解码准循环量子LDPC码
Decoding Quasi-Cyclic Quantum LDPC Codes
AI总结：
本文提出一种针对基于提升乘积构造的准循环量子LDPC码的高效解码算法，可纠正Θ(N/log N)个对抗性误差，并将解码问题归约为经典扩张器LDPC码的含噪综合征解码。
摘要：
量子低密度奇偶校验（qLDPC）码是追求量子容错过程中的重要组成部分。近期关于qLDPC码的显著进展催生了渐近良好码的构造，这些码允许线性时间解码器来纠正影响码字量子比特恒定比例的误差。这些构造虽然理论上明确，但依赖于仅通过概率论证证明存在性的强内码，导致码长过大，难以实际应用。在实践中，由两个重复码乘积得到的曲面/环面码仍是常被选用的qLDPC码。\n 先前基于扩张器经典LDPC码与重复码的提升乘积的构造（Panteleev & Kalachev, 2020）实现了近线性距离（$\Omega(N/\log N)$，其中$N$为码字量子比特数），并避免了此类难以处理的内码的需求。我们的主要成果是为这些码提供一种高效解码算法，可纠正$\Theta(N/\log N)$个对抗性误差。在此过程中，我们为这些码的超图乘积版本给出了此类算法，该版本具有较弱的$\Theta(\sqrt{N})$距离（但更简单）。我们的解码算法利用了所考虑的码是准循环的这一事实，即它们尊重循环群对称性。\n 由于重复码并非基于扩张器，先前针对基于扩张器的qLDPC码的解码方法（通常通过贪婪翻转码比特以降低某个势函数）不适用于我们的场景。相反，我们将解码问题（以黑盒方式）归约为在含噪奇偶校验综合征下解码经典基于扩张器的LDPC码。为完整性，我们还包含了足以应用于量子场景的经典含噪综合征解码的处理。
英文摘要：
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 $Ω(N/\log N)$ 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 $Θ(N/\log N)$ adversarial errors. En route, we give such an algorithm for the hypergraph product version these codes, which have weaker $Θ(\sqrt{N})$ 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.
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