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 Z. Applying this procedure to chain complexes obtained by "lifting" recently developed quantum codes, which correspond to chain complexes over Z2, we construct the first examples of power law Z2 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.
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