Quantum error correction codes (QECCs) play a central role in both quantum communications and quantum computation. Practical quantum error correction codes, such as stabilizer codes, are generally structured to suit a specific use, and present rigid code lengths and code rates. This paper shows that it is possible to both construct and decode QECCs that can attain the maximum performance of the finite blocklength regime, for any chosen code length when the code rate is sufficiently high. A recently proposed strategy for decoding classical codes called GRAND (guessing random additive noise decoding) opened doors to efficiently decode classical random linear codes (RLCs) performing near the maximum rate of the finite blocklength regime. By using noise statistics, GRAND is a noise-centric efficient universal decoder for classical codes, provided that a simple code membership test exists. These conditions are particularly suitable for quantum systems, and therefore the paper extends these concepts to quantum random linear codes (QRLCs), which were known to be possible to construct but whose decoding was not yet feasible. By combining QRLCs and a newly proposed quantum-GRAND, this work shows that it is possible to decode QECCs that are easy to adapt to changing conditions. The paper starts by assessing the minimum number of gates in the coding circuit needed to reach the QRLCs' asymptotic performance, and subsequently proposes a quantum-GRAND algorithm that makes use of quantum noise statistics, not only to build an adaptive code membership test, but also to efficiently implement syndrome decoding.

翻译：量子纠错码在量子通信和量子计算中均发挥着核心作用。实际量子纠错码（如稳定子码）通常针对特定用途而设计，具有固定的码长和码率。本文证明，在码率足够高的条件下，对于任意选定码长，可以构造并译码能实现有限块长区域最优性能的量子纠错码。近期提出的经典码译码策略——猜测随机加性噪声译码（GRAND）——为高效译码性能接近有限块长区域最大码率的经典随机线性码（RLCs）开辟了道路。基于噪声统计特性，GRAND是一种以噪声为中心的通用高效经典码译码器，前提是存在简单的码成员测试。这些条件特别适用于量子系统，因此本文将上述概念拓展至量子随机线性码（QRLCs）。尽管已知QRLCs可被构造，但其译码此前始终不可行。通过结合QRLCs与新提出的量子GRAND，本研究表明，能够译码易于适应条件变化的量子纠错码。本文首先评估达到QRLCs渐近性能所需的编码电路最小门数，随后提出一种利用量子噪声统计特性的量子GRAND算法，该算法不仅可构建自适应码成员测试，还能高效实现伴随式译码。