Anyons And Topological Quantum Computation

Topological quantum computation is a novel interdisciplinary field devel-oped in the past decades, involving quantum computation, topology, topological quantum field theory, and condensed matter physics. In topological quantum computing, we store and control quantum information using topological states in quantum many-body systems. Topological quantum computation is intrinsically fault-tolerant, which brings hope to the physical realization of quantum compu-tation, and pushes us to explore the topological quantum properties of condensed matters.In a strongly correlated system with topological orders, there might be a class of exotic particles called anyons. Different from the Bosons and fermions in three-dimensional space, anyons obey anyonic statistics either Abelian or non-Abelian. Non-Abelian anyons can be used to encode quantum bits, i.e., qubits, and store quantum information topologically. The world-lines of non-Abelian anyons in-tertwining in (2+1)-dimensional space-time form braids, which can be used to construct universal topological quantum gates, in order to carry out arbitrary topological operations. Due to the discreteness of braid topology, topological quantum computation is intrinsically fault-tolerant, and local disturbances do not affect the storage and process of topological quantum information. However, it is not easy at all to effectively construct universal quantum gates from the huge space of braids. We can decompose two-qubit gates into single-qubit gates and thus make it possible to construct two-qubit gates. The discreteness of the braid topology suggests that we introduce geometric errors into redundant degrees of freedom and thus we obtain low-leakage topological quantum computation. We further decompose the degrees of freedom of single-qubit gates, and using the idea of error reduction by error introduction, we obtain highly accurate single-qubit gates. We also exploit the degrees of freedom in topological quantum comput-ing algorithms. Besides, we propose a renormalization-group-like algorithm for quantum compiling, and thus theoretically obtain arbitrarily accurate topological quantum computation. The discussions in this thesis can also be generalized and applied to fault-tolerant quantum computation in general quantum computation and mathematical fields involving universal construction of matrices.