Hyperbolic and semi-hyperbolic surface codes for quantum storage

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 with $2.9 \% $ for the toric code). In this code family, parity checks are of weight 4 and 5, while each qubit participates in four different parity checks. We introduce a family of semi-hyperbolic codes that 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