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Entangling Qubits by Heisenberg Spin Exchange and Anyon Braiding

Title: Entangling Qubits by Heisenberg Spin Exchange and Anyon Braiding.
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Name(s): Zeuch, Daniel, author
Bonesteel, N. E., professor directing dissertation
Sussman, Mark, university representative
Hill, S. (Stephen Olof), committee member
Piekarewicz, Jorge, 1956-, committee member
Florida State University, degree granting institution
College of Arts and Sciences, degree granting college
Department of Physics, degree granting department
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2016
Publisher: Florida State University
Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource (142 pages)
Language(s): English
Abstract/Description: As the discovery of quantum mechanics signified a revolution in the world of physics more than one century ago, the notion of a quantum computer in 1981 marked the beginning of a drastic change of our understanding of information and computability. In a quantum computer, information is stored using quantum bits, or qubits, which are described by a quantum-mechanical superposition of the quantum states 0 and 1. Computation then proceeds by acting with unitary operations on these qubits. These operations are referred to as quantum logic gates, in analogy to classical computation where bits are acted on by classical logic gates. In order to perform universal quantum computation it is, in principle, sufficient to carry out single-qubit gates and two-qubit gates, where the former act on individual qubits and the latter, acting on two qubits, are used to entangle qubits with each other. The present thesis is divided into two main parts. In the first, we are concerned with spin-based quantum computation. In a spin-based quantum computer, qubits are encoded into the Hilbert space spanned by spin-½ particles, such as electron spins trapped in semiconductor quantum dots. For a suitable qubit encoding, turning on-and-off, or "pulsing," the isotropic Heisenberg exchange Hamiltonian JSi · Sj allows for universal quantum computation and it is this scheme, known as exchange-only quantum computation, which we focus on. In the second part of this thesis, we consider a topological quantum computer in which qubits are encoded using so-called Fibonacci anyons, exotic quasiparticle excitations that obey non-Abelian statistics, and which may emerge in certain two-dimensional topological systems such as fractional quantum-Hall states. Quantum gates can then be carried out by moving these particles around one another, a process that can be viewed as braiding their 2+1 dimensional worldlines. The subject of the present thesis is the development and theoretical understanding of procedures used for entangling qubits. We begin by presenting analytical constructions of pulse sequences which can be used to carry out two-qubit gates that are locally equivalent to a controlled-PHASE gate. The corresponding phase can be arbitrarily chosen, and for one particular choice this gate is equivalent to controlled-NOT. While the constructions of these sequences are relatively lengthy and cumbersome, we further provide a straightforward and intuitive derivation of the shortest known two-qubit pulse sequence for carrying out a controlled-NOT gate. This derivation is carried out completely analytically through a novel "elevation" of a simple three-spin pulse sequence to a more complicated five-spin pulse sequence. In the case of topological quantum computation with Fibonacci anyons, we present a new method for constructing entangling two-qubit braids. Our construction is based on an iterative procedure, established by Reichardt, which can be used to systematically generate braids whose corresponding operations quickly converge towards an operation that has a diagonal matrix representation in a particular natural basis. After describing this iteration procedure we show how the resulting braids can be used in two explicit constructions for two-qubit braids. Compared to two-qubit braids that can be found using other methods, the braids generated here are among the most efficient and can be obtained straightforwardly without computational overhead.
Identifier: FSU_2016SU_Zeuch_fsu_0071E_13323 (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 2016.
Date of Defense: May 31, 2016.
Keywords: entanglement, non-Abelian anyons, quantum computation, quantum dots
Bibliography Note: Includes bibliographical references.
Advisory Committee: Nicholas E. Bonesteel, Professor Directing Dissertation; Mark Sussman, University Representative; Vladimir Dobrosavljevic, Committee Member; Stephen Hill, Committee Member; Jorge Piekarewicz, Committee Member.
Subject(s): Physics
Quantum theory
Condensed matter
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_2016SU_Zeuch_fsu_0071E_13323
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 rights-holder(s). The copyright in theses and dissertations completed at Florida State University is held by the students who author them.
Host Institution: FSU

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Zeuch, D. (2016). Entangling Qubits by Heisenberg Spin Exchange and Anyon Braiding. Retrieved from http://purl.flvc.org/fsu/fd/FSU_2016SU_Zeuch_fsu_0071E_13323