论文标题
从量子代码建造歧管
Building manifolds from quantum codes
论文作者
论文摘要
我们给出了一个“逆向工程”的过程，该过程是封闭的，简单地连接的，riemannian的歧管，其本地几何有限制的局部几何形状，从$ \ mathbb {z} $上方的稀疏链复合物。将此过程应用于通过“提升”最近开发的量子代码获得的链络合物，该量子代码对应于$ \ Mathbb {Z} _2 $上的链络合物，我们构建了Power Law $ \ Mathbb {z} _2 _2 $ Systolot Freedom的第一个示例。 结果，在图理论中可能具有独立感兴趣的结果，我们给出了一种有效的随机算法来为图构建一个弱基本周期的基础，以便每个边缘仅在基础上以各种聚类为单位。我们使用此结果来使我们构建的流形的基本组变得微不足道。
We give a procedure for "reverse engineering" a closed, simply connected, Riemannian manifold with bounded local geometry from a sparse chain complex over $\mathbb{Z}$. Applying this procedure to chain complexes obtained by "lifting" recently developed quantum codes, which correspond to chain complexes over $\mathbb{Z}_2$, we construct the first examples of power law $\mathbb{Z}_2$ systolic freedom. As a result that may be of independent interest in graph theory, we give an efficient randomized algorithm to construct a weakly fundamental cycle basis for a graph, such that each edge appears only polylogarithmically times in the basis. We use this result to trivialize the fundamental group of the manifold we construct.
