Quantum error correction must detect and correct errors without directly reading the quantum information. Classical low-density parity check (LDPC) codes have a well-established design theory: by choosing how many checks connect to each bit, retaining appropriate randomness, and avoiding short loops, designers can pursue both a large minimum distance and a threshold phenomenon in which the decoding failure rate drops sharply below a predicted noise level. Quantum LDPC codes must additionally make two types of error checks orthogonal so that they do not interfere. Applying this quantum mechanical constraint throughout the full design can create short loops and weak structures, making it difficult to retain both classical advantages.
Associate Professor Kenta Kasai of Institute of Science Tokyo (Science Tokyo) in Japan asked whether classical LDPC design theory could be transferred to quantum codes while preserving both distance and threshold behavior. He constructed a quantum LDPC code that uses affine permutation matrices and applies orthogonality only to the selected, or active, rows used for error correction. The complementary, or latent, parent rows retain randomness, allowing connection degrees and short loops to be designed according to classical LDPC principles.
His study is published in the journal Quantum on September 9, 2026.
“The quantum constraint is essential, but it does not have to govern every part of the design. By applying it only where error correction uses it, we can preserve the classical LDPC design freedom needed to pursue both a large minimum distance and threshold behavior,” says Kasai.
The method produced a (3,12)-regular, girth-8 code written as [[9216,4612,d]] with d≤48. It protects 4,612 logical qubits using 9,216 physical qubits—a ratio of nearly one logical qubit for every two physical qubits—indicating the potential to reduce the hardware overhead of quantum error correction. Weight-48 logical operators can be constructed explicitly, and the X- and Z-type distances associated with the latent structure are both exactly 48. Detailed searches and low-error-rate simulations found no lower-weight logical errors. Although a global lower bound on d remains open, these results provide strong evidence that the overall minimum distance is near 48. Eliminating 4- and 6-cycles also reduces trapping sets that can stall BP decoding.
“The significance is not only that one high-rate code performs well. Threshold, minimum distance, short loops, and hard-to-decode error patterns can now be considered within the same design framework developed for classical LDPC codes,” says Kasai.
The code was evaluated using BP decoding with low-complexity post-processing. A concrete code protects 4,612 logical qubits with 9,216 physical qubits, shows strong evidence of a minimum distance near 48, and exhibits a clear decoding waterfall close to the density-evolution prediction for the corresponding classical, nonorthogonal random (3,12)-regular LDPC ensemble. That classical BP benchmark is p≈0.05702; notably, this is not a measured threshold of the proposed quantum code. With post-processing, its frame error rate reached 10⁻⁸ at 4% depolarizing noise, corresponding to approximately one failure per 100 million trials.
The high encoding rate—roughly one logical qubit per two physical qubits—could substantially reduce the hardware required for quantum error correction. More broadly, the waterfall's approach to the classical prediction suggests that the classical methodology of predicting decoder limits from connection degrees and designing toward those limits can also guide quantum LDPC codes. The present results are based on theoretical design and numerical experiments; performance on a specific quantum processor must be evaluated separately.
“Classical LDPC coding has spent decades learning how to predict decoding limits from connection degrees and design toward those limits. Seeing the quantum code's waterfall approach the same benchmark suggests that this methodology can guide quantum LDPC design,” explains Kasai.
Kasai released the preprint on arXiv on January 13, 2026. While it was still under review, researchers at Harvard University, MIT, and QuEra Computing adapted the construction to reconfigurable neutral-atom quantum computers. Their paper, posted on April 17, designs codes and error-detection procedures around atom rearrangement with acousto-optic deflectors (AODs) and refers to the resulting family as “Kasai codes.” QuEra highlighted this development in official posts, including one titled “Kasai Code Breakthrough in Quantum Error Correction.” Kasai's paper was accepted by Quantum on August 3, 2026.
Other independent teams have extended the design. A team including Tsinghua University's Center for Quantum Information posted Cornucopia codes on August 3, 2026, emphasizing regular structures suited to simultaneous atom movements. Willers Yang and colleagues posted GALA codes on August 7, 2026; their general framework recovers previously developed rate-1/2 Kasai constructions as special cases and incorporates AOD-compatible atom movements and logical operations into code design. Between February and July 2026, Kasai presented the preprint in 11 invited talks at universities, companies, and international workshops and conferences. These included QID2026 at the Korea Institute for Advanced Study (KIAS) in Seoul, Korea, where he gave an invited talk titled “Advancing Quantum LDPC Codes with Classical LDPC Design Principles.” Further invited talks were scheduled for LG Electronics' Quantum Journal Club in August; the 2026 YITP Fault-Tolerant Logical Processing Workshop at Kyoto University's Yukawa Institute for Theoretical Physics in September–October; the international conference Frontiers in Scaling Superconducting Quantum Computing in Italy in October; the Workshop on Quantum Error Correction at CWI in Amsterdam, the Netherlands, in October; the RQC Colloquium at RIKEN's Center for Quantum Computing in October; and the 3rd RIKEN-Harvard Joint Quantum Workshop at RIKEN in December. In January 2027, Kasai is also scheduled to give an invited talk in the special session “Coding Theory: From Foundations to Frontiers” at the Joint Mathematics Meetings (JMM 2027) in Chicago, Illinois.
“Independent teams are now adapting the same design principles to hardware layouts and more general code families. This suggests that Kasai codes are becoming a common foundation for international work on scalable quantum error correction,” concludes Kasai.
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About Institute of Science Tokyo (Science Tokyo)
Institute of Science Tokyo (Science Tokyo) was established on October 1, 2024, following the merger between Tokyo Medical and Dental University (TMDU) and Tokyo Institute of Technology (Tokyo Tech), with the mission of “Advancing science and human wellbeing to create value for and with society.”
Reference
Author: Kenta Kasai
Title: Breaking the Orthogonality Barrier in Quantum LDPC Codes
Journal: Quantum
DOI: https://doi.org/10.22331/q-2026-09-09-2205
Affiliation: Department of Information and Communications Engineering, Institute of Science Tokyo, Japan
Quantum
Computational simulation/modeling
Not applicable
Breaking the Orthogonality Barrier in Quantum LDPC Codes
9-Sep-2026