Geometric quantum computation is the idea that geometric phases can be used
to implement quantum gates, i.e., the basic elements of the Boolean network
that forms a quantum computer. Although originally thought to be limited to
adiabatic evolution, controlled by slowly changing parameters, this form of
quantum computation can as well be realized at high speed by using nonadiabatic
schemes. Recent advances in quantum gate technology have allowed for
experimental demonstrations of different types of geometric gates in adiabatic
and nonadiabatic evolution. Here, we address some conceptual issues that arise
in the realizations of geometric gates. We examine the appearance of dynamical
phases in quantum evolution and point out that not all dynamical phases need to
be compensated for in geometric quantum computation. We delineate the relation
between Abelian and non-Abelian geometric gates and find an explicit physical
example where the two types of gates coincide. We identify differences and
similarities between adiabatic and nonadiabatic realizations of quantum
computation based on non-Abelian geometric phases.Comment: Revised version; journal reference adde
