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The Need for Structure in Quantum LDPC Codes

机译：量子LDPC码对结构的需求

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摘要

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.
机译：量子距离中线性距离最小的量子LDPC码的存在是量子信息中的一个中心问题。尽管进行了多年研究，但尚不存在良好的量子LDPC码，但至少可以知道，它们无法在非常规则的拓扑（例如低维网格）上定义。在这项工作中，我们建立了一个互补的结果，表明稀疏生成的良好量子CSS代码需要以本地术语的“结构”来约束代码空间，以免在定义明确的意义上“过于随机”。为了证明这一点，我们证明了与经典代码的权重分布上的Krasikov和Litsyn定理的弱相反，这可能是由于它们具有独立的利益：对偶空间中权重分布近似为二项式的子空间几乎没有的码字。重量轻，相当于具有不可忽略的“近似”最小距离。尽管它们可能没有很大的最小非零权重，但是仍然只有很少的汉明权重低的单词。

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