Building manifolds from quantum codes - Freedman, Michael; Hastings, Matthew (Geometric and functional analysis, 2021)
We give a procedure for “reverse engineering" a closed, simply connected, Riemannian manifold with bounded local geometry from a sparse chain complex over Z . Applying this procedure to chain complexes obtained by “lifting" recently developed quantum codes, which correspond to chain complexes over Z 2 , we construct the first examples of power law 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.