| Description |
The subject of this dissertation is the presentation of a surface code, the string-net surface code, based on the exactly-solvable Levin-Wen model for doubled topological phases. We construct the circuits needed to encode quantum information in the many-body states of a two-dimensional network of qudits, as well as circuits to measure and manipulate the encoded states. This framework serves as both a quantum error-correcting code and a quantum simulator of Abelian doubled topological phases. What distinguishes the string-net surface code from present surface code prototypes is the feature of topological symmetry in the Levin-Wen model. We will use a discrete formulation of the topological symmetry to construct quantum circuits that realize these transformations. This enables encoded quantum gates to be, in principle, achieved solely in terms of quantum circuits, contrasting with the current methods utilizing code deformation and lattice surgery. We describe the encoding of quantum information using gapped boundaries and demonstrate how to perform gates from the generalized Clifford group in a topologically protected manner, including the use of defect lines. Our proposal suggests that from the quantum information perspective, the fusion algebra is the proper generalization of the Pauli algebra. |
