Figure 3

(a) Pulse sequences (square Z bias and sinusoid microwave for each qubit) for the two-qubit phase-gate operation with 2Q-DD, where both qubits are dynamically tuned on resonance at the gate point for interaction with the big square pulses. Two phase adjustments involving three parameters are critical to validate Eq. (5): A small square pulse with an adjustable amplitude, which is 15 ns in width, is applied to Q 1 right before the square pulse to align its x axis of the Bloch sphere to that of Q 2 ’s in the rotating frame of the on-resonance frequency; the initial phase φ j of each drive Ω j e − i φ j , which is latter inverted at τ / 2 , is aligned to the x axis of each qubit, so that φ 1 = φ 2 for the maximum coupling strength. Here Ω 1 / 2 π ≈ 3.6 and Ω 2 / 2 π ≈ 6.9 MHz . (b) Experimental data (dots) and numerical simulations (solid lines) for the evolution of populations of the four computational states, | 00 ⟩ (blue), | 01 ⟩ (red), | 10 ⟩ (green), and | 11 ⟩ (cyan), with the initial input state in | 00 ⟩ . For numerical simulations, we use the Lindblad master equation and include the microwave crosstalk effect [30], where the pure dephasing times of the two qubits are set to the T ϕ values obtained in the 2Q-DD procedures as exemplified in d . (c) Similar data as in b with the initial input state in | 01 ⟩ . The nonsinusoidal effects in b and c are due to the experimental nonideality with respect to the requirement | Ω 1 − Ω 2 | ≫ | λ | , which leads to nonvanishing transition probabilities between different dressed states. (d) Ramsey interference data of Q 1 , while Q 2 is maintained in the dressed state | + φ , 2 ⟩ throughout the 2Q-DD procedure. Q 1 ’s pure dephasing time during the 2Q-DD procedure is estimated using the Ramsey decay envelope with the equations provided in the caption of Fig. 1. All probability data are corrected for readout errors [35].