Poincar'e Duality and Multiplicative Structures on Quantum Codes
This research develops sheaf codes as a unified framework for quantum LDPC codes by generalizing Poincaré duality to cell complexes. The work establishes rigorous dualities between different code aspects using cap and cup products, leading to explicit isomorphisms between cohomology groups. This mathematical foundation is then used to construct practical, transversal quantum gates like logical CZ gates.