Stabilizer-assisted Decoding Achieves 20% Complexity Reduction for Quantum Error Correction

Quantum error correction represents a crucial hurdle in building practical quantum computers, and researchers are continually seeking more efficient decoding methods. Giulio Pech, Mert Gökduman, Hanwen Yao, and Henry D. Pfister, all from the Duke Quantum Center at Duke University, have now presented a novel approach to decoding quantum low-density parity-check (QLDPC) codes with erasures, significantly reducing computational complexity. Their work introduces a ‘stabilizer-assisted inactivation decoding’ method, combining classical peeling with a new ‘dual peeling’ procedure to identify and fix bits within the erased set without compromising the decoded result. This innovation not only matches the performance of maximum likelihood decoding but also demonstrably reduces the number of symbolic guesses required , exceeding a 20% reduction for certain codes at high erasure rates , paving the way for more scalable and practical quantum computation. Erasure Decoding via Stabilizer Matrix Reduction Scientists have developed a new, reduced-complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes operating over erasures, a significant step towards practical fault-tolerant quantum computation. This breakthrough combines established inactivation decoding, integrating peeling with symbolic guessing, with a novel dual peeling procedure to dramatically improve efficiency. The research team achieved this by performing row operations on the stabilizer matrix, revealing stabilizer generators and their linear combinations that are entirely supported by the set of erased qubits. Identifying these fully erased stabilizers allows the researchers to fix bits without affecting the logical state of the decoded result, effectively reducing the number of variables requiring symbolic guesses and shrinking the size of the final linear system needing to be solved. Crucially, the study unveils that combining dual peeling with standard peeling alone, without inactivation, is sufficient to achieve maximum likelihood decoding for surface codes, demonstrating a streamlined approach for a prominent quantum computing architecture. Experiments across several QLDPC code families confirm that this new decoder matches the performance of ML decoding in terms of logical failure rates, while simultaneously reducing computational complexity. Specifically, the team demonstrated a greater than 20% reduction in symbolic guesses for the B1 lifted product code at high erasure rates, a substantial improvement in decoding speed and resource usage. This reduction in symbolic guesses directly translates to a smaller linear system to solve, further accelerating the decoding process. The core innovation lies in the dual peeling procedure, which efficiently identifies fully erased stabilizers by strategically manipulating the stabilizer matrix. These stabilizers, while not directly contributing to error correction, previously created bottlenecks in the peeling process and unnecessarily inflated the number of symbolic guesses required during inactivation decoding. By proactively identifying and leveraging these fully erased stabilizers, the researchers bypass these bottlenecks, streamlining the entire decoding process. This integrated approach, dual peeling followed by peeling and inactivation, provides a powerful and efficient method for achieving maximum likelihood erasure decoding for QLDPC codes, paving the way for more practical and scalable quantum error correction. Furthermore, the work establishes that for surface codes, the dual peeling technique, when used in conjunction with standard peeling, eliminates the need for inactivation altogether, resulting in an even simpler and faster decoding algorithm. Simulations consistently demonstrate that the decoder achieves the same logical failure performance as the computationally intensive ML decoding, but with significantly reduced complexity. This achievement is particularly relevant given the increasing focus on erasure-based error correction strategies, driven by advancements in qubit architectures and erasure conversion proposals that translate physical errors into erasures, making this noise model increasingly practical.

The team will present their findings as a poster at QIP 2026.

