Constructions and Noise Threshold of Hyperbolic Surface Codes

TL;DR

This paper presents new hyperbolic surface codes with improved encoding rates and analyzes their noise thresholds, offering a promising alternative to traditional surface codes for quantum error correction.

Contribution

It provides concrete constructions of hyperbolic surface codes and estimates their noise thresholds, enhancing quantum error correction methods.

Findings

Hyperbolic surface codes have better rate vs. protection tradeoff.

Numerical estimates of noise thresholds are provided.

Codes require variable length connections based on hyperbolic geometry.

Abstract

We show how to obtain concrete constructions of homological quantum codes based on tilings of 2D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in two-dimensional quantum codes whose tradeoff of encoding rate versus protection is more favorable than for the surface code. These surface codes would require variable length connections between qubits, as determined by the hyperbolic geometry. We provide numerical estimates of the value of the noise threshold and logical error probability of these codes against independent X or Z noise, assuming noise-free error correction.