Building manifolds from quantum codes

TL;DR

This paper introduces a method to construct Riemannian manifolds from quantum codes, demonstrating new systolic freedom phenomena and providing algorithms for graph basis construction that impact manifold topology.

Contribution

It presents a novel procedure to build manifolds from quantum code chain complexes and introduces an efficient algorithm for graph basis construction.

Findings

Constructed manifolds exhibiting power law $ ext{Z}_2$ systolic freedom.

Developed an efficient randomized algorithm for graph cycle basis.

Trivialized the fundamental group of the constructed manifolds.

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.