quantum-computing

Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes

John Blue, Harshil Avlani, Zhiyang He, Liu Ziyin, and Isaac L. Chuang

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AbstractFault-tolerant quantum computers will depend crucially on the performance of the classical decoding algorithm which takes in the results of measurements and outputs corrections to the errors inferred to have occurred. Machine learning models have shown great promise as decoders for the surface code; however, this promise has not yet been substantiated for the more challenging task of decoding quantum low-density parity-check (QLDPC) codes. In this paper, we present a recurrent, transformer-based neural network designed to decode circuit-level noise on Bivariate Bicycle (BB) codes. For the $[[72,12,6]]$ BB code, at a physical error rate of $p=0.1\%$, our model achieves logical error rates almost $5$ times lower than belief propagation with ordered statistics decoding (BP-OSD), and roughly $5$ times larger than a most-likely error decoder. Moreover, while BP-OSD has a wide distribution of runtimes with significant outliers, our model has a consistent runtime and is an order-of-magnitude faster than the worst-case times from a benchmark BP-OSD implementation. On the $[[144,12,12]]$ BB code, our model obtains worse logical error rates but maintains the speed advantage. These results provide initial evidence that machine learning decoders can out-perform conventional decoders on small QLDPC codes, but suggest more complex architectures and/or training procedures are necessary to scale to larger code sizes.► BibTeX data@article{Blue2026machinelearning, doi = {10.22331/q-2026-06-30-2149}, url = {https://doi.org/10.22331/q-2026-06-30-2149}, title = {Machine {L}earning {D}ecoding of {C}ircuit-{L}evel {N}oise for {B}ivariate {B}icycle {C}odes}, author = {Blue, John and Avlani, Harshil and He, Zhiyang and Ziyin, Liu and Chuang, Isaac L.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2149}, month = jun, year = {2026} }► References [1] Barbara M. Terhal. ``Quantum error correction for quantum memories''. Rev. Mod. Phys. 87, 307–346 (2015). https://doi.org/10.1103/RevModPhys.87.307 [2] Poulami Das, Christopher A. 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The above citations are from SAO/NASA ADS (last updated successfully 2026-06-30 13:00:50). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-06-30 13:00:48: Could not fetch cited-by data for 10.22331/q-2026-06-30-2149 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractFault-tolerant quantum computers will depend crucially on the performance of the classical decoding algorithm which takes in the results of measurements and outputs corrections to the errors inferred to have occurred. Machine learning models have shown great promise as decoders for the surface code; however, this promise has not yet been substantiated for the more challenging task of decoding quantum low-density parity-check (QLDPC) codes. In this paper, we present a recurrent, transformer-based neural network designed to decode circuit-level noise on Bivariate Bicycle (BB) codes. For the $[[72,12,6]]$ BB code, at a physical error rate of $p=0.1\%$, our model achieves logical error rates almost $5$ times lower than belief propagation with ordered statistics decoding (BP-OSD), and roughly $5$ times larger than a most-likely error decoder. Moreover, while BP-OSD has a wide distribution of runtimes with significant outliers, our model has a consistent runtime and is an order-of-magnitude faster than the worst-case times from a benchmark BP-OSD implementation. On the $[[144,12,12]]$ BB code, our model obtains worse logical error rates but maintains the speed advantage. These results provide initial evidence that machine learning decoders can out-perform conventional decoders on small QLDPC codes, but suggest more complex architectures and/or training procedures are necessary to scale to larger code sizes.► BibTeX data@article{Blue2026machinelearning, doi = {10.22331/q-2026-06-30-2149}, url = {https://doi.org/10.22331/q-2026-06-30-2149}, title = {Machine {L}earning {D}ecoding of {C}ircuit-{L}evel {N}oise for {B}ivariate {B}icycle {C}odes}, author = {Blue, John and Avlani, Harshil and He, Zhiyang and Ziyin, Liu and Chuang, Isaac L.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2149}, month = jun, year = {2026} }► References [1] Barbara M. Terhal. ``Quantum error correction for quantum memories''. Rev. Mod. Phys. 87, 307–346 (2015). https://doi.org/10.1103/RevModPhys.87.307 [2] Poulami Das, Christopher A. 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Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes

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