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Quantum Holonomy for Many-Body Systems and Quantum Computation
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. (Condensed Matter Physics)
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The research of this Ph. D. thesis is in the field of Quantum Computation and Quantum

Information. A key problem in this field is the fragile nature of quantum states. This

becomes increasingly acute when the number of quantum bits (qubits) grows in order to

perform large quantum computations. It has been proposed that geometric (Berry) phases

may be a useful tool to overcome this problem, because of the inherent robustness of such

phases to random noise. In the thesis we investigate geometric phases and quantum

holonomies (matrix-valued geometric phases) in many-body quantum systems, and elucidate

the relationship between these phases and the quantum correlations present in the systems.

An overall goal of the project is to assess the feasibility of using geometric phases and

quantum holonomies to build robust quantum gates, and investigate their behavior when the

size of a quantum system grows, thereby gaining insights into large-scale quantum

computation.

In a first project we study the Uhlmann holonomy of quantum states for hydrogen-like

atoms. We try to get into a physical interpretation of this geometric concept by analyzing its

relation with quantum correlations in the system, as well as by comparing it with different

types of geometric phases such as the standard pure state geometric phase, Wilczek-Zee

holonomy, Lévay geometric phase and mixed-state geometric phases. In a second project we

establish a unifying connection between the geometric phase and the geometric measure of

entanglement in a generic many-body system, which provides a universal approach to the

study of quantum critical phenomena. This approach can be tested experimentally in an

interferometry setup, where the geometric measure of entanglement yields the visibility of

the interference fringes, whereas the geometric phase describes the phase shifts. In a third

project we propose a scheme to implement universal non-adiabatic holonomic quantum

gates, which can be realized in novel nano-engineered systems such as quantum dots,

molecular magnets, optical lattices and topological insulators. In a fourth project we propose

an experimentally feasible approach based on “orange slice” shaped paths to realize non-

Abelian geometric phases, which can be used particularly for geometric manipulation of

qubits. Finally, we provide a physical setting for realizing non-Abelian off-diagonal

geometric phases. The proposed setting can be implemented in a cyclic chain of four qubits

with controllable nearest-neighbor interactions. Our proposal seems to be within reach in

various nano-engineered systems and therefore opens up for first experimental test of the

non-Abelian off-diagonal geometric phase.

Place, publisher, year, edition, pages
Växjö: Linnaeus University Press , 2013. , 140 p.
Series
Linnaeus University Dissertations, 141/2013
Keyword [en]
Quantum holonomy, geometric phase, quantum correlations, quantum phase transitions, quantum computation
National Category
Condensed Matter Physics Other Physics Topics
Research subject
Natural Science; Natural Science, Physics
Identifiers
URN: urn:nbn:se:lnu:diva-28311ISBN: 978-91-87427-38-1 (print)OAI: oai:DiVA.org:lnu-28311DiVA: diva2:642198
Public defence
2013-08-26, Ny227, Kalmar Nyckel, Kalmar, 13:00 (English)
Opponent
Supervisors
Available from: 2013-09-10 Created: 2013-08-21 Last updated: 2013-09-10Bibliographically approved
List of papers
1. Non-Abelian geometric phases in a system of coupled quantum bits
Open this publication in new window or tab >>Non-Abelian geometric phases in a system of coupled quantum bits
2014 (English)Manuscript (preprint) (Other academic)
Abstract [en]

A common strategy to measure the Abelian geometric phase for a qubit is to let it evolve along an ‘orange slice’ shaped path connecting two antipodal points on the Bloch sphere by two different semi- great circles. Since the dynamical phases vanish for such paths, this allows for direct measurement of the geometric phase. Here, we generalize the orange slice setting to the non-Abelian case. The proposed method to measure the non-Abelian geometric phase can be implemented in a cyclic chain of four qubits with controllable interactions.

National Category
Physical Sciences
Research subject
Social Sciences
Identifiers
urn:nbn:se:lnu:diva-28087 (URN)000332330800001 ()
Available from: 2013-08-13 Created: 2013-08-13 Last updated: 2014-04-29Bibliographically approved
2. Unifying geometric entanglement and geometric phase in a quantum phase transition
Open this publication in new window or tab >>Unifying geometric entanglement and geometric phase in a quantum phase transition
2013 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 88, no 1, Article ID: 012310- p.Article in journal (Refereed) Published
Abstract [en]

Geometric measure of entanglement and geometric phase have recently been used to analyze quantum phase transition in the XY spin chain. We unify these two approaches by showing that the geometric entanglement and the geometric phase are respectively the real and imaginary parts of a complex-valued geometric entanglement, which can be investigated in typical quantum interferometry experiments. We argue that the singular behavior of the complex-valued geometric entanglement at a quantum critical point is a characteristic of any quantum phase transition, by showing that the underlying mechanism is the occurrence of level crossings associated with the underlying Hamiltonian.

