Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Applied Sciences

Resource-Efficient Decoding of Topological Color Codes via Neural-Guided Union-Find Optimization

Minghao Fu, Cewen Tian, Zhe-Cheng Fan, Hongyang Ma

Published Aug 13, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.3390/app15168937

Related papers

Probing Threshold Behavior of Adaptive Cascaded Quantum Codes Under Variable Biased Noise for Practical Fault-Tolerant Quantum Computing

Yongnan Chen, Zaixu Fan, Haopeng Wang +2 more

This paper proposes a resource optimized cascaded quantum surface repetition code architecture integrated with a Union Find (UF) enhanced hybrid decoder, which suppresses biased noise and improves the scalability of quantum error correction through synergistic inner outer quantum code collaboration. The hybrid architecture employs inner quantum repetition codes for local error suppression and outer rotated quantum surface codes for topological robustness, reducing auxiliary quantum qubits by 12.5% via shared stabilizers and compact lattice embedding. An optimized UF decoder employing path compression and adaptive cluster merging achieves near-linear time complexity O(nα(n)), outperforming minimum-weight perfect matching (MWPM) decoders O(n2.5). Under Z-biased noise η=10, simulations demonstrate a 28.2% error threshold, 2.6% higher than standard quantum surface codes, and 15% lower logical error rates via dynamic boundary expansion. At code distance d=7, resource savings reach 9.3% with maximum relative error below 8.5%, fulfilling fault-tolerance criteria. The UF decoder exhibits 38% threshold advantage over MWPM at low bias η≈10−3 and 15% less degradation at high noise p=0.5, enabling scalable real-time decoding. This framework bridges theoretical thresholds with practical resource constraints, offering a noise-adaptive QEC solution for near-term quantum devices including photonic quantum systems referenced in the paper’s background on repetition cat qubits.

Electronics•Jan 19, 2026•0 citations

A scalable and real-time neural decoder for topological quantum codes

Andrew W. Senior, Thomas Edlich, Francisco J. H. Heras +21 more

Fault-tolerant quantum computing will require error rates far below those achievable with physical qubits. Quantum error correction (QEC) bridges this gap, but depends on decoders being simultaneously fast, accurate, and scalable. This combination of requirements remains unmet by a machine-learning decoder, nor by any decoder for promising resource-efficient codes such as the color code. Here we introduce AlphaQubit 2, a neural-network decoder that achieves near-optimal logical error rates for both surface and color codes at scale under realistic noise. For the color code, it is orders of magnitude faster than other high-accuracy decoders. We demonstrate real-time decoding faster than 1μs per cycle on commercial accelerators: for the surface code to distance 11, with better accuracy than leading real-time decoders; and the first real-time decoding of the color code to distance 9. These results support the practical application of a wider class of promising QEC codes, and establish a credible path towards high-accuracy, real-time neural decoding at the scales required for fault-tolerant quantum computation.

quant-pharXiv:2512.07737•Dec 9, 2025•2 citations

Fault-Tolerant Hybrid Decoder for Quantum Surface Codes on Probabilistic Inference and Topological Clustering

Xingyu Qiao, Xiaomei Xu, Hongyang Ma +1 more

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.

Applied Sciences•Mar 8, 2026•0 citations

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

Yuhang Gu, Hui Li

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.

Asian Hardware Oriented Security and Trust Symposium (AsianHOST)•Dec 19, 2025•1 citation

Ising model formulation for highly accurate topological color codes decoding

Yugo Takada, Yusaku Takeuchi, Keisuke Fujii

Quantum error correction is an essential ingredient for reliable quantum computation for theoretically provable quantum speedup. Topological color codes, one of the quantum error correction codes, have an advantage against the surface codes in that all Clifford gates can be implemented transversally. However, the hardness of decoding makes the color codes not suitable as the best candidate for experimentally feasible implementation of quantum error correction. Here we propose an Ising model formulation that enables highly accurate decoding of the color codes. In this formulation, we map stabilizer operators to classical spin variables to represent an error satisfying the syndrome. Then we construct an Ising Hamiltonian that counts the number of errors and formulate the decoding problem as an energy minimization problem of an Ising Hamiltonian, which is solved by simulated annealing. In numerical simulations on the (4.8.8) lattice, we find an error threshold of 10.36(5)% for bit-flip noise model, 18.47(5)% for depolarizing noise model, and 2.90(4)% for phenomenological noise model (bit-flip error is located on each of data and measurement qubits), all of which are higher than the thresholds of existing efficient decoding algorithms. Furthermore, we verify that the achieved logical error rates are almost optimal in the sense that they are almost the same as those obtained by exact optimizations by CPLEX with smaller decoding time in many cases. Since the decoding process has been a bottleneck for performance analysis, the proposed decoding method is useful for further exploration of the possibility of the topological color codes. Published by the American Physical Society 2024

