Geometrically Local Quantum and Classical Codes from Subdivision

Ting-Chun Lin

Adam Wills

Min-Hsiu Hsieh

arXiv preprint, 2023. DOI: 10.48550/arXiv.2309.16104

Date of Publication

September 27, 2023

Abstract

A geometrically local quantum code is an error correcting code situated within ℝD, where the checks only act on qubits within a fixed spatial distance. The main question is: What is the optimal dimension and distance for a geometrically local code? Recently, Portnoy made a significant breakthrough with codes achieving optimal dimension and distance up to polylogs. However, the construction invokes a somewhat advanced mathematical result that involves lifting a chain complex to a manifold. This paper bypasses this step and streamlines the construction by noticing that a family of good quantum low-density parity-check codes, balanced product codes, naturally carries a two-dimensional structure. Together with a new embedding result that will be shown elsewhere, this quantum code achieves the optimal dimension and distance in all dimensions. In addition, we show that the code has an optimal energy barrier. We also discuss similar results for classical codes.

Centers

Quantum Computing Research Center

Table of Contents