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Asian Hardware Oriented Security and Trust Symposium (AsianHOST)

NNUF: Nearest Neighbor Union-Find Decoder for Quantum Error Correction

Yuhang Gu, Hui Li

Published Dec 19, 2025• 1 citation
Publisher

Abstract

The realization of fault-tolerant quantum computers is critically dependent on efficient and accurate quantum error correction. For topological codes such as the surface code, the decoder often becomes a performance bottleneck. This paper presents an optimized nearest-neighbor union-find decoder, named NNUF, a novel algorithm designed to accelerate the traditional union-find (UF) decoder. NNUF leverages pre-computed nearest-neighbor information to form preliminary error clusters, which streamlines the subsequent decoding stages. We evaluated NNUF against standard benchmarks and demonstrated that it reduces the decoding time while maintaining the high error correction performance of the original UF decoder. Specifically, for large code distances where $d \geq 7$, our results reveal a speedup of up to $1.14 \times$ compared to a standard UF implementation. This establishes NNUF as a valuable tool for quantum systems requiring fast and reliable decoding, representing a practical step towards real-time error correction in large-scale quantum processors.

DOI
10.1109/asianhost68425.2025.11370357

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The realization of fault-tolerant quantum computers is critically dependent on efficient and accurate quantum error correction. For topological codes such as the surface code, the decoder often becomes a performance bottleneck. This paper presents an optimized nearest-neighbor union-find decoder, named NNUF, a novel algorithm designed to accelerate the traditional union-find (UF) decoder. NNUF leverages pre-computed nearest-neighbor information to form preliminary error clusters, which streamlines the subsequent decoding stages. We evaluated NNUF against standard benchmarks and demonstrated that it reduces the decoding time while maintaining the high error correction performance of the original UF decoder. Specifically, for large code distances where d≥7d \geq 7d≥7, our results reveal a speedup of up to 1.14×1.14 \times1.14× compared to a standard UF implementation. This establishes NNUF as a valuable tool for quantum systems requiring fast and reliable decoding, representing a practical step towards real-time error correction in large-scale quantum processors.
Real-time Quantum Error Correction (QEC) for Fault-Tolerant Quantum Computing (FTQC) demands immediate surface code decoding with both high accuracy and low latency. The highly accurate Union-Find (UF) algorithm has traditionally been limited to batch-processing, as its iterative cluster growth is incompatible with the concurrent initiation of decoding with each syndrome measurement round, required for real-time QEC. This research presents a novel online UF decoder microarchitecture designed to overcome these limitations. Our key innovation decomposes UF cluster growth into incremental, per-cycle steps managed by a dedicated SHIFT stage. ASIC evaluation of our implementation demonstrates a102.38\mathbf{1 0 2. 3 8}102.38ns latency for a surface code with a distance of 23 and an approximate 1.4% threshold, meeting critical QEC performance targets. By demonstrating a viable online UF decoder with competitive performance, this research offers a crucial building block for the practical realization of scalable FTQC.
Fast decoders that achieve strong error suppression are essential for fault-tolerant quantum computation (FTQC) from both practical and theoretical perspectives. The union-find (UF) decoder for the surface code is widely regarded as a promising candidate, offering almost-linear time complexity and favorable empirical error suppression supported by numerical evidence. However, the lack of a rigorous threshold theorem has left open whether the UF decoder can achieve fault tolerance beyond the error models and parameter regimes tested in numerical simulations. Here, we provide a rigorous proof of a finite threshold for the UF decoder on the surface code under the circuit-level local stochastic error model. To this end, we develop a refined error-clustering framework that extends techniques previously used to analyze cellular-automaton and renormalization-group decoders, by showing that error clusters can be separated by substantially larger buffers, thereby enabling analytical control over the behavior of the UF decoder. Using this guarantee, we further prove a quasi-polylogarithmic upper bound on the average runtime of a parallel UF decoder in terms of the code size. We also show that this framework yields a finite threshold for the greedy decoder, a simpler decoder with lower complexity but weaker empirical error suppression. These results provide a solid theoretical foundation for the practical use of UF-based decoders in the development of fault-tolerant quantum computers, while offering a unified framework for studying fault tolerance across these practical decoders.
