[Paper Review] Efficient Decoding of Surface Code Syndromes for Error Correction in Quantum Computing

Debasmita Bhoumik, Pinaki Sen+4 more|arXiv (Cornell University)|Oct 21, 2021

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a two-level machine learning decoder for surface code quantum error correction, using feedforward neural networks and convolutional neural networks to correct physical and logical errors under symmetric and asymmetric depolarizing noise. The method achieves approximately 10× higher pseudo-threshold and 2× higher threshold than Minimum Weight Perfect Matching, with robust performance even at low training data ratios (40:60).

ABSTRACT

Errors in surface code have typically been decoded by Minimum Weight Perfect Matching (MWPM) based method. Recently, neural-network-based Machine Learning (ML) techniques have been employed for this purpose. Here we propose a two-level (low and high) ML-based decoding scheme, where the first level corrects errors on physical qubits and the second one corrects any existing logical errors, for different noise models. Our results show that our proposed decoding method achieves $\\sim10 \ imes$ and $\\sim2 \ imes$ higher values of pseudo-threshold and threshold respectively, than for MWPM. We show that usage of more sophisticated ML models with higher training/testing time, do not provide significant improvement in the decoder performance. Finally, data generation for training the ML decoder requires significant overhead hence lower volume of training data is desirable. We have shown that our decoder maintains a good performance with the train-test-ratio as low as $40:60$.

Motivation & Objective

• Address the limitations of Minimum Weight Perfect Matching (MWPM) in decoding surface code syndromes for quantum error correction.
• Develop a machine learning-based decoder that improves threshold and pseudo-threshold performance under realistic noise models.
• Reduce the training data overhead required for effective ML-based decoding in near-future quantum devices.
• Evaluate the impact of model complexity on decoding performance and inference time.
• Assess the robustness of the decoder across varying train-test data ratios, especially with limited training data.

Proposed method

• Propose a two-level ML decoding architecture: a low-level decoder corrects physical qubit errors using FFNN or CNN, followed by a high-level decoder that detects and corrects residual logical errors.
• Train the low-level decoder on syndrome measurements and error configurations for surface codes of distance d = 3, 5, and 7 under symmetric and asymmetric depolarizing noise.
• Use fully connected networks (FFNN) and convolutional neural networks (CNN) with 3–4 dense layers and 3–64 filter convolutions, respectively, for error prediction.
• Apply a two-stage inference process: first predict physical errors, then use the residual syndrome to detect logical errors via the high-level network.
• Optimize for low data usage by evaluating performance across varying train-test ratios, including as low as 40:60.
• Compare decoder performance against MWPM using pseudo-threshold and threshold metrics under different noise models and code distances.

Experimental results

Research questions

• RQ1 Can a two-level ML decoder outperform MWPM in terms of pseudo-threshold and threshold for surface code error correction?
• RQ2 How does model complexity (e.g., FFNN vs. deeper CNN) affect decoding performance and inference time?
• RQ3 What is the minimum amount of training data required to achieve optimal decoding accuracy, especially under low-data regimes?
• RQ4 How does the decoder perform under asymmetric depolarizing noise, which is more realistic in current quantum hardware?
• RQ5 Does the performance of the ML decoder degrade significantly with increasing physical error probability, and how does this affect variance in predictions?

Key findings

• The proposed ML decoder achieves approximately 10× higher pseudo-threshold and 2× higher threshold compared to MWPM under both symmetric and asymmetric depolarizing noise.
• More complex neural network architectures (e.g., deeper CNNs) do not yield significant performance improvements over simpler models (e.g., FFNN or shallow CNN), despite higher training and inference times.
• The decoder maintains high accuracy even with a train-test ratio as low as 40:60, demonstrating strong generalization with minimal training data.
• Performance variance (standard deviation) increases with higher physical error probabilities, indicating greater uncertainty in predictions as noise levels rise.
• The decoder shows robustness to asymmetric noise, which is more representative of real-world quantum hardware, without performance degradation.
• The two-level design effectively reduces logical error rates by correcting residual errors missed by the low-level decoder, enhancing overall error suppression.