发表机构
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本工作首次显式构造了达到列表解码容量且具有LDPC性质的量子码，并给出近线性时间列表解码算法，基于扩展图和量子AEL放大。
AI 中文摘要
在经典编码理论中，寻找达到列表解码容量的显式码一直是重要的驱动力。虽然随机码很容易被证明能达到容量，但显式构造直到数十年后才在开创性工作[Guruswami和Rudra，STOC 2006]中被发现。在量子编码理论中，非常期望码族是LDPC的。虽然好的量子码已知数十年，但获得额外的LDPC性质一直难以捉摸。事实上，直到最近，好的量子LDPC码才在突破性工作[Panteleev和Kalachev，STOC 2022]中被发现。在此背景下，一个自然的问题是寻求一个显式的量子码族，既能达到列表解码容量，又具有重要的LDPC性质。在本工作中，我们提供了（据我们所知）第一个达到列表解码容量的量子LDPC码的显式构造，即列表解码半径接近量子Singleton界且列表大小恒定。此外，我们提供了接近容量的近线性时间（关于块长）列表解码算法。我们的显式码构造基于扩展图，通过Alon-Edmonds-Luby（AEL）放大的量子类比[Bergamaschi，Golowich和Gunn，STOC 2024]，这实现了重要的LDPC性质。我们的高效列表解码算法通过将经典的基于扩展图的弱正则性列表解码算法[Srivastava和Tulsiani，FOCS 2025][Jeronimo和Singh，2025]推广到量子AEL的适当实例化而获得。
英文摘要
In classical coding theory, the quest for explicit codes achieving list decoding capacity has been an important driving force. While random codes are easily shown to achieve capacity, an explicit construction was only discovered decades later in the seminal work [Guruswami and Rudra, STOC 2006]. In quantum coding theory, it is highly desirable that a code family be LDPC. While good quantum codes were known for decades, obtaining the additional LDPC property was elusive. In fact, only very recently that good quantum LDPC codes were discovered in a breakthrough work [Panteleev and Kalachev, STOC 2022]. In this context, a natural question is to ask for an explicit family of quantum codes achieving list decoding capacity while also possessing the important LDPC property. In this work, we provide (to the best of our knowledge) the first explicit constructions of quantum LDPC codes achieving list decoding capacity, namely, with a list decoding radius approaching the quantum Singleton bound with constant list sizes. Furthermore, we provide near-linear time (in the block-length) list decoding algorithms approaching capacity. Our explicit code construction are based on expander graphs via the quantum analogue of Alon-Edmonds-Luby (AEL) amplification [Bergamaschi, Golowich and Gunn, STOC 2024], and this enables the important LDPC property. Our efficient list decoding algorithms are obtained by generalizing the classical expander-based weak-regularity list decoding algorithms [Srivastava and Tulsiani, FOCS 2025] [Jeronimo and Singh, 2025] to suitable instantiations of quantum AEL.
CommentsSODA 2027, to Appear
