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

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arxiv 1310.5555 v1 pith:QFJI3QA4 submitted 2013-10-21 math.DG

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

classification math.DG
keywords codesarithmeticdimensionaldistancehomologicalhyperbolicmanifoldsquantum
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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 Z\'emor, who asked whether homological codes with such parameters could exist at all.

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