Quantum error correction is a prerequisite for quantum computing; however, the performance critically depends on the accuracy of the decoding algorithm. To address these challenges, we propose a hybrid decoding architecture, BP + UF + BP. The protocol initiates with a truncated global BP stage to extract probabilistic gradients without requiring full convergence. This soft information guides a reliability-based Union-Find (UF) algorithm to prioritize high-likelihood error mechanisms. Finally, a local subgraph BP refinement maximizes correction accuracy. Numerical simulations on rotated surface codes under circuit-level depolarizing noise demonstrate a fault-tolerance threshold of approximately 0.72%. This significantly outperforms standard Minimum Weight Perfect Matching (MWPM) and Union-Find (UF) baselines. Notably, our method significantly reduces the logical error rate compared to the conventional decoders. With its empirically near-linear scaling under fixed iteration, the proposed architecture presents a scalable solution for real-time fault-tolerant quantum computing.
Quantum error correction (QEC) is crucial for achieving reliable quantum computation. Among topological QEC codes, color codes can correct bit-flip and phase-flip errors simultaneously, enabling efficient resource utilization. However, existing decoders such as the Union–Find (UF) algorithm exhibit limited accuracy under high noise levels. We propose a hybrid decoding framework that augments a modified UF algorithm—enhanced with a secondary growth strategy—with a lightweight recurrent neural network (RNN). The RNN refines the error chains identified by UF, improving resolution without significantly increasing computational overhead. The simulation results show that our method achieves notable accuracy gains over baseline UF decoding, particularly in high-error regimes, while preserving the near-linear runtime scaling and low memory footprint of UF. At higher physical error rates, RNN-based path optimization improves UF decoding accuracy by approximately 4.7%. The decoding threshold of the color code reaches 0.1365, representing an increase of about 2% compared to UF without RNN optimization. With its simple data structure and low space complexity, the proposed method is well suited for low-latency, resource-constrained quantum computing environments.
In fault-tolerant quantum computing, soft outputs from real-time decoders play a crucial role in improving decoding accuracy, post-selecting magic states, and accelerating lattice surgery. A recent paper by Meister et al. [arXiv:2405.07433 (2024)] proposed an efficient method to evaluate soft outputs for cluster-based decoders, including the Union-Find (UF) decoder. However, in parallel computing environments, its computational complexity is comparable to or even surpasses that of the UF decoder itself, resulting in a substantial overhead. Furthermore, this method requires global information about the decoding graph, making it poorly suited for existing hardware implementations of the UF decoder on Field-Programmable Gate Arrays (FPGAs). In this paper, to alleviate these issues, we develop more efficient methods for evaluating high-quality soft outputs in cluster-based decoders by introducing several early-stopping techniques. Our central idea is that the precise value of a large soft output is often unnecessary in practice. Based on this insight, we introduce two types of novel soft-outputs: the bounded cluster gap and the extra-cluster gap. The former reduces the computational complexity of Meister's method by terminating the calculation at an early stage. Our numerical simulations show that this method achieves improved scaling with code distance ddd compared to the original proposal. The latter, the extra-cluster gap, quantifies decoder reliability by performing a small, additional growth of the clusters obtained by the decoder. This approach offers the significant advantage of enabling soft-output computation without modifying the existing architecture of FPGA-implemented UF decoders. These techniques offer lower computational complexity and higher hardware compatibility, laying a crucial foundation for future real-time decoders with soft outputs.
Efficient and accurate decoders are essential for quantum error correction. Surface codes are due to their high error threshold and scalability. The Minimum Weight Perfect Matching (MWPM) decoder and Union-Find (UF) decoder are the two most commonly used decoders for quantum surface codes, known for their accuracy and decoding speed, respectively. In this paper, we design an iteratively reweighted UF (IRUF) decoder considering the correlation of bit flip error and phase flip error in the depolarization noise channel. It has higher decoding accuracy than the MWPM decoder and UF decoder, and the decoding threshold is 11.5% and 6.3% higher respectively. Notably the IRUF decoder almost retains the same linear time complexity as the UF decoder, which is far better than the MWPM decoder.