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 and Terhal (2009 New J. Phys. 11 043029) (BT) and Bravyi et al (2010 Phys. Rev. Lett. 104 050503) (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 kd2/(D−1)⩽O(n). Baspin and Krishna (2022 Quantum 6 711) 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.

中文翻译：

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改进了连接受限的量子代码的速率-距离权衡


要实现量子纠错码，受代码约束约束的量子比特必须表现出某种形式的有限连接。Bravyi 和 Terhal （2009 New J. Phys.11 043029） （BT） 以及 Bravyi 等人 （2010 Phys. Rev. Lett.104 050503） （BPT） 的工作确定了几何位置性限制代码属性——例如，由 D 维晶格上的局部检查定义的 [[n，k，d]] 量子代码必须服从 kd2/（D−1）⩽O（n）。Baspin 和 Krishna （2022 Quantum6， 711） 研究了与量子代码相关的连通性图如何约束代码参数的更普遍的问题。与 BPT 和 BT 边界相比，这些权衡适用于更丰富的代码类别，后者仅捕获几何本地代码。我们扩展并改进了这项工作，根据连通性图中 separator 的大小建立了更严格的尺寸-距离权衡。我们还获得了一个距离边界，该距离边界覆盖了具有特定分离曲线的所有稳定器代码，而不仅仅是 LDPC 代码。