Classical Code Design Theory Yields High-Rate Quantum LDPC Codes

Quantum computers promise to solve problems that no classical machine could ever hope to tackle, yet the fragile quantum bits at their heart are extraordinarily susceptible to noise. Every serious roadmap for building a useful quantum computer therefore depends on quantum error correction, the art of protecting quantum information without ever measuring it directly. Now, Associate Professor Kenta Kasai of Institute of Science Tokyo (Science Tokyo) has shown that one of the most successful design frameworks in classical communications can be carried over into the quantum realm almost intact, producing a quantum low-density parity-check (LDPC) code with a remarkably high encoding rate and strong evidence of excellent error-correcting properties. The work, published in the journal Quantum on September 9, 2026, is already reshaping how research groups around the world approach quantum code design.

Classical LDPC codes are the quiet workhorses of modern digital life. They protect data in 5G wireless links, satellite broadcasts, Wi-Fi signals, and solid-state storage drives. Their power comes from a mature and well-understood design theory. Engineers choose how many parity checks connect to each bit, retain just the right amount of randomness in the connection pattern, and carefully avoid short cycles in the graph that describes the code. When these ingredients are balanced correctly, the resulting codes achieve two prized properties simultaneously: a large minimum distance, which guarantees that many errors must accumulate before information is corrupted, and a threshold phenomenon, in which the decoding failure rate drops sharply once the noise falls below a predictable level. This threshold behavior is what makes LDPC codes approach the theoretical limits of communication.

Quantum LDPC codes face an additional constraint that has no classical analogue. Because quantum mechanics forbids the direct measurement of the encoded information, two different families of error checks, one for bit-flip errors and one for phase-flip errors, must be made orthogonal to each other so that they do not interfere when both are measured. In previous constructions, imposing this orthogonality requirement across the entire design process tended to destroy the very features that make classical LDPC codes work so well. Short loops crept back into the structure, connection degrees became difficult to control, and designers were forced to choose between a good minimum distance and a good threshold, rarely achieving both at once.

Kasai’s insight was to ask whether the orthogonality constraint truly needs to govern every part of the design. His answer was no. In the construction described in his paper, titled Breaking the Orthogonality Barrier in Quantum LDPC Codes, he built the code from affine permutation matrices and applied the orthogonality condition only to the selected, or active, rows that are actually used for error correction. The complementary, or latent, parent rows remain free of the quantum constraint and retain their randomness. This separation means that connection degrees, randomness, and the avoidance of short loops can all be engineered according to classical LDPC principles, while the quantum-mechanical requirement is satisfied exactly where it matters.

The quantum constraint is essential, but it does not have to govern every part of the design, Kasai explains. By applying it only where error correction uses it, the classical LDPC design freedom needed to pursue both a large minimum distance and threshold behavior can be preserved. The resulting code is a (3,12)-regular construction with girth 8, meaning the shortest loops in its graphical structure contain eight edges, and it is written in standard quantum notation as [[9216,4612,d]] with d at most 48. That notation means the code protects 4,612 logical qubits using 9,216 physical qubits, a ratio of nearly one logical qubit for every two physical qubits. For comparison, the surface codes used in many current hardware proposals typically demand hundreds or thousands of physical qubits per logical qubit, so a high-rate code of this kind could dramatically reduce the hardware overhead of fault-tolerant quantum computing.

The evidence for a large minimum distance is particularly strong. Weight-48 logical operators can be constructed explicitly, and the X-type and Z-type distances associated with the latent structure of the code are both exactly 48. Detailed searches and simulations at low error rates found no logical errors of lower weight anywhere in the code. Although a rigorous global lower bound on the minimum distance remains an open mathematical question, these results together provide compelling evidence that the overall minimum distance sits near 48. In addition, the deliberate elimination of 4-cycles and 6-cycles reduces the trapping sets, those troublesome small structures in the code graph that can stall belief-propagation decoding and cause otherwise correctable errors to slip through.

Decoding performance was evaluated using belief propagation with low-complexity post-processing. The code exhibits a clear decoding waterfall that sits close to the density-evolution prediction for the corresponding classical, non-orthogonal random (3,12)-regular LDPC ensemble. That classical benchmark, which describes the best a belief-propagation decoder could do on an idealized infinite version of the code, is approximately 0.05702, and Kasai emphasizes that this figure is a reference point rather than a measured threshold of the quantum code itself. With post-processing, the quantum code achieved a frame error rate of 10 to the power of minus 8 at 4 percent depolarizing noise, corresponding to roughly one decoding failure per 100 million trials. The significance, Kasai notes, is not merely that one high-rate code performs well, but that threshold behavior, minimum distance, short loops, and hard-to-decode error patterns can now all be considered within the same design framework that classical coding theory spent decades perfecting.

Classical LDPC coding has spent decades learning how to predict decoding limits from connection degrees and to design codes that approach those limits, Kasai explains. Seeing the quantum code’s waterfall approach the same classical benchmark suggests that this predictive methodology can now guide quantum LDPC design as well. He cautions, however, that the present results rest on theoretical analysis and numerical experiments, and that performance on a specific quantum processor must be evaluated separately before any hardware claims can be made.

The wider community has not waited for that hardware evaluation. Kasai released a preprint on arXiv on January 13, 2026, and while the paper 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 using acousto-optic deflectors, and refers to the resulting family of codes as Kasai codes. QuEra highlighted the 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, and between February and July he presented the work in eleven invited talks at universities, companies, and international workshops, including QID2026 at the Korea Institute for Advanced Study in Seoul.

Independent teams have continued to extend the framework. A group including Tsinghua University’s Center for Quantum Information posted a family of Cornucopia codes on August 3, 2026, emphasizing regular structures suited to simultaneous atom movements, and Willers Yang and colleagues posted GALA codes on August 7, 2026, offering a general framework that recovers the earlier rate-1/2 Kasai constructions as special cases while incorporating atom movements and logical operations directly into the code design. Further invited presentations are scheduled at LG Electronics, Kyoto University’s Yukawa Institute for Theoretical Physics, a superconducting quantum computing conference in Italy, CWI in Amsterdam, RIKEN’s Center for Quantum Computing, and the Joint Mathematics Meetings in Chicago in January 2027. As Kasai concludes, independent teams adapting the same design principles to hardware layouts and more general code families suggests that these codes are becoming a common foundation for international work on scalable quantum error correction, a development that could bring fault-tolerant quantum computers significantly closer to practical reality.

Subject of Research: Quantum low-density parity-check error-correcting codes designed using classical LDPC design theory

Article Title: Bringing classical LDPC code design theory to quantum computers

Article References: Bringing classical LDPC code design theory to quantum computers. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: quantum error correction, LDPC codes, Kenta Kasai, Science Tokyo, minimum distance, belief propagation, neutral-atom quantum computing, encoding rate, density evolution, Quantum journal, trapping sets, fault tolerance