Stabilizing the trajectory of a superconducting qubit by projective measurement feedback

Making a system state follow a prescribed trajectory despite fluctuations and errors commonly consists in monitoring an observable (temperature, blood-glucose level…) and reacting on its controllers (heater power, insulin amount …). In the quantum domain, there is a change of paradigm in feedback since measurements modify the state of the system, most dramatically when the trajectory goes through superpositions of measurement eigenstates. Here, we demonstrate the stabilization of an arbitrary trajectory of a superconducting qubit by measurement based feedback. The protocol benefits from the long coherence time ($T_2>10 mu$s) of the 3D transmon qubit, the high efficiency (82%) of the phase preserving Josephson amplifier, and fast electronics ensuring less than 500 ns delay. At discrete time intervals, the state of the qubit is measured and corrected in case an error is detected. For Rabi oscillations, where the discrete measurements occur when the qubit is supposed to be in the measurement pointer states, we demonstrate an average fidelity of 85% to the targeted trajectory. For Ramsey oscillations, which does not go through pointer states, the average fidelity reaches 75%. Incidentally, we demonstrate a fast reset protocol allowing to cool a 3D transmon qubit down to 0.6% in the excited state.