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arXiv:2506.15636 · IEEE Transactions on Information Theory

Quantum Error Correction Exploiting Degeneracy to Approach the Hashing Bound

Kenta Kasai

Published Jan 1, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum error correction is essential for realizing scalable quantum computation. Among various approaches, low-density parity-check codes over higher-order Galois fields have shown promising performance due to their structured sparsity and compatibility with iterative decoding algorithms whose computational complexity scales linearly with the number of physical qubits. In this work, we demonstrate that explicitly exploiting the degeneracy of quantum errors can significantly enhance the decoding performance. Simulation results over the depolarizing channel indicate that the proposed method, at a coding rate of 1/3, achieves a frame error rate as low as $10^{-4}$ at a physical error rate of 9.45% for a code with 104,000 logical qubits and 312,000 physical qubits, approaching the quantum hashing bound. These findings highlight the critical role of degeneracy in closing the gap to the fundamental limits of quantum error correction.

DOI
10.1109/tit.2026.3747884

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Quantum error correction is essential for realizing scalable quantum computation. Among various approaches, low-density parity-check codes over higher-order Galois fields have shown promising performance due to their structured sparsity and compatibility with iterative decoding algorithms whose computational complexity scales linearly with the number of physical qubits. In this work, we demonstrate that explicitly exploiting the degeneracy of quantum errors can significantly enhance the decoding performance. Simulation results over the depolarizing channel indicate that the proposed method, at a coding rate of 1/3, achieves a frame error rate as low as 10−410^{-4}10−4 at a physical error rate of 9.45% for a code with 104,000 logical qubits and 312,000 physical qubits, approaching the quantum hashing bound. These findings highlight the critical role of degeneracy in closing the gap to the fundamental limits of quantum error correction.
Quantum error correction is essential for building scalable quantum computers. Within the stabilizer formalism, the Calderbank-Shor-Steane framework constructs quantum codes from pairs of classical linear codes. A distinctive feature in this setting is degeneracy, where multiple equivalent error estimates exist-a phenomenon that has no classical counterpart, and the lack of a meaningful classical coding-theoretic interpretation of which has remained a gap in the literature. In this paper, we demonstrate that degeneracy is closely related to the classical operation of shortening of a linear block code. Interestingly, the shortening here takes place at the decoder rather than at the encoder. Leveraging this insight, we present a parallel decoding scheme for quantum low-density parity-check codes, which we term impulse decoding, that significantly outperforms belief propagation with ordered statistics decoding, as well as several other existing techniques, under both code-capacity and circuit-level noise, with significantly lesser complexity. We then present another algorithm based on decoding of residual errors, which when combined with impulse decoding achieves further performance improvement under circuit-level noise.
Abstract Recent progress in quantum computing has enabled systems with tens of reliable logical qubits, built from thousands of noisy physical qubits1. However, many impactful applications demand quantum computations with millions of logical qubits2, necessitating highly scalable quantum error correction. In classical information theory, low-density parity-check (LDPC) codes3 can approach channel capacity efficiently4. Yet, no quantum error-correcting codes with efficient decoding have been shown to approach the hashing bound—a fundamental limit on quantum capacity—despite decades of research5–7. Here, we present quantum LDPC codes that not only approach the hashing bound but also allow decoding with computational cost linear in the number of physical qubits. This breakthrough paves the way for large-scale, fault-tolerant quantum computation. Combined with emerging hardware that manages many qubits, our approach brings quantum solutions to important real-world problems significantly closer to reality.
Quantum low-density parity-check (qLDPC) codes are promising for realizing scalable fault-tolerant quantum computation due to their potential for low-overhead protocols. A common approach to decoding qLDPC codes is to use the belief propagation (BP) decoder, followed by a post-processing step to enhance decoding accuracy. For fast decoding, the post-processing algorithm is desirable to have a small computational cost and rely only on local operations on the Tanner graph to facilitate parallel implementation. To address this requirement, we propose degeneracy cutting (DC), an efficient post-processing technique for the BP decoder that operates on information restricted to the support of each stabilizer generator. DC selectively removes one variable node with the lowest error probability for each stabilizer generator, significantly improving decoding performance while retaining the favorable computational scaling and structure amenable to parallelization inherent to BP. We further extend our method to realistic noise models, including phenomenological and circuit-level noise models, by introducing the detector degeneracy matrix, which generalizes the notion of stabilizer-induced degeneracy to these settings. Numerical simulations demonstrate that BP+DC achieves decoding performance approaching that of BP followed by ordered statistics decoding (BP+OSD) in several settings, while requiring significantly less computational cost. Our results present BP+DC as a promising decoder for fault-tolerant quantum computing, offering a valuable balance of accuracy, efficiency, and suitability for parallel implementation.
Quantum low-density parity-check codes are promising candidates for low-overhead fault-tolerant quantum computing, but degeneracy is known to impair the convergence of belief-propagation (BP) decoding of these codes. In this work, we show that appending linearly independent rows to a check matrix of a stabilizer code can reduce the search space for a valid degenerate solution. Motivated by this, we extend the recently proposed affine subcode ensemble decoding technique from the classical to the quantum setting. Moreover, we employ overcomplete matrices for each decoding path. Monte-Carlo simulations on toric and generalized bicycle codes demonstrate improved convergence and reduced logical error rate.
We present a general framework for applying linear quantum error mitigation (QEM) techniques directly to physical qubits within a logical qubit to suppress logical errors. By exploiting the linearity of quantum error correction (QEC), we demonstrate that any linear QEM method$\unicode{x2014}$including probabilistic error cancellation (PEC), zero-noise extrapolation (ZNE), and symmetry verification$\unicode{x2014}$can be integrated into the physical layer without requiring modifications to the subsequent QEC decoder. Applying this framework to memory experiments using PEC, we analytically prove and numerically verify that the leading-order contribution to the logical error can be removed, increasing the effective code distance by 2. Our simulations on repetition and rotated surface codes show that a distance-3 code with physical-level PEC achieves logical error rates lower than or similar to a distance-5 unmitigated code while using 40% and 64% fewer qubits, respectively. These results establish physical-level QEM as a widely compatible and resource-efficient strategy for enhancing logical performance in early fault-tolerant architectures.
Efficient and accurate decoding of quantum error-correcting codes is essential for fault-tolerant quantum computation, however, it is challenging due to the degeneracy of errors, the complex code topology, and the large space for logical operators in high-rate codes. In this work, we propose a decoding algorithm utilizing generative modeling in machine learning. We employ autoregressive neural networks to learn the joint probability of logical operators and syndromes in an unsupervised manner, eliminating the need for labeled training data. The learned model can approximately perform maximum likelihood decoding by directly generating the most likely logical operators for kkk logical qubits with O(2k)\mathcal O(2k)O(2k) computational complexity. Thus, it is particularly efficient for decoding high-rate codes with many logical qubits. The proposed approach is general and applies to a wide spectrum of quantum error-correcting codes including surface codes and quantum low-density parity-check codes (qLDPC), under noise models ranging from code capacity noise to circuit level noise. We conducted extensive numerical experiments to demonstrate that our approach achieves significantly higher decoding accuracy compared to the minimum weight perfect matching and belief propagation with ordered statistics on the surface codes and high-rate quantum low-density parity-check codes. Our approach highlights generative artificial intelligence as a potential solution for the real-time decoding of realistic and high-rate quantum error correction codes.