  Trapped atomic ions have become one of the most promising architectures for a
quantum computer, and current effort is now devoted to the transport of trapped
ions through complex segmented ion trap structures in order to scale up to much
larger numbers of trapped ion qubits. This paper covers several important
issues relevant to ion transport in any type of complex multidimensional rf
(Paul) ion trap array. We develop a general theoretical framework for the
application of time-dependent electric fields to shuttle laser-cooled ions
along any desired trajectory, and describe a method for determining the effect
of arbitrary shuttling schedules on the quantum state of trapped ion motion. In
addition to the general case of linear shuttling over short distances, we
introduce issues particular to the shuttling through multidimensional
junctions, which are required for the arbitrary control of the positions of
large arrays of trapped ions. This includes the transport of ions around a
corner, through a cross or T junction, and the swapping of positions of
multiple ions in a laser-cooled crystal. Where possible, we make connections to
recent experimental results in a multidimensional T junction trap, where
arbitrary 2-dimensional transport was realized.
