  We investigate quantum control of an oscillator mode off-resonantly coupled
to an ancillary qubit. In the strong dispersive regime, we may drive the qubit
conditioned on number states of the oscillator, which together with
displacement operations can achieve universal control of the oscillator. Based
on our proof of universal control, we provide explicit constructions for
arbitrary state preparation and arbitrary unitary operation of the oscillator.
Moreover, we present an efficient procedure to prepare the number state
$\left|n\right\rangle$ using only $O\left(\sqrt{n}\right)$ operations. We also
compare our scheme with known quantum control protocols for coupled
qubit-oscillator systems. This universal control scheme of the oscillator can
readily be implemented using superconducting circuits.
