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We present a detailed analysis of the modulated-carrier quantum phase gateimplemented with Wigner crystals of ions confined in Penning traps. Weelaborate on a recent scheme, proposed by two of the authors, to engineertwo-body interactions between ions in such crystals. We analyze for the firsttime 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 thepresence of the magnetic field in the rotating frame the many-body (classical)Hamiltonian describing small oscillations from the ion equilibrium positionscan be recast in canonical form. As a consequence, we are able to demonstratethat fast and robust two-qubit gates are achievable within the currentexperimental limitations. Moreover, we describe a realization of thestate-dependent sign-changing dipole forces needed to realize the investigatedquantum computing scheme.

Fast and robust quantum computation with ionic Wigner crystals