Building manifolds from quantum codes

Abstract

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.