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 log1/2 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.
|Original language||English (US)|
|Title of host publication||Mathematics of Quantum Computation|
|Number of pages||34|
|ISBN (Print)||1584882824, 9781584882824|
|State||Published - Jan 1 2002|
All Science Journal Classification (ASJC) codes
- Computer Science(all)