  We show that with the assistance of a third level of the qubits an n-qubit
phase gate can be constructed from $2n-4$ two-qutrit conditional swap gates, a
single qutrit-qubit controlled phase gate, and two single-qutrit operations.
Unlike previous schemes, our scheme uses the auxiliary level to "expose" some
state to the qutrit-qubit controlled phase gate, instead of using it to "hide"
states from the conditional dynamics. Neither the number of the additional
levels nor that of single-qutrit operations needs to increase with $n$. We
propose a physical implementation of the required elementary gates in cavity
QED, and show that the total gate time may be greatly reduced as compared with
that required in the previous methods.
