Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds
Author(s)Lubotzky, Alexander; Guth, Lawrence
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Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are low density parity check codes with linear rate and distance n [superscript ε]. Their rate is evaluated via Euler characteristic arguments and their distance using Z[subscript 2] -systolic geometry. This construction answers a question of Zémor [“On Cayley graphs, surface codes, and the limits of homological coding for quantum error correction,” in Proceedings of Second International Workshop on Coding and Cryptology (IWCC), (Lecture Notes in Computer Science). Vol. 5557 (2009), pp. 259–273], who asked whether homological codes with such parameters could exist at all.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal of Mathematical Physics
American Institute of Physics (AIP)
Guth, Larry, and Alexander Lubotzky. “Quantum Error Correcting Codes and 4-Dimensional Arithmetic Hyperbolic Manifolds.” Journal of Mathematical Physics 55, no. 8 (August 2014): 082202.