Building manifolds from quantum codes

  title={Building manifolds from quantum codes},
  author={Michael H. Freedman and Matthew B. Hastings},
  journal={Geometric and Functional Analysis},
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… 
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