arXiv:2303.01348•Physical Review Research•Jan 24, 2024•10 citations

Design of an Online Surface Code Decoder Using Union-Find Algorithm

Takuya Kasamura, Junichiro Kadomoto, Hidetsugu Irie

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 a$\mathbf{1 0 2. 3 8}$ns 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.

IEEE International Conference on Computer Design: VLSI in Computers and Processors (ICCD)•Nov 10, 2025•0 citations
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.
This paper proposes a resource optimized cascaded quantum surface repetition code architecture integrated with a Union Find (UF) enhanced hybrid decoder, which suppresses biased noise and improves the scalability of quantum error correction through synergistic inner outer quantum code collaboration. The hybrid architecture employs inner quantum repetition codes for local error suppression and outer rotated quantum surface codes for topological robustness, reducing auxiliary quantum qubits by 12.5% via shared stabilizers and compact lattice embedding. An optimized UF decoder employing path compression and adaptive cluster merging achieves near-linear time complexity O(nα(n)), outperforming minimum-weight perfect matching (MWPM) decoders O(n2.5). Under Z-biased noise η=10, simulations demonstrate a 28.2% error threshold, 2.6% higher than standard quantum surface codes, and 15% lower logical error rates via dynamic boundary expansion. At code distance d=7, resource savings reach 9.3% with maximum relative error below 8.5%, fulfilling fault-tolerance criteria. The UF decoder exhibits 38% threshold advantage over MWPM at low bias η≈10−3 and 15% less degradation at high noise p=0.5, enabling scalable real-time decoding. This framework bridges theoretical thresholds with practical resource constraints, offering a noise-adaptive QEC solution for near-term quantum devices including photonic quantum systems referenced in the paper’s background on repetition cat qubits.
Fault-tolerant quantum computing will require error rates far below those achievable with physical qubits. Quantum error correction (QEC) bridges this gap, but depends on decoders being simultaneously fast, accurate, and scalable. This combination of requirements remains unmet by a machine-learning decoder, nor by any decoder for promising resource-efficient codes such as the color code. Here we introduce AlphaQubit 2, a neural-network decoder that achieves near-optimal logical error rates for both surface and color codes at scale under realistic noise. For the color code, it is orders of magnitude faster than other high-accuracy decoders. We demonstrate real-time decoding faster than 1μs per cycle on commercial accelerators: for the surface code to distance 11, with better accuracy than leading real-time decoders; and the first real-time decoding of the color code to distance 9. These results support the practical application of a wider class of promising QEC codes, and establish a credible path towards high-accuracy, real-time neural decoding at the scales required for fault-tolerant quantum computation.
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.
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.
Quantum error correction is an essential ingredient for reliable quantum computation for theoretically provable quantum speedup. Topological color codes, one of the quantum error correction codes, have an advantage against the surface codes in that all Clifford gates can be implemented transversally. However, the hardness of decoding makes the color codes not suitable as the best candidate for experimentally feasible implementation of quantum error correction. Here we propose an Ising model formulation that enables highly accurate decoding of the color codes. In this formulation, we map stabilizer operators to classical spin variables to represent an error satisfying the syndrome. Then we construct an Ising Hamiltonian that counts the number of errors and formulate the decoding problem as an energy minimization problem of an Ising Hamiltonian, which is solved by simulated annealing. In numerical simulations on the (4.8.8) lattice, we find an error threshold of 10.36(5)% for bit-flip noise model, 18.47(5)% for depolarizing noise model, and 2.90(4)% for phenomenological noise model (bit-flip error is located on each of data and measurement qubits), all of which are higher than the thresholds of existing efficient decoding algorithms. Furthermore, we verify that the achieved logical error rates are almost optimal in the sense that they are almost the same as those obtained by exact optimizations by CPLEX with smaller decoding time in many cases. Since the decoding process has been a bottleneck for performance analysis, the proposed decoding method is useful for further exploration of the possibility of the topological color codes. Published by the American Physical Society 2024
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.