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Title: Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds

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{sup ε}. Their rate is evaluated via Euler characteristic arguments and their distance using Z{sub 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.
Authors:
 [1] ;  [2]
  1. Department of Mathematics, MIT, Cambridge, Massachusetts 02139 (United States)
  2. Institute of Mathematics, Hebrew University, Jerusalem 91904 (Israel)
Publication Date:
OSTI Identifier:
22306087
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; COMPUTER CODES; FOUR-DIMENSIONAL CALCULATIONS; GEOMETRY; GRAPH THEORY; MATHEMATICAL MANIFOLDS; PARITY