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THEORETICAL AND NUMERICAL STUDIES OF PHASE TRANSITIONS AND ERROR THRESHOLDS IN TOPOLOGICAL QUANTUM MEMORIES
 Date Issued:
 2014
 Abstract/Description:
 This dissertation is the collection of a progressive research on the topic of topological quantum computation and information with the focus on the error threshold of the wellknown models such as the unpaired Majorana, the toric code, and the planar code.We study the basics of quantum computation and quantum information, and in particular quantum error correction. Quantum error correction provides a tool for enhancing the quantum computation fidelity in the noisy environment of a real world. We begin with a brief introduction to stabilizer codes. The stabilizer formalism of the theory of quantum error correction gives a welldefined description of quantum codes that is used throughout this dissertation. Then, we turn our attention to a quite new subject, namely, topological quantum codes. Topological quantum codes take advantage of the topological characteristics of a physical manybody system. The physical manybody systems studied in the context of topological quantum codes are of two essential natures: they either have intrinsic interaction that selfcorrects errors, or are actively corrected to be maintainedin a desired quantum state. Examples of the former are the toric code and the unpaired Majorana, while an example for the latter is the surface code.A brief introduction and history of topological phenomena in condensed matter is provided. The unpaired Majorana and the Kitaev toy model are briefly explained. Later we introduce a spin model that maps onto the Kitaev toy model through a sequence of transformations. We show how this model is robust and tolerates local perturbations. The research on this topic, at the time of writing this dissertation, is still incomplete and only preliminary results are represented.As another example of passive error correcting codes with intrinsic Hamiltonian, the toric code is introduced. We also analyze the dynamics of the errors in the toric code known as anyons. We show numerically how the addition of disorder to the physical system underlying the toric code slows down the dynamics of the anyons. We go further and numerically analyze the presence of timedependent noise and the consequent delocalization of localized errors.The main portion of this dissertation is dedicated to the surface code. We study the surface code coupled to a noninteracting bosonic bath. We show how the interaction between the code and the bosonic bath can effectively induce correlated errors. These correlated errors may be corrected up to some extend. The extension beyond which quantum error correction seems impossible is the error threshold of the code. This threshold is analyzed by mapping the effective correlated error model onto a statistical model. We then study the phase transition in the statistical model. The analysis is in two parts. First, we carry out derivation of the effective correlated model, its mapping onto a statistical model, and perform an exact numerical analysis. Second, we employ a Monte Carlo method to extend the numerical analysis to large system size.We also tackle the problem of surface code with correlated and singlequbit errors by an exact mapping onto a twodimensional Ising model with boundary fields. We show how the phase transition point in one model, the Ising model, coincides with the intrinsic error threshold of the other model, the surface code.
Title:  THEORETICAL AND NUMERICAL STUDIES OF PHASE TRANSITIONS AND ERROR THRESHOLDS IN TOPOLOGICAL QUANTUM MEMORIES. 
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Name(s): 
Jouzdani, Pejman, Author Mucciolo, Eduardo, Committee Chair Chang, Zenghu, Committee Member Leuenberger, Michael, Committee Member Abouraddy, Ayman, Committee Member University of Central Florida, Degree Grantor 

Type of Resource:  text  
Date Issued:  2014  
Publisher:  University of Central Florida  
Language(s):  English  
Abstract/Description:  This dissertation is the collection of a progressive research on the topic of topological quantum computation and information with the focus on the error threshold of the wellknown models such as the unpaired Majorana, the toric code, and the planar code.We study the basics of quantum computation and quantum information, and in particular quantum error correction. Quantum error correction provides a tool for enhancing the quantum computation fidelity in the noisy environment of a real world. We begin with a brief introduction to stabilizer codes. The stabilizer formalism of the theory of quantum error correction gives a welldefined description of quantum codes that is used throughout this dissertation. Then, we turn our attention to a quite new subject, namely, topological quantum codes. Topological quantum codes take advantage of the topological characteristics of a physical manybody system. The physical manybody systems studied in the context of topological quantum codes are of two essential natures: they either have intrinsic interaction that selfcorrects errors, or are actively corrected to be maintainedin a desired quantum state. Examples of the former are the toric code and the unpaired Majorana, while an example for the latter is the surface code.A brief introduction and history of topological phenomena in condensed matter is provided. The unpaired Majorana and the Kitaev toy model are briefly explained. Later we introduce a spin model that maps onto the Kitaev toy model through a sequence of transformations. We show how this model is robust and tolerates local perturbations. The research on this topic, at the time of writing this dissertation, is still incomplete and only preliminary results are represented.As another example of passive error correcting codes with intrinsic Hamiltonian, the toric code is introduced. We also analyze the dynamics of the errors in the toric code known as anyons. We show numerically how the addition of disorder to the physical system underlying the toric code slows down the dynamics of the anyons. We go further and numerically analyze the presence of timedependent noise and the consequent delocalization of localized errors.The main portion of this dissertation is dedicated to the surface code. We study the surface code coupled to a noninteracting bosonic bath. We show how the interaction between the code and the bosonic bath can effectively induce correlated errors. These correlated errors may be corrected up to some extend. The extension beyond which quantum error correction seems impossible is the error threshold of the code. This threshold is analyzed by mapping the effective correlated error model onto a statistical model. We then study the phase transition in the statistical model. The analysis is in two parts. First, we carry out derivation of the effective correlated model, its mapping onto a statistical model, and perform an exact numerical analysis. Second, we employ a Monte Carlo method to extend the numerical analysis to large system size.We also tackle the problem of surface code with correlated and singlequbit errors by an exact mapping onto a twodimensional Ising model with boundary fields. We show how the phase transition point in one model, the Ising model, coincides with the intrinsic error threshold of the other model, the surface code.  
Identifier:  CFE0005512 (IID), ucf:50314 (fedora)  
Note(s): 
20141201 Ph.D. Sciences, Physics Doctoral This record was generated from author submitted information. 

Subject(s):  Quantum computation  topological quantum memories  Majorana mode  Ising model  phase transition  error threshold  statistical mechanics  Monte Carlo  
Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0005512  
Restrictions on Access:  public 20141215  
Host Institution:  UCF 