Geometrically Local Quantum and Classical Codes from Subdivision

TL;DR

This paper presents a streamlined construction of geometrically local quantum and classical codes that achieve optimal dimension, distance, and energy barrier, improving upon previous methods by leveraging the inherent structure of balanced product codes.

Contribution

It simplifies the construction of optimal geometrically local quantum codes by utilizing the natural two-dimensional structure of balanced product codes, bypassing complex mathematical lifting procedures.

Findings

Achieves optimal dimension and distance in all dimensions.

Codes have an optimal energy barrier.

Results extend to classical codes with similar properties.

Abstract

A geometrically local quantum code is an error correcting code situated within $\mathbb{R}^D$, 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.