  We present a detailed analysis of the modulated-carrier quantum phase gate
implemented with Wigner crystals of ions confined in Penning traps. We
elaborate on a recent scheme, proposed by two of the authors, to engineer
two-body interactions between ions in such crystals. We analyze for the first
time the situation in which the cyclotron (w_c) and the crystal rotation (w_r)
frequencies do not fulfill the condition w_c=2w_r. It is shown that even in the
presence of the magnetic field in the rotating frame the many-body (classical)
Hamiltonian describing small oscillations from the ion equilibrium positions
can be recast in canonical form. As a consequence, we are able to demonstrate
that fast and robust two-qubit gates are achievable within the current
experimental limitations. Moreover, we describe a realization of the
state-dependent sign-changing dipole forces needed to realize the investigated
quantum computing scheme.
