Explore automorphism gadgets within homological product codes through this 35-minute conference talk delivered at the Kavli Institute for Theoretical Physics. Delve into advanced quantum error correction concepts as part of a broader conference examining quantum many-body systems through the framework of quantum error correction and fault-tolerance. Learn about novel theoretical constructions in quantum coding theory that challenge conventional understanding of quantum phases of matter, with particular emphasis on non-local geometries and structured noise models. Discover how these developments create new opportunities in quantum many-body physics for describing properties and evolution of encoded quantum matter, with relevance to near-term quantum computing experiments and scalable quantum computation implementations.