Computational and control aspects of quantum holonomy Tarn, T. J.
ISBN: 0496313851
Summary:
Geometric phases have long been a source of fascination and insight
into classical and quantum physical theories. Geometric methods have
also made a profound impact in the field of engineering. Inspired by
the appearance of geometric phases in biology, engineers have sought
to create motion in machines via cyclic variations in shape space.
Recently, geometric phases have been proposed as a way of
constructing logic gates in a quantum computer. Geometric quantum
computation employs non-Abelian holonomies to do quantum logic
processing.
In this dissertation, a comprehensive theory for the control of
quantum systems with non-Abelian holonomy is presented. By exploiting
the rich geometry of principal bundles with connection, insight may
be gained into the control theoretic properties of such systems. The
well developed theory of control systems evolving on principal
bundles is useful in treating the controllability as well as the
constructive controllability problems for quantum holonomic systems.
Problems arising in geometric quantum computation are cast within
this framework to obtain new results for systems with the conditional
Berry phase and systems based on squeezed coherent states. For these
systems, we characterize the reachable set and then use the Cartan
decomposition of compact Lie groups to completely solve the
constructive controllability problem. Control laws, obtained in this
way, can then be used to implement quantum algorithms. Robustness of
the model and entanglement as a computational resource are touched
upon.
Notes:
Director: T. J. Tarn.
Thesis (D.Sc.)--Washington University, 2002.
Electronic reproduction.Ann Arbor, MI: ProQuest Information and
Learning Company,2007.System requirements: Adobe Acrobat
Reader.Available via World Wide Web.Digital version of: Computational
and control aspects of quantum holonomy.
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