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Entanglement and Quantum Computation from a Geometric and Topological Perspective
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Chemistry - Ångström, Theoretical Chemistry.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we investigate geometric and topological structures in the context of entanglement and quantum computation.

A parallel transport condition is introduced in the context of Franson interferometry based on the maximization of two-particle coincidence intensity. The dependence on correlations is investigated and it is found that the holonomy group is in general non-Abelian, but Abelian for uncorrelated systems. It is found that this framework contains a parallel transport condition developed by Levay in the case of two-qubit systems undergoing local SU(2) evolutions.

Global phase factors of topological origin, resulting from cyclic local SU(2) evolution, called topological phases, are investigated in the context of multi-qubit systems. These phases originate from the topological structure of the local SU(2)-orbits and are an attribute of most entangled multi-qubit systems. The relation between topological phases and SLOCC-invariant polynomials is discussed. A general method to find the values of the topological phases in an n-qubit system is described.

A non-adiabatic generalization of holonomic quantum computation is developed in which high-speed universal quantum gates can be realized by using non-Abelian geometric phases. It is shown how a set of non-adiabatic holonomic one- and two-qubit gates can be implemented by utilizing transitions in a generic three-level Λ configuration. The robustness of the proposed scheme to different sources of error is investigated through numerical simulation. It is found that the gates can be made robust to a variety of errors if the operation time of the gate can be made sufficiently short. This scheme opens up for universal holonomic quantum computation on qubits characterized by short coherence times.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. , 66 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 930
Keyword [en]
Quantum Information, Geometric Phases, Topological Phases, Entanglement, Quantum Computation
National Category
Physical Sciences
Identifiers
ISBN: 978-91-554-8364-7OAI: oai:DiVA.org:uu-173120DiVA: diva2:516636
Public defence
2012-06-07, Häggsalen, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Supervisors
Available from: 2012-05-14 Created: 2012-04-18 Last updated: 2012-08-01Bibliographically approved
List of papers
1. Correlation-induced non-Abelian quantum holonomies
Open this publication in new window or tab >>Correlation-induced non-Abelian quantum holonomies
2011 (English)In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, Vol. 44, no 14, 145301- p.Article in journal (Refereed) Published
Abstract [en]

In the context of two-particle interferometry, we construct a parallel transport condition that is based on the maximization of coincidence intensity with respect to local unitary operations on one of the subsystems. The dependence on correlation is investigated and it is found that the holonomy group is generally non-Abelian, but Abelian for uncorrelated systems. It is found that our framework contains the Lévay geometric phase (2004 J. Phys. A: Math. Gen. 37 1821) in the case of two-qubit systems undergoing local SU(2) evolutions.

Keyword
Quantum holonomy, quantum correlations, quantum interferometry
National Category
Physical Sciences
Physics
Identifiers
urn:nbn:se:uu:diva-148641 (URN)10.1088/1751-8113/44/14/145301 (DOI)000288597500011 ()
Note
Also in IOP Select, http://Select.iop.org. Additional address (E. Sjöqvist): Centre for Quantum Technologies, NUS, Singapore. Additional address (M. S. Williamson): Erwin Schrödinger International Institute for Mathematical Physics, Wien, AustriaAvailable from: 2011-03-09 Created: 2011-03-09 Last updated: 2012-08-01Bibliographically approved
2. Topological phases and multiqubit entanglement
Open this publication in new window or tab >>Topological phases and multiqubit entanglement
2012 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 85, no 3, 032112-1-032112-11 p.Article in journal (Refereed) Published
Abstract [en]

Global phase factors of topological origin, resulting from cyclic local $\rm{SU}$ evolution, called topological phases, were first described in [Phys. Rev. Lett. {\bf 90}, 230403 (2003)], in the case of entangled qubit pairs. In this paper we investigate topological phases in multi-qubit systems as the result of cyclic local $\rm{SU(2)}$ evolution. These phases originate from the topological structure of the local $\rm{SU(2)}$-orbits and are an attribute of most entangled multi-qubit systems. We discuss the relation between topological phases and SLOCC-invariant polynomials and give examples where topological phases appear. A general method to find the values of the topological phases in an $n$-qubit system is described and a complete list of these phases for up to seven qubits is given.

Keyword
Topological phase, multipartite entanglement, quantum information
National Category
Other Physics Topics
Physics
Identifiers
urn:nbn:se:uu:diva-169141 (URN)10.1103/PhysRevA.85.032112 (DOI)000301333700003 ()
Funder
Swedish Research Council
Note

Available from: 2012-02-23 Created: 2012-02-23 Last updated: 2015-08-11Bibliographically approved
Open this publication in new window or tab >>Non-Adiabatic Holonomic Quantum Computation
2012 (English)In: New Journal of Physics, ISSN 1367-2630, E-ISSN 1367-2630, Vol. 14, 103035Article in journal (Refereed) Published
Abstract [en]

We develop a non-adiabatic generalization of holonomic quantum computation in which high-speed universal quantum gates can be realized using non-Abelian geometric phases. We show how a set of non-adiabatic holonomic one- and two-qubit gates can be implemented by utilizing optical transitions in a generic three-level Λ configuration. Our scheme opens up the possibility of realizing universal holonomic quantum computation on qubits characterized by short coherence time.

Keyword
Quantum computation, geometric phase, quantum gates
National Category
Physical Sciences Atom and Molecular Physics and Optics
Physics
Identifiers
urn:nbn:se:uu:diva-157181 (URN)10.1088/1367-2630/14/10/103035 (DOI)000310439000001 ()
Note

Available from: 2011-08-18 Created: 2011-08-18 Last updated: 2016-02-19Bibliographically approved
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