The Physics of (good) LDPC Codes I. Gauging and dualities

Understand

Low-depth parity check (LDPC) codes are a paradigm of error correction that allow for spatially non-local interactions between (qu)bits, while still enforcing that each (qu)bit interacts only with finitely many others.

• On expander graphs, they can give rise to ``good codes'' that combine a finite encoding rate with an optimal scaling of the code distance, which governs the code's robustness against noise.
• Such codes have garnered much recent attention due to two breakthrough developments: the construction of good quantum LDPC codes and good locally testable classical LDPC codes, using similar methods.
• Here we explore these developments from a physics lens, establishing connections between LDPC codes and ordered phases of matter defined for systems with non-local interactions and on non-Euclidean geometries.