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Electronics

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

Yongnan Chen, Zaixu Fan, Haopeng Wang, Cewen Tian, Hongyang Ma

Published Jan 19, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.3390/electronics15020436

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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.
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.
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.
Fault-tolerant quantum computation critically depends on the ability to detect and correct physical qubit errors faster than they accumulate. The surface code (SC) has emerged as the most experimentally promising quantum error correction (QEC) framework due to its high threshold and local stabilizer structure, yet its practical performance is bottlenecked by decoder efficiency. Traditional decoding approaches such as minimum-weight perfect matching (MWPM) achieve near-optimal accuracy under independent noise but struggle with correlated errors, circuit-level noise, and real-time latency constraints. This paper investigates how neural network (NN)-based decoders reduce logical error rates in SC quantum systems by learning complex syndrome-to-correction mappings from data. We propose a hybrid convolutional-recurrent neural architecture that processes stabilizer measurement syndromes across both spatial and temporal dimensions, enabling the decoder to capture correlated error patterns inherent in realistic hardware. Systematic simulations under depolarizing, biased, and circuit-level noise models demonstrate that the proposed neural decoder achieves a logical error rate reduction of up to 41.3% compared to MWPM at a physical error rate of p = 0.008, while maintaining sub-millisecond decoding latency compatible with superconducting qubit cycle times. These results establish that data-driven neural decoders represent a scalable and hardware-compatible pathway toward achieving the logical error rates required for practical 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.
Biased-noise qubits, in which one type of error (e.g. XXX- and YYY-type errors) is significantly suppressed relative to the other (e.g. ZZZ-type errors), can significantly reduce the overhead of quantum error correction. Codes such as the rectangular surface code or XZZX code substantially reduce the qubit overhead under biased noise, but they still face challenges. The rectangular surface code suffers from a relatively low threshold, while the XZZX code requires twice as many physical qubits to maintain the same code distance as the surface code. In this work, we introduce a 2D local code construction that outperforms these codes for noise biases η≥7×104η\ge 7\times10^{4}η≥7×104, reducing the qubit overhead by over 50% at pZ=10−3p_Z=10^{-3}pZ​=10−3 and η=2×106η= 2 \times 10^6η=2×106 to achieve a logical error rate of 10−1210^{-12}10−12. Our construction relies on the concatenation of two classical codes. The inner codes are repetition phase-flip codes while the outer codes are high-rate bit-flip codes enabled by their implementation at the logical level, which circumvents device connectivity constraints. These results indicate that under sufficiently biased noise, it is advantageous to address phase-flip and bit-flip errors at different layers of the coding scheme. The inner code should prioritize a high threshold for phase-flip errors, while the bit-flip outer code should optimize for encoding rate efficiency. In the strong biased-noise regime, high-rate outer codes keep the overhead for correcting residual bit-flip errors comparable to that of the repetition code itself, meaningfully lower than that required by earlier approaches.
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.