Quantum low-density parity-check (qLDPC) codes are an important component in the quest for quantum fault tolerance. Dramatic recent progress on qLDPC codes has led to constructions which are asymptotically good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. These constructions, while theoretically explicit, rely on inner codes with strong properties only shown to exist by probabilistic arguments, resulting in lengths that are too large to be practically relevant. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice. A previous construction based on the lifted product of an expander-based classical LDPC code with a repetition code (Panteleev & Kalachev, 2020) achieved a near-linear distance (of $\Omega(N/\log N)$ where $N$ is the number of codeword qubits), and avoids the need for such intractable inner codes. Our main result is an efficient decoding algorithm for these codes that corrects $\Theta(N/\log N)$ adversarial errors. En route, we give such an algorithm for the hypergraph product version these codes, which have weaker $\Theta(\sqrt{N})$ distance (but are simpler). Our decoding algorithms leverage the fact that the codes we consider are quasi-cyclic, meaning that they respect a cyclic group symmetry. Since the repetition code is not based on expanders, previous approaches to decoding expander-based qLDPC codes, which typically worked by greedily flipping code bits to reduce some potential function, do not apply in our setting. Instead, we reduce our decoding problem (in a black-box manner) to that of decoding classical expander-based LDPC codes under noisy parity-check syndromes. For completeness, we also include a treatment of such classical noisy-syndrome decoding that is sufficient for our application to the quantum setting.

翻译：量子低密度奇偶校验（qLDPC）码是实现量子容错的重要组件。近期qLDPC码研究取得突破性进展，已构建出渐近性能优良的编码方案，并具备线性时间解码器以纠正影响恒定比例码字量子比特的错误。这些构造虽然在理论上具有显式性，但其依赖的内码需具备仅通过概率论证证明存在的强性质，导致编码长度过大而缺乏实际应用价值。在实际应用中，基于两个重复码乘积的表面码/环面码仍是常用的qLDPC码选择。先前基于扩展图经典LDPC码与重复码提升乘积的构造（Panteleev & Kalachev, 2020）实现了近线性距离（达到$\Omega(N/\log N)$，其中$N$为码字量子比特数），且避免了此类难处理内码的需求。本文主要成果是针对这类编码的高效解码算法，能够纠正$\Theta(N/\log N)$个对抗性错误。在研究过程中，我们首先为这类编码的超图乘积版本提供了相应算法，该版本具有较弱的$\Theta(\sqrt{N})$距离（但结构更简单）。我们的解码算法利用了所研究编码的准循环特性——即它们保持循环群对称性。由于重复码不基于扩展图，以往解码扩展图qLDPC码的典型方法（通常通过贪婪翻转码比特来降低势函数）不适用于本场景。为此，我们将解码问题（以黑盒方式）归约为含噪声奇偶校验校验子的经典扩展图LDPC码解码问题。为保持完整性，我们还提供了适用于量子场景的经典噪声校验子解码理论框架。