# Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage

## TL;DR

This paper explores hyperbolic surface codes for quantum storage, demonstrating their efficiency and thresholds, and introduces semi-hyperbolic codes that interpolate between toric and hyperbolic codes, with practical methods for qubit manipulation.

## Contribution

It introduces semi-hyperbolic codes that balance encoding rate and threshold, and demonstrates their advantages over toric codes in quantum storage overhead.

## Key findings

- Hyperbolic surface codes have a noise threshold of 1.3%.
- Semi-hyperbolic codes interpolate between toric and hyperbolic codes.
- Hyperbolic codes outperform toric codes in qubit overhead for target error rates.

## Abstract

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.