Learn about circuit complexity of code states in this 56-minute lecture from Microsoft Research's Jeongwan Haah, delivered as part of the IAS Park City Mathematics Institute's 2023 Quantum Computation program. Explore the surface code's significance in quantum computing, both for practical applications and its connection to topological phases of matter. Examine key concepts including the limitations of local codes in one dimension, the relationship between code distance and linear system size in two-dimensional local codes, logical qubit constraints in d-dimensional local codes, and the equivalence of two-dimensional translation invariant codes to surface codes. Delve into triorthogonal codes and their applications in magic state distillation as part of a comprehensive exploration of quantum error-correcting codes and their topological aspects.