从随机量子码到基于局部性质的显式qLDPC码

中文摘要 AI 辅助

构造与随机码参数匹配的显式码一直是编码理论中的核心且很大程度上悬而未决的问题。量子设置更具挑战性，因为量子码最好是LDPC码。局部坐标线性（LCL）见证[Levi, Mosheiff, 和 Shagrithaya, FOCS 2025]为许多编码理论性质提供了统一语言，从距离到列表解码和列表恢复。特别地，它为研究随机线性码的性质提供了框架，这些随机线性码在许多线性码性质上达到最优参数。然而，对于CSS量子码，局部见证有两个不同的秩：在商去稳定子之前的物理秩和商去之后的逻辑秩。我们为嵌套空间$S \subseteq C$开发了LCL框架的量子版本，其中局部约束施加在物理代表上，而独立性在逻辑商$C/S$中测量。由此产生的理论给出了随机CSS码的阈值定理，并因此表明每扇区速率阈值等于经典速率阈值。我们还定义了子空间设计[Guruswami和Xing, STOC 2013]的量子类比，并表明它们可以在量子-LCL框架内以自然方式描述。最后，我们以类似于[Jeronimo和Shagrithaya, STOC 2026]的LCL去随机化方式，为任意折叠量子-LCL性质给出显式构造。因此，我们获得了首批具有最优列表大小的量子列表可解码码和列表可恢复码的显式构造，以及显式量子子空间设计码。我们注意到，我们所有的显式构造都是qLDPC码，这是量子纠错码的一个重要性质。

英文摘要

Constructing explicit codes matching the parameters of random codes has been a central and largely elusive question in coding theory. The quantum setting is even more challenging since it is highly desirable that the quantum code be an LDPC code. Local coordinate-wise linear (LCL) [Levi, Mosheiff, and Shagrithaya, FOCS 2025] witnesses provide a unifying language for many coding-theoretic properties, from distance to list decoding and list recovery. In particular, it provides a framework to study properties of random linear codes, which achieve optimal parameters for many properties of linear codes. For CSS quantum codes, however, a local witness has two distinct ranks: its physical rank before quotienting by stabilizers and its logical rank after quotienting. We develop a quantum version of the LCL framework for nested spaces $S \subseteq C$, in which local constraints are imposed on physical representatives while independence is measured in the logical quotient $C/S$. The resulting theory gives a threshold theorem for random CSS codes, and as a consequence shows that the per-sector rate threshold is equal to the classical rate threshold. We also define a quantum analogue of subspace design [Guruswami and Xing, STOC 2013] and show that they can be described in a natural manner within the quantum-LCL framework. Finally, we give explicit constructions for arbitrary folded quantum-LCL properties, in a manner similar to the LCL derandomization of [Jeronimo and Shagrithaya, STOC 2026]. As a consequence, we obtain the first explicit constructions of quantum list-decodable codes and list-recoverable codes that have optimal list sizes, in addition to explicit quantum subspace design codes. We note that all our explicit constructions are qLDPC codes, an important property for quantum error-correcting codes.

发表机构

• University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）

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