  We propose to use a cryogenic nonlinear resonator for the projective readout,
classical memory, and feedback for a superconducting qubit. This approach
sidesteps many of the inefficiencies inherent in two-way communication between
temperature stages in typical systems with room temperature controllers, and
avoids increasing the cryogenic heat load. This controller may find a broad
range of uses in multi-qubit systems, but here we analyze two specific
demonstrative cases in single qubit-control. In the first case, the nonlinear
controller is used to initialize the qubit in a definite eigenstate. And in the
second case, the qubit's state is read into the controller's classical memory,
where it is stored for an indefinite period of time, and then used to reinstate
the measured state after the qubit has decayed. We analyze the properties of
this system and we show simulations of the time evolution for the full system
dynamics.
