Fig. 7.

Detuned-KCQ readout, initialization, and gate calibration. (A) The histogram of CQR. (B) The collapse of wave-functions corresponding to CQR. (C) The QNDness optimization of CQR with various readout strengths. (D) The SNAIL nonlinear oscillator spectrum in the rotating frame as a function of detuning. (E) The sequence for calibrating $X(\pi /2)$ gate with 400-μs relaxation time in the initialization. (F) The tunneling between the ground states when the relaxation time is long enough in the initialization. (G) The measured $|\pm Z⟩$ population oscillation. (H) The numerical simulation of the results above it. (I) The sequence for observing the excited states tunneling with 5-μs relaxation time in initialization. (J) The tunneling between the excited states when the relaxation time is short. (K) The measured population oscillation between excited states. (L) The numerical simulation of the results above.