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 paper 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。实用的量子纠错码（如稳定子码）通常针对特定用途设计，具有固定的码长和码率。本文证明，在码率足够高的情况下，对于任意选定的码长，可以构造并解码达到有限块长度区域最大性能的量子纠错码。近期提出的一种名为GRAND（猜测随机加性噪声解码）的经典码解码策略，为高效解码性能接近有限块长度区域最大速率的经典随机线性码打开了大门。通过利用噪声统计特性，只要存在简单的码成员测试，GRAND便是一种以噪声为中心的经典码高效通用解码器。这些条件特别适用于量子系统，因此本文将此类概念扩展至量子随机线性码——这类码已知可构造，但此前其解码尚不可行。通过结合量子随机线性码与新提出的量子GRAND算法，本文证明能够解码易于适应变化条件的量子纠错码。本文首先评估了达到量子随机线性码渐近性能所需的编码电路最小门数，随后提出了一种量子GRAND算法，该算法利用量子噪声统计特性，不仅能构建自适应的码成员测试，还能高效实现伴随式解码。