Quantum Error Correction near the Coding Theoretical Bound

Abstract: Recent progress in quantum computing has enabled systems with tens of reliable logical qubits, built from thousands of noisy physical qubits. However, many impactful applications demand quantum computations with millions of logical qubits, necessitating highly scalable quantum error correction. In classical information theory, low-density parity-check (LDPC) codes can approach channel capacity efficiently. Yet, no quantum error-correcting codes with efficient decoding have been shown to approach the hashing bound - a fundamental limit on quantum capacity - despite decades of research. Here, we present quantum LDPC codes that not only approach the hashing bound but also allow decoding with computational cost linear in the number of physical qubits. This breakthrough paves the way for large-scale, fault-tolerant quantum computation. Combined with emerging hardware that manages many qubits, our approach brings quantum solutions to important real-world problems significantly closer to reality.

• The paper presents quantum LDPC codes that approach the hashing bound while ensuring linear decoding complexity.
• It introduces advanced protograph-based construction, employing permutation matrices and finite field extensions to minimize error floors.
• Experiments confirm significant error rate reduction, making the approach viable for scalable, fault-tolerant quantum systems.

Quantum Error Correction near the Coding Theoretical Bound

Introduction

The paper "Quantum Error Correction near the Coding Theoretical Bound" addresses the pivotal challenge of devising quantum error-correcting codes approaching the hashing bound with scalable decoding complexity. As quantum computing advances, achieving fault-tolerant quantum computation for systems with millions of logical qubits becomes a critical necessity. Traditional quantum error-correcting codes fail to combine proximity to the hashing bound with efficient decoding. This research introduces quantum LDPC codes capable of reaching the hashing bound while maintaining decoding complexity linear to the number of physical qubits, accomplished through innovative construction and decoding techniques.

Code Construction

The cornerstone of this study is the development of LDPC-CSS codes constructed from orthogonal protograph matrix pairs. This involves extending traditional LDPC paradigms by leveraging permutation matrices beyond circulant forms, enhancing structural randomness to mitigate short cycle issues typically capped at a girth of 12 in QC-LDPC codes. By satisfying specified commutativity and cycle avoidance conditions, the construction achieves quantum codes with larger girth, minimizing error floors. The use of finite field extension further optimizes these matrices for decoding, ultimately allowing for binary orthogonal matrix construction critical for CSS code definition.

Decoding Methodology

The paper employs a sum-product algorithm (SP) adept at handling the underlying code structure, which accounts for the intricate correlations of quantum errors. This decoding approach, extending beyond classical decodings, processes the syndrome information to reliably estimate quantum error vectors. By integrating FFT for computational efficiency and treating errors comprehensively, the SP-based decoding ensures performance that closely approaches the theoretical hashing bound.

Numerical Analysis

Extensive numerical analyses conducted on various code configurations demonstrate significant strides towards achieving the hashing bound. The paper presents results for multiple code rates, showing substantial elimination of error floors down to frame error rates (FER) of $10^{-4}$. Notably, the codes’ performance approaches the hashing bound, showcasing the effectiveness of the proposed construction and decoding strategies. Comparison with traditional methods highlights marked improvements in error correction capabilities, underscoring the potential for practical application in large-scale quantum systems.

Conclusion

This research sets a precedent in quantum error correction by simultaneously achieving proximity to the hashing bound and maintaining efficient decoding complexity. The introduction of generalized protograph-based code construction and enhancement of error floor mitigation through optimized decoding strategies offer a viable path for deploying scalable quantum error correction essential for future quantum information processing applications. Future investigation is recommended to extend the methodology, especially concerning further reductions in error floors and addressing potential scalability challenges in more complex quantum networks.