuracy.

Diffusion Models Decode Quantum LDPC Codes

This research explores the application of diffusion models, a powerful class of generative models, as decoders for Quantum Low-Density Parity-Check (LDPC) codes. Traditionally, decoding these codes has relied on computationally expensive methods, but diffusion models offer a potentially more robust and scalable alternative. These models learn to reverse a diffusion process, starting with noise and gradually reconstructing data, and are particularly well-suited for this task.

Quantum LDPC codes are promising for building fault-tolerant quantum computers, defined by sparse parity-check matrices. Key aspects include forward diffusion, which adds noise to data over time, and reverse diffusion, which learns to remove noise and reconstruct the original data. Score matching is a technique used to train these models by estimating the gradient of the data distribution.

Machine learning is used to learn the optimal decoding strategy for error-correcting codes. Transformers, a type of neural network architecture, have achieved state-of-the-art results in many areas and are used in some diffusion model approaches. Researchers are exploring techniques such as classifier-free guidance and flow matching, an alternative to diffusion models. Denoising Diffusion Implicit Models (DDIMs) offer a faster sampling method, and Variational Autoencoders (VAEs) provide another generative model option.

Current research directions include combining diffusion models with traditional decoders, developing entirely new diffusion-based decoders, and scaling these methods to larger codes. Improving robustness to different types of noise, exploring the trade-offs between classical and quantum resources, and creating general-purpose foundation models are also key areas of investigation. Researchers are adapting diffusion models, typically used for continuous data, to discrete data like quantum states and employing them for state preparation.

Diffusion Decoding Surpasses Quantum Code Limits

This research demonstrates a new approach to decoding quantum low-density parity-check codes, employing a diffusion model framework as an alternative to traditional belief propagation methods. The team successfully implemented a masked diffusion decoder that outperforms existing state-of-the-art decoders, including belief propagation with ordered statistics decoding and autoregressive neural decoders, in terms of both accuracy and decoding speed, particularly in worst-case scenarios. Importantly, the decoder achieves improved performance while requiring fewer computational steps during inference, offering a valuable trade-off between speed and accuracy.

Further analysis of the trained neural network revealed that the diffusion decoder learns the underlying structure of the quantum codes, despite being trained only on paired syndrome and logical error samples. Comparisons between masked and continuous diffusion decoders indicate that the masked approach scales more effectively for decoding under realistic noise conditions.