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A new class of highly efficient and scalable quantum low-density parity-check error correction codes, capable of performance approaching the theoretical hashing bound, has been developed by scientists at the Institute of Science, Tokyo, Japan. These novel error correction codes can handle quantum codes with hundreds of thousands of qubits, potentially enabling large-scale fault-tolerant quantum computing, with applications in diverse fields, including quantum chemistry and optimization problems.

In recent years, quantum computers have begun to handle double-digit quantum bits, or qubits. However, many essential applications targeted by quantum computers, such as quantum chemistry, cryptography, etc., demand millions or even more logical qubits. Scaling to such numbers is a major challenge, as quantum computers suffer from inherent errors that increase rapidly with the number of qubits.

For practical quantum computing, highly efficient quantum correction methods that can accommodate a vast number of logical qubits are necessary. Unfortunately, current quantum error correction methods are extremely resource-intensive, relying on essentially zero-rate codes. As a result, only a tiny fraction of reliable logical qubits can be extracted from an overwhelmingly large number of physical qubits.

The biggest obstacles to scaling up quantum computers are engineering challenges related to device stability and control technologies.

These include the short coherence times of qubits, high error rates in gate operations and measurements, limited interaction ranges between qubits, and difficulties associated with large-scale integration and cooling technologies. Each of these represents a fundamental bottleneck in building large numbers of reliable logical qubits.

However, even in an idealized setting where such device-level limitations are removed, the design of quantum error correction codes has faced major unresolved challenges. These include low coding rates and the lack of sharp threshold phenomena that limit performance improvements. Additional challenges include performance stagnation in the high-reliability region due to error floors—a significant gap from the theoretical hashing bound, and the need for costly post-processing after belief-propagation decoding.

While scaling up is essential for solving practical problems with quantum computers, it is also theoretically known that quantum error correction improves with scale. Yet, until now, no quantum codes had been discovered that could truly harness the benefits of large-scale quantum systems.

In classical information theory, there exist highly efficient error correction codes, known as low-density parity-check (LDPC) codes, that can approach the theoretical performance limit.

While many studies have attempted to develop quantum LDPC error correction codes, to date, none have been able to approach the hashing bound, the maximum theoretical amount of information that can be transmitted over a quantum channel, which serves as the benchmark for quantum error correction performance.

In a recent breakthrough, Associate Professor Kenta Kasai and Master's student Mr. Daiki Komoto from the Department of Information and Communications Engineering, School of Engineering at Institute of Science Tokyo (Science Tokyo), Japan, have successfully developed novel LDPC quantum error correction codes, capable of approaching the hashing bound, while maintaining high efficiency.

"Our quantum error-correcting code has a greater than 1/2 code rate, targeting hundreds of thousands of logical qubits," explains Kasai. "Moreover, its decoding complexity is proportional to the number of physical qubits, which is a significant achievement for quantum scalability."

Their study is published in the journal npj Quantum Information.

The researchers started by constructing protograph LDPC codes, which are known to have excellent error correction performance. They introduced a novel construction technique based on affine permutations, which improves diversity in code structure and avoids short cycles, which are known to negatively affect decoding performance.

Unlike conventional LDPC codes, which are defined over binary finite fields, these new codes are defined over non-binary finite fields. This means that these codes can carry more information, improving the decoding performance. These protograph codes were then transformed into Calderbank-Shor-Steane codes, a well-known family of quantum error correction codes.

Additionally, the researchers developed a new, efficient decoding method using the well-known sum-product algorithm. This method handles both bit-flip (X) and phase-flip (Z) errors, the two fundamental errors in quantum computing, simultaneously. Any error in a quantum state can be corrected by addressing these errors. For comparison, most previous error-correction codes handle only one type of error at a time.

As a result, in large-scale numerical simulations, these new error-correction codes achieved extremely high decoding performance, demonstrating a frame error rate as low as 10^−4, even for codes with hundreds of thousands of qubits.

This performance is quite close to the hashing bound, representing a significant achievement. Importantly, these error correction codes have a decoding complexity proportional to the number of qubits, making them highly efficient and scalable for large-scale quantum computing.

"Our quantum LDPC error correction codes can potentially enable quantum computers to scale up to millions of logical qubits," remarks Kasai.

"This will significantly improve the reliability and scalability of quantum computers for practical applications while also paving the way for future research."

Overall, this study represents a major step forward for the development of fault-tolerant quantum computers for practical applications, benefiting many fields.