Geometrically Local Quantum and Classical Codes from Subdivision

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? This question was recently answered by Portnoy which constructed codes with optimal dimension and distance up to polylogs. This paper extends Portnoy's work by constructing a code which additionally has an optimal energy barrier up to polylogs. The key ingredient is a simpler code construction obtained by subdividing the balanced product codes. We also discuss applications to classical codes.