Building manifolds from quantum codes

• URL: http://arxiv.org/abs/2012.02249v3
• Date: Mon, 24 May 2021 23:47:23 GMT
• Title: Building manifolds from quantum codes
• Authors: Michael Freedman and Matthew B. Hastings
• Abstract summary: We construct the first examples of power law $mathbbZ$ systolic freedom. We give an efficient randomized algorithm to construct a weakly fundamental cycle basis for a graph. We use this result to trivialize the fundamental group of the manifold we construct.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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