Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds

TL;DR

This paper introduces a new family of quantum LDPC codes derived from regular tessellations of hyperbolic 4-manifolds, achieving non-zero rate and polynomial minimum distance, with an efficient decoding algorithm exploiting the regular structure.

Contribution

It adapts a hyperbolic tessellation-based construction to develop quantum LDPC codes from hypercubes, enhancing decoding efficiency and code parameters.

Findings

Codes have non-vanishing rate and distance scales as n^{0.1}

Regular tessellation of hypercubes enables efficient decoding

The construction extends previous hyperbolic code frameworks

Abstract

We adapt a construction of Guth and Lubotzky [arXiv:1310.5555] to obtain a family of quantum LDPC codes with non-vanishing rate and minimum distance scaling like $n^{0.1}$ where $n$ is the number of physical qubits. Similarly as in [arXiv:1310.5555], our homological code family stems from hyperbolic 4-manifolds equipped with tessellations. The main novelty of this work is that we consider a regular tessellation consisting of hypercubes. We exploit this strong local structure to design and analyze an efficient decoding algorithm.