Celebratio Mathematica — Freedman

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incollection M. H. Freedman, D. A. Meyer, and F. Luo: “\( Z_2 \)-systolic freedom and quantum codes,” pp. 287–320 in Mathematics of quantum computation. Edited by R. K. Brylinski and G. Chen. Computational Mathematics 3. Chapman & Hall/CRC (Boca Raton, FL), 2002. MR 2007952 Zbl 1075.81508

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 \( \mathbb{S}^2\times\mathbb{S}^1 \), 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.

Goong Chen	Related
Feng Luo	Related
David A. Meyer	Related
Ranee K. Brylinski	Related