  We study analytically and numerically the properties of phonon modes in an
ion quantum computer. The ion chain is placed in a harmonic trap with an
additional periodic potential which dimensionless amplitude $K$ determines
three main phases available for quantum computations: at zero $K$ we have the
case of Cirac-Zoller quantum computer, below a certain critical amplitude
$K<K_c$ the ions are in the Kolmogorov-Arnold-Moser (KAM) phase, with
delocalized phonon modes and free chain sliding, and above the critical
amplitude $K>K_c$ ions are in the pinned Aubry phase with a finite frequency
gap protecting quantum gates from temperature and other external fluctuations.
For the Aubry phase, in contrast to the Cirac-Zoller and KAM phases, the phonon
gap remains independent of the number of ions placed in the trap keeping a
fixed ion density around the trap center. We show that in the Aubry phase the
phonon modes are much better localized comparing to the Cirac-Zoller and KAM
cases. Thus in the Aubry phase the recoil pulses lead to local oscillations of
ions while in other two phases they spread rapidly over the whole ion chains
making them rather sensible to external fluctuations. We argue that the
properties of localized phonon modes and phonon gap in the Aubry phase provide
advantages for the ion quantum computations in this phase with a large number
of ions.
