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We propose to use a cryogenic nonlinear resonator for the projective readout,classical memory, and feedback for a superconducting qubit. This approachsidesteps many of the inefficiencies inherent in two-way communication betweentemperature stages in typical systems with room temperature controllers, andavoids increasing the cryogenic heat load. This controller may find a broadrange of uses in multi-qubit systems, but here we analyze two specificdemonstrative cases in single qubit-control. In the first case, the nonlinearcontroller is used to initialize the qubit in a definite eigenstate. And in thesecond 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 reinstatethe measured state after the qubit has decayed. We analyze the properties ofthis system and we show simulations of the time evolution for the full systemdynamics.

Closing a quantum feedback loop inside a cryostat: Autonomous state preparation and long-time memory of a superconducting qubit