Extended Data Fig. 5: Two-phonon phase gate measurements.

a, Two-qubit joint excitation probability P_ee as a function of Q₂ scattering detuning, as shown in Fig. 3d, and here including the single-phonon phase gate data shown in Fig. 2c, in log scale without visibility matrix correction. The two maxima in \({P}_{ee,\max }\) align in Q₂ frequency with the single qubit excitation minima \({P}_{Q1,\min }\) and \({P}_{Q2,\min }\) (blue and orange dashed line), indicating the two-phonon phase shift is approximately twice that of the single-phonon phase shift. b, For fixed Q₂ detuning at -9.2 MHz (red dashed line in panel a), the two-qubit joint excitation P_ee is shown as a function of the coupler G₂ control pulse amplitude. The minimum \({P}_{ee,\min }\) is reached for a pulse amplitude of -0.5 (a.u.), corresponding to maximum coupling. The coupler is turned off when the pulse amplitude is near zero, with no phase shift applied, where P_ee approaches its maximum. The average visibility \({{\mathcal{V}}}_{{\rm{ee,raw}}}\equiv ({P}_{ee,\max }-{P}_{ee,\min })/{P}_{ee,\max }=0.968\pm 0.004\).