The existence of quantum LDPC codes with minimal distance scaling linearly in the number of qubits is a central open problem in quantum information. Despite years of research good quantum LDPC codes are not known to exist, but at the very least it is known they cannot be defined on very regular topologies, like low-dimensional grids. In this work we establish a complementary result, showing that good quantum CSS codes which are sparsely generated require “structure” in the local terms that constrain the code space so as not to be “too-random” in a well-defined sense. To show this, we prove a weak converse to a theorem of Krasikov and Litsyn on weight distributions of classical codes due to which may be of independent interest: subspaces for which the distribution of weights in the dual space is approximately binomial have very few codewords of low weight, tantamount to having a non-negligible “approximate” minimal distance. While they may not have a large minimal non-zero weight, they still have very few words of low Hamming weight.

中文翻译：

---

对量子 LDPC 码结构的需求

在量子比特数中具有最小距离线性缩放的量子 LDPC 码的存在是量子信息中的一个核心开放问题。尽管经过多年的研究，尚不知道是否存在良好的量子 LDPC 码，但至少已知它们不能在非常规则的拓扑结构（如低维网格）上定义。在这项工作中，我们建立了一个补充结果，表明稀疏生成的良好量子 CSS 代码需要在局部术语中具有“结构”，以限制代码空间，以免在明确定义的意义上“过于随机”。为了证明这一点，我们证明了 Krasikov 和 Litsyn 关于经典代码权重分布的定理的弱逆逆，因此可能具有独立意义：对偶空间中权重分布近似二项式的子空间具有很少的低权重码字，相当于具有不可忽略的“近似”最小距离。虽然它们可能没有很大的最小非零权重，但它们仍然很少有低汉明权重的词。