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Topological Quantum Computation and the Stability of Topological Memories

Mohseninia, Razieh | 2016

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 48598 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Karimipour, Vahid
  7. Abstract:
  8. antum computers are necessary to simulate quantum systems. e fragility of qubits in presence of decoherence and external noise is the biggest obstacle in realizing a quantum computer. To overcome su problems, topological quantum computation has been introduced by Kitaev that combines the main quantum feature of the quantum world, namely, superposition of states, with the robustness of classical bits whi is the result of a macroscopic number of very small entities, comprising ea bit. In this way, topological features whi are robust against local perturbations are used for storing information. Kitaev’s model is a famous many-body model, whi has topologically degenerate ground states, robust against local perturbations. As is well-known Z3 Kitaev model can be used to perform universal quantum computation. Due to unwanted interactions in the system, for e xample interactions with electrostatic origin, a perturbation may be added to the Hamiltonian of the system. In this thesis, stability of the Z3 Kitaev model in the presence of external perturbation in the form of Pos interaction is studied. We show that the low-energy sector of the Kitaev-Pos model is mapped to the Pos model in the presence of transverse magnetic field. Our study relies on two high-order series expansions based on continuous unitary transformations in the limits of small and large Pos couplings as well as mean-field approximation. Our analysis reveals that the topological phase of the Z3 Kitaev model breaks down to the Pos model through a first-order phase transition. We capture the phase transition by analysis of the ground-state energy, one-quasiparticle gap, and geometric measure of entanglement.Another example of topological memories is the Topological Color Code. In 2 dimensions, one can implement the Clifford group, whi is enough for doing quantum distillation of entanglement, in a fully topological manner, using color code, without any need to address single bits. In this thesis, thermal stability of this model in presence of a thermal bath is studied. We study the Lindblad evolution of the observables in the weak coupling limit of the Born-Markov approximation. e auto-correlation functions of the observables are used as a figure of merit for the thermal stability. We show that all of the observables auto-correlation functions decay exponentially in time. By finding a lower bound on the decay rate, whi is a constant independent of the system size, we show that the Topological Color Code is unstable against thermal fluctuations from the bath at finite temperature, even though it is stable at T = 0 against local quantum perturbations. e full spectrum of a 1 dimensional non-abelian Kitaev model is studied in the last apter of this thesis, as a first step of finding the full spectrum of non-abelian Kitaev model in 2 dimensions. We show that the 1 dimensional model is equivalent to a generalization of the classical one-dimensional Pos model, where the symmetry groupis a non-Abelian finite group. It turns out that this model has a quantum nature in that its spectrum of energy eigenstates consists of entangled states. We determine the complete energy spectrum, i.e., the ground states and all the excited states with their degeneracy structure. We calculate the partition function by two different algebraic and combinatorial methods. We also determine the entanglement properties of its ground states
  9. Keywords:
  10. Topological Quantum Computation ; Kitaev Model ; Color Code ; Non-Abelian Potts Model

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