For quantum error-correcting codes to be realizable, it is important that the qubits subject to the code constraints exhibit some form of limited connectivity. The works of Bravyi & Terhal (BT) and Bravyi, Poulin & Terhal (BPT) established that geometric locality constrains code properties -- for instance $[[n,k,d]]$ quantum codes defined by local checks on the $D$-dimensional lattice must obey $k d^{2/(D-1)} \le O(n)$. Baspin and Krishna studied the more general question of how the connectivity graph associated with a quantum code constrains the code parameters. These trade-offs apply to a richer class of codes compared to the BPT and BT bounds, which only capture geometrically-local codes. We extend and improve this work, establishing a tighter dimension-distance trade-off as a function of the size of separators in the connectivity graph. We also obtain a distance bound that covers all stabilizer codes with a particular separation profile, rather than only LDPC codes.

翻译：为使量子纠错码可实现，受码约束的量子比特需具有某种形式的有限连通性，这一点至关重要。Bravyi和Terhal（BT）以及Bravyi、Poulin和Terhal（BPT）的研究表明，几何局域性约束了码的性质——例如，由D维晶格上的局域校验定义的[[n,k,d]]量子码必须满足k d^{2/(D-1)} ≤ O(n)。Baspin和Krishna研究了更一般的问题：量子码相关的连通图如何约束码参数。与仅适用于几何局域码的BPT和BT界相比，这些权衡适用于更丰富的码类。我们扩展并改进了这项工作，建立了基于连通图中分隔符大小的更紧的维度-距离权衡。我们还得到了一个距离界，该界涵盖了具有特定分离轮廓的所有稳定子码，而不仅仅是LDPC码。