  We show how a hyperbolic surface code could be used for overhead-efficient
quantum storage. We give numerical evidence for a noise threshold of 1.3% for
the {4,5}-hyperbolic surface code in a phenomenological noise model (as
compared to 2.9% for the toric code). In this code family parity checks are of
weight 4 and 5 while each qubit participates in 4 different parity checks. We
introduce a family of semi-hyperbolic codes which interpolate between the toric
code and the {4,5}-hyperbolic surface code in terms of encoding rate and
threshold. We show how these hyperbolic codes outperform the toric code in
terms of qubit overhead for a target logical error probability. We show how
Dehn twists and lattice code surgery can be used to read and write individual
qubits to this quantum storage medium.
