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Braiding and Berry's Phases in NonAbelian Quantum Hall States
Title:  Braiding and Berry's Phases in NonAbelian Quantum Hall States. 
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Name(s): 
Zikos, Georgios, author Bonesteel, Nicholas, professor directing dissertation Aldrovandi, Ettore, outside committee member Schlottmann, Pedro, committee member Reina, Laura, committee member Chiorescu, Irinel, committee member Department of Physics, degree granting department Florida State University, degree granting institution 

Type of Resource:  text  
Genre:  Text  
Issuance:  monographic  
Date Issued:  2009  
Publisher: 
Florida State University Florida State University 

Place of Publication:  Tallahassee, Florida  
Physical Form: 
computer online resource 

Extent:  1 online resource  
Language(s):  English  
Abstract/Description:  If one could be built, a quantum computer would be capable of storing and manipulating quantum states with sufficient accuracy to carry out computations that no classical computer can do (most notably factoring integers in polynomial time). The greatest obstacle to building such a device is the problem of error and decoherence. Classical computers can exploit the physical robustness of ordered states to protect classical information (as in, for example, the magnetically ordered state of a hard drive). Remarkably, a type of quantum order known as topological order can, in principle, play the same role for quantum information. The best studied topologically ordered states are quantum Hall states. These states arise when a twodimensional electron gas is placed in a strong magnetic field and cooled to low temperatures. Under the right conditions, the electrons condense into an incompressible quantum liquid whose excitations are particlelike objects with fractional charge (quasiparticles). Certain quantum Hall states are thought to be non Abelian. This means that when a finite number of quasiparticles are present and fixed in space there is a low energy Hilbert space with finite dimension, rather than a unique state. Unitary operations can then be carried out on this Hilbert space by adiabatically dragging quasiparticles around one another so that their worldlines sweep out braids in 2+1 dimensional space time. A quantum computer which stores quantum information in this Hilbert space and computes by braiding is known as a topological quantum computer. In this thesis I review our work on determining precisely how one would carry out a computation on a topological quantum computer. I focus on the socalled Fibonacci anyonsquasiparticles which may exist in the experimentally observed quantum Hall state at Landau level filling fraction ν = 12/5. I give explicit prescriptions for encoding qubits (quantum bits) using Fibonacci anyons, and show how one would carry out a universal set of quantum gates (the quantum analogs of Boolean logic gates) by braiding them. I then focus in particular on my work developing algorithms for performing brute force searches over the space of braids to find braids which produce unitary operations close to any desired operation. These brute force searches are a crucial part of our quantum gate construction, and I show that by using a socalled "load balanced" bidirectional search I can find braids which approximate any desired operation to an accuracy of 1 part in 10 5 . I then turn to my work calculating the Berry's phase obtained when quasiparticles are moved around one another in the MooreRead state, a non Abelian state generally believed to describe the ν = 5/2 quantum Hall effect. This work is done using variational Monte Carlo, a method which allows one to numerically evaluate the Berry's phase for finite size systems. By exploiting certain properties of the MooreRead state I have been able to study systems consisting of as many as 150 electrons. In so doing I have verified the conjectured connection between the Berry's phase produced by physically moving quasiparticles around one another and the mathematical phase one obtains by simply analytically continuing the quasiparticle coordinates. An added benefit of these calculations is that we can deduce the length scale which determines the size of the quasiparticles. This length scale dictates how far apart the quasiparticles must be in order to prevent errors when they are used for topological quantum computation.  
Identifier:  FSU_migr_etd0503 (IID)  
Submitted Note:  A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.  
Degree Awarded:  Summer Semester, 2009.  
Date of Defense:  April 27, 2009.  
Keywords:  Variational Monte Carlo, NonAbelian Statistics, Fractional Quantum Hall Effect, Topological Quantum Computation  
Bibliography Note:  Includes bibliographical references.  
Advisory Committee:  Nicholas Bonesteel, Professor Directing Dissertation; Ettore Aldrovandi, Outside Committee Member; Pedro Schlottmann, Committee Member; Laura Reina, Committee Member; Irinel Chiorescu, Committee Member.  
Subject(s):  Physics  
Persistent Link to This Record:  http://purl.flvc.org/fsu/fd/FSU_migr_etd0503  
Use and Reproduction:  This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rightsholder(s). The copyright in theses and dissertations completed at Florida State University is held by the students who author them.  
Host Institution:  FSU 
Zikos, G. (2009). Braiding and Berry's Phases in NonAbelian Quantum Hall States. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd0503