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

We propose to use a nonlinear resonator for projective readout, classical memory, and feedback for a superconducting qubit. Keeping the classical controller at cryogenic temperatures 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 multiqubit 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 and 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.