National Category
Condensed Matter Physics
Research subject
Physics, Condensed Matter Physics
Identifiers
urn:nbn:se:lnu:diva-28081 (URN)10.1103/PhysRevA.88.012310 (DOI)000321833000004 ()2-s2.0-84880606305 (Scopus ID)
Available from: 2013-08-13 Created: 2013-08-13 Last updated: 2017-12-06Bibliographically approved
3. Non-Abelian off-diagonal geometric phases in nano-engineered four-qubit systems
Open this publication in new window or tab >>Non-Abelian off-diagonal geometric phases in nano-engineered four-qubit systems
2013 (English)In: Europhysics letters, ISSN 0295-5075, E-ISSN 1286-4854, Vol. 103, no 6, 60011- p.Article in journal (Refereed) Published
Abstract [en]

The concept of off-diagonal geometric phase (GP) has been introduced in order to recover interference information about the geometry of quantal evolution where the standard GPs are not well-defined. In this Letter, we propose a physical setting for realizing non-Abelian off-diagonal GPs. The proposed non-Abelian off-diagonal GPs can be implemented in a cyclic chain of four qubits with controllable nearest-neighbor interactions. Our proposal seems to be within reach in various nano-engineered systems and therefore opens up for first experimental test of the non-Abelian off-diagonal GP.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2013
National Category
Physical Sciences
Research subject
Natural Science, Physics
Identifiers
urn:nbn:se:lnu:diva-28085 (URN)10.1209/0295-5075/103/60011 (DOI)000326280200011 ()2-s2.0-84887094158 (Scopus ID)
Available from: 2013-08-13 Created: 2013-08-13 Last updated: 2017-12-06Bibliographically approved
4. Non-Abelian quantum holonomy of hydrogen-like atoms
Open this publication in new window or tab >>Non-Abelian quantum holonomy of hydrogen-like atoms
2011 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, ISSN 1050-2947, Vol. 84, no 3, Article ID: 032111- p.Article in journal (Refereed) Published
Abstract [en]

We study the Uhlmann holonomy [Rep. Math. Phys. 24, 229 (1986)] of quantum states for hydrogen-like atoms, where the intrinsic spin and orbital angular momentum are coupled by the spin-orbit interaction and subject to a slowly varying magnetic field. We show that the holonomy for the orbital angular momentum and spin subsystems is non-Abelian, while the holonomy of the whole system is Abelian. Quantum entanglement in the states of the whole system is crucially related to the non-Abelian gauge structure of the subsystems. We analyze the phase of the Wilson loop variable associated with the Uhlmann holonomy, and find a relation between the phase of the whole system with corresponding marginal phases. Based on the result for the model system we provide evidence that the phase of the Wilson loop variable and the mixed-state geometric phase [Phys. Rev. Lett. 85, 2845 (2000)] are in general inequivalent.

Keyword
Quantum holonomy, spin-orbit coupling, hydrogen-like atoms, quantum entanglement
National Category
Natural Sciences
Research subject
Natural Science, Physics
Identifiers
urn:nbn:se:lnu:diva-11495 (URN)10.1103/PhysRevA.84.032111 (DOI)2-s2.0-80053118405 (Scopus ID)
Available from: 2011-04-28 Created: 2011-04-28 Last updated: 2016-11-01Bibliographically approved
5. Universal Non-adiabatic Holonomic Gates in Quantum Dots and Single-Molecule Magnets
Open this publication in new window or tab >>Universal Non-adiabatic Holonomic Gates in Quantum Dots and Single-Molecule Magnets
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Geometric manipulation of a quantum system offers a method for fast, universal, and robust quantum information processing. Here, we propose a scheme for universal all-geometric quantum computation using non-adiabatic quantum holonomies. We propose three different realizations of the scheme based on an unconventional use of quantum dot and single-molecule magnet devices,which offer promising scalability and robust efficiency.

National Category
Natural Sciences Physical Sciences Condensed Matter Physics
Research subject
Physics, Condensed Matter Physics
Identifiers
urn:nbn:se:lnu:diva-19220 (URN)
Available from: 2012-06-01 Created: 2012-06-01 Last updated: 2016-11-01Bibliographically approved

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Azimi Mousolou, Vahid

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