Deterministic nonlinear phase gates induced by a single qubit

We propose deterministic realizations of nonlinear phase gates by repeating non-commuting Rabi interactions feasible between a harmonic oscillator and {\em only} a single two-level ancillary qubit. We show explicitly that the key nonclassical features of the states after the ideal cubic phase gate and the quartic phase gate are reproduced faithfully by the engineered operators. This theoretical proposal completes the universal set of operators in continuous variable quantum computation.