Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds

Larry Guth, Alexander Lubotzky

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57 Scopus citations

Abstract

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 nε. Their rate is evaluated via Euler characteristic arguments and their distance using Z2-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.

Original languageEnglish
Article number082202
JournalJournal of Mathematical Physics
Volume55
Issue number8
DOIs
StatePublished - 6 Aug 2014

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