TOPOLOGICAL PHASE OF MATTER AND FLOQUET CODES

Topological phase of matter is a special quantum phase of gapped Hamiltonian that is beyond the Landau's symmetry breaking paradigm. Topological phase of matter exhibit extraordinary topology-depedent properties and provide significant resources for quantum computation, as it can support anyons as lowest excitations, which contribute to fault-tolerant topological quantum computation, while its ground state are naturally quantum error correcting codes.

This thesis focused on studying topological phase of matter and its potential contribution to quantum computation. The author first works on the Ribbon operators in the Kitaev Quantum Double model with semisimple Hopf algebra $H$, which captures the anyonic excitations of $\mathbf{D}(H)$, Second, twist defects are studied in the Kitaev spin liquid context and shows the potential contribution to quantum computations by manipulating defects. Third, two classes of topological floquet code are introduced, to overcome the high cost of many-body syndrome operators and also gives new construction of topological orders.