Z₂-systolic freedom and quantum codes

A closely coupled pair of conjectures/questions—one in differential geometry (by M. Gromov), the other in quantum information theory—are both answered in the negative. The answer derives from a certain metrical flexibility of manifolds and a corresponding improvement to the theoretical efficiency of existing local quantum codes. We exhibit this effect by constructing a family of metrics on S² × S¹, and other three and four dimensional manifolds. Quantitatively, the explicit “freedom” exhibited is too weak (a log^1/2 factor in the natural scaling) to yield practical codes but we cannot rule out the possibility of other families of geometries with more dramatic freedom.