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Introduction to Categorical Aspects of Topological Quantum Computation

Ahmadi, Fatimah | 2015

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 46875 (01)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Karimipour, Vahid
  7. Abstract:
  8. One of the problems facing quantum computation is errors due to interaction with the environment which destroy coherence of quantum states. Most schemes to design a quantum computer therefore focus on finding ways to minimize the interactions of the qubits with the environment. Constructing such systems with large numbers of qubits which are infallible is a hard task and far from being achieved in the near future. There is another quantum computational model which is called topological quantum computation, proposing a different solution. Qubits of this model are quasiparticles of a 2-dimensional topologically ordered system that are called Anyons. In this model gates are nonabelian representations of the braid group. Since topological properties are invariant under local perturbations, topological quantum computation is a fault-tolerant model. Unlike local order, topological order not only cannot be described by Landau theory, but also no comprehensive theory has been proposed to describe completely its features. Since the lowenergy effective theory of topologically ordered systems is a topological quantum field theory and considering the close relationship between topological quantum field theory and category theory, it seems the mathematical framework describing topological order is category theory. Category theory gives us a unified picture and helps us to understand unfamiliar features of topological order. In this thesis we first introduce topological quantum computation and then we try to clarify its relation with category theory
  9. Keywords:
  10. Quantum Computation ; Anyon ; Topological Order ; Category Theory ; MOnoidal Category ; Topological Quantum Computation ; Fusion Space ; Fibonacci Anyon ; Braid Group

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