We present new constructions of quantum codes of linear or close-to-linear distance and dimension with low-weight stabilizers. Only a few constructions of such codes were previously known, and were primarily based on a specific operation from homological algebra, namely the balanced product. In contrast, our constructions are based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes). Our results help address the natural question: When do homological products preserve good code distance? Our first main result constructs asymptotically good $[[N,\Theta(N),\Theta(N)]]$ quantum codes with small polynomial stabilizer weight from homological products of codes with a property called product-expansion. This notion was recently introduced and used to bound the distance of balanced product quantum codes; we apply it instead to homological products. For every $\epsilon>0$, our second main result constructs close-to-linear distance $[[N,N^{1-\epsilon},N^{1-\epsilon}]]$ (subsystem) quantum LDPC codes with constant stabilizer weight from iterated homological products of a constant-sized quantum locally testable code. The key insight here is that by using subsystem codes (but still with constant-weight stabilizers), we can circumvent a particular obstruction that limited the distance of many prior product code constructions to at most $\tilde{O}(\sqrt{N})$.

翻译：本文提出了具有线性或近似线性距离与维度、且含低权重稳定子的量子码新构造方法。此前仅有少数此类构造为人所知，且主要基于同调代数中的特定运算——即平衡积。相比之下，我们的构造基于更基础且广泛使用的积运算——同调积（即链复形的张量积）。我们的研究有助于回答一个自然问题：同调积何时能保持优良的码距？我们的第一个主要结果通过具有乘积扩张性质的码进行同调积，构造出渐近优良的$[[N,\Theta(N),\Theta(N)]]$量子码，其稳定子权重为小多项式量级。乘积扩张概念近期被提出并用于界定平衡积量子码的距离，我们将其应用于同调积场景。对于任意$\epsilon>0$，我们的第二个主要结果通过对恒定规模的量子局部可测试码进行迭代同调积，构造出具有恒定稳定子权重的近似线性距离$[[N,N^{1-\epsilon},N^{1-\epsilon}]]$（子系统）量子LDPC码。此处的关键洞见在于：通过采用子系统码（仍保持恒定权重稳定子），我们能够规避先前许多乘积码构造中距离上限被限制在$\tilde{O}(\sqrt{N})$的内在障碍。