TY - JOUR
T1 - Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds
AU - Guth, Larry
AU - Lubotzky, Alexander
N1 - Publisher Copyright:
© 2014 AIP Publishing LLC.
PY - 2014/8/6
Y1 - 2014/8/6
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84925780718&partnerID=8YFLogxK
U2 - 10.1063/1.4891487
DO - 10.1063/1.4891487
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AN - SCOPUS:84925780718
SN - 0022-2488
VL - 55
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
IS - 8
M1 - 082202
ER -