UC San Diego

The local-to-global principle—the idea that global structure is determined by, or can be certified from, local constraints—appears across mathematics, theoretical computer science, and physics under different disguises. This thesis studies two manifestations of this principle: in quantum error-correcting codes, where local parity checks certify the integrity of the global encoded information, and in the entanglement bootstrap, where local reduced density matrices determine the global structure of quantum phases of matter.Part I constructs quantum low-density parity check (qLDPC) codes. We first build a family of quantum locally testable codes (qLTCs) achieving linear dimension, near-linear distance, and near-constant soundness, by generalizing square-complex qLDPC codes to higher-dimensional cubical chain complexes. We then extend the cubical construction to the sheaf cochain complexes, on which we define Poincaré duality and a cup product. Poincaré duality clarifies the duality between the X- and Z-distances of the corresponding quantum code, while the cup product yields three sheaf codes that jointly support a transversal CCZ gate, providing a general route to non-Clifford gates on qLDPC codes.Part II develops an entanglement-based characterization of conformal field theories (CFTs). We first show that the ground state of any unitary 1+1D CFT is a critical point of a particular linear combination of subsystem entropies, and we verify numerically that this condition is stringent yet robust across known critical lattice models. We then leverage this characterization to perform a computational search over four-qubit and four-qutrit systems, which recovers known CFTs and identifies roughly 20 new candidate critical points, potentially including new irrational CFTs.