Stabilizer Row Operations for QLDPC Decoding offer significant Scientists have engineered a reduced complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes operating over erasures, significantly advancing error correction capabilities. This work combines established inactivation decoding, integrating peeling with symbolic guessing, with a novel dual peeling procedure to tackle the complexities of decoding erased quantum information. The core innovation lies in performing row operations on the stabilizer matrix, efficiently revealing stabilizer generators and their linear combinations that exclusively affect the erased qubit set. Crucially, identifying these stabilizers allows the researchers to freely fix the value of an erased bit without altering the logical state of the decoded result, effectively reducing the number of variables requiring symbolic guesses. This reduction directly translates to a smaller linear system needing to be solved, dramatically decreasing computational burden. The study demonstrates that combining dual peeling with standard peeling, even without inactivation, is sufficient to achieve maximum likelihood decoding for surface codes, a significant theoretical result. Experiments employed several QLDPC code families to validate the decoder’s performance, confirming it matches the logical failure performance of ML decoding while substantially reducing the complexity of inactivation decoding, achieving over a 20% reduction in symbolic guesses for the B1 lifted product code at high erasure rates. The methodology centres on maximising the number of fully erased stabilizers, then fixing one erased bit within each of their supports to eliminate degrees of freedom that would otherwise necessitate symbolic guesses. Following this, peeling plus inactivation decoding is applied to the diminished erasure pattern, with symbolic guesses determined by solving a system of linear equations using Gaussian elimination. This integrated approach markedly decreases both the number of symbolic guesses and the size of the final linear system, providing an efficient pathway to maximum likelihood erasure decoding for QLDPC codes. For surface codes specifically, the team showed that applying dual peeling followed by standard peeling, without inactivation, is sufficient for achieving ML decoding over erasures, streamlining the process further. Simulations across diverse QLDPC code families consistently confirmed that the decoder attains equivalent logical failure performance to ML decoding, but with a significantly lower computational cost.

Dual Peeling Simplifies QLDPC Decoding Significantly, improving performance Scientists have developed a reduced complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes designed for erasure channels. The research team combined classical inactivation decoding, integrating peeling with symbolic guessing, with a novel dual peeling procedure to achieve this breakthrough. During the dual peeling stage, row operations were performed on the stabilizer matrix, efficiently revealing stabilizer generators and their linear combinations solely supported by the erased set. Each identified stabilizer allowed the team to freely fix a bit within its support without impacting the logical state of the decoded result, effectively reducing the number of inactivated variables and the size of the final linear system needing to be solved. Experiments revealed that combining dual peeling with standard peeling alone, without inactivation, is sufficient to achieve maximum likelihood (ML) decoding for erasure correction of surface codes. Across several QLDPC code families, the decoder matched ML logical failure performance while significantly reducing the complexity of inactivation decoding; notably, a greater than 20% reduction in symbolic guesses was observed for the B1 lifted product code at high erasure rates. The work demonstrates that if a nonzero portion of x′ E is an erased X-logical operator, then ML decoding must fail, confirming the algorithm’s success on the same set of erasures as ML decoding. Data shows that the combined dual-primal peeling decoder requires only O(n) operations, a substantial improvement over inactivation decoding, which uses Gaussian elimination on the matrix C with a complexity of O(|I|3). Measurements confirm that dual peeling substantially improves performance over primal peeling across all code families tested. Specifically, the expected number of inactivations was reduced for the B1 lifted product code, HGP2025 code, and various surface codes (11 and 13) at different erasure rates.

The team measured logical failure probability as a function of erasure probability (p) for several code families, comparing the ML inactivation decoder, primal peeling, and the combined dual-primal peeling decoder. Figure 2 plots these results, demonstrating the effectiveness of the new approach. Further analysis showed that dual peeling reduces the size of the matrix requiring inversion, an effect that remains significant even at higher erasure rates, though it varies between code families. The dual peeling stage identifies stabilizer generators within the erased set through row operations on the stabilizer matrix, allowing for the direct fixing of bits without impacting the decoded result. Researchers demonstrated that combining dual peeling with standard peeling alone achieves ML decoding for surface codes, and across several QLDPC code families, their decoder matches ML logical failure performance while significantly reducing decoding complexity, including a greater than 20% reduction in symbolic guesses for the B1 lifted product code at high erasure rates. The decoder’s advantage lies in its ability to reduce the size of the linear system needing to be solved, offering an O(n) operation count in some cases, a substantial improvement over inactivation decoding’s O(|I|3) complexity. The authors acknowledge a limitation in that while the combined dual-primal peeling decoder doesn’t always match ML performance, it still offers a significant computational benefit. Future work will focus on extending this stabilizer-assisted decoding to bit-flip channels and comparing it with more complex decoders like BPGD on the erasure channel. 👉 More information 🗞 Stabilizer-Assisted Inactivation Decoding of Quantum Error-Correcting Codes with Erasures 🧠 ArXiv: https://arxiv.org/abs/2601.14236 Tags: