Geometric methods for construction of quantum gates |
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Authors: | Z Giunashvili |
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Institution: | (1) Department of Theoretical Physics, Institute of Mathematics, Georgian Academy of Sciences, Tbilisi, Georgia |
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Abstract: | The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations
are investigated. These methods are shown to be very useful for the problem of constructing a universal set of gates for quantum
computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal
is considered from the control theory viewpoint.
Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold,
the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold, and the corresponding
dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action, and the symplectic
form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection
on the principal bundle over the Grassmannian and its relation with the Berry phase is considered and investigated for the trajectories of Hamiltonian dynamical systems.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 44, Quantum
Computing, 2007. |
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Keywords: | |
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