Building manifolds from quantum codes

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 comple...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 31; no. 4; pp. 855 - 894
Main Authors Freedman, Michael, Hastings, Matthew
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.08.2021
Springer Nature B.V
Subjects
Algorithms
Analysis
Chains
Graph theory
Mathematics
Mathematics and Statistics
Reverse engineering
Riemann manifold
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Summary: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.
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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-021-00567-3