  Decoherence is studied in an attractive proposal for an actual implementation
of a quantum computer based on trapped ions. Emphasis is placed on the
decoherence arising from the vibrational motion of the ions, which is compared
with that due to spontaneous emission from excited states of the ions. The
calculation is made tractable by exploiting the vast difference in time scales
between the vibrational excitations and the intra-ionic electronic excitations.
Since the latter are several orders of magnitude faster, an adiabatic
approximation is used to integrate them out and find the inclusive probability
P(t) for the elec- tronic state of the ions to evolve as it would in the
absence of vibrational coupling, and the ions to evolve into any state
whatsoever. The decoherence time is found at zero temperature and for any
number of ions N in the computer. Comparison is made with the spontaneous
emission decoherence, and the implications for how trap voltages and other
parameters should be scaled with N are discussed.
