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

Abstract

Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are LDPC codes with linear rate and distance $n^{\epsilon }$. Their rate is evaluated via Euler characteristic arguments and their distance using ${\mathbb{Z}}_2$-systolic geometry. This construction answers a queston of Zémor, who asked whether homological codes with such parameters could exist at all.

21 pages

References in corpus (1)

• Normal Subgroup Growth of Linear Groups: the (G2; F4;E8)-Theorem

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