Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds

Journal Article · Fri Aug 15 00:00:00 EDT 2014 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4891487· OSTI ID:22306087

Guth, Larry ^[1]; Lubotzky, Alexander ^[2]

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1. Department of Mathematics, MIT, Cambridge, Massachusetts 02139 (United States)
2. Institute of Mathematics, Hebrew University, Jerusalem 91904 (Israel)

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.

• OSTI ID: 22306087
• Journal Information: Journal of Mathematical Physics, Vol. 55, Issue 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
97 MATHEMATICAL METHODS AND COMPUTING
COMPUTER CODES
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MATHEMATICAL MANIFOLDS
PARITY