Fig. 3
QPT of the geometric two-qubit CZ gate, obtained with the drive amplitude \(\Omega {\rm{/}}2\pi \approx \sqrt {7.0}\) MHz and detuning δ/2π = 4 MHz. a Pulse sequences illustrated in three dimensions (left) and projected to two dimensions (right), with the axes as labeled. For each qubit, the first sinusoid with a Gaussian envelope is for state preparation, which is varied to generate one of the four states \(\left\{ {\left| 0 \right\rangle } \right.\), \(\left( {\left| 0 \right\rangle - i\left| 1 \right\rangle } \right){\rm{/}}\sqrt 2\), \(\left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right){\rm{/}}\sqrt 2\), \(\left. {\left| 1 \right\rangle } \right\}\)}; the second sinusoid with a Gaussian envelope is also variable, acting as the rotation pulse needed in QST; sandwiched in between the two sinusoids is the big square pulse used to adjust the qubit energy levels of Q5 (there is no frequency adjustment on Q1), which combines with the resonator microwave drive to fulfill the CZ gate; the next small square pulse produces a single-qubit rotation on each qubit to partially compensate for the dynamical phase accumulated during the CZ gate; finally qubits are measured by demodulation of the two-tone microwave through the TL readout line (light brown lines with color-coded sinusoids). Here the readout and gate frequencies of Q5 are different for minimizing the Q1–Q5 interaction during readout. b Ideal (χ id, left) and experimental (χ exp, right) quantum process matrices. The color code for Pauli basis {I, X, Y, Z} is shown at the top-left corner. Imaginary components of χ exp are measured to be no larger than 0.015 in magnitude. χ exp has a fidelity F = Tr(χ id χ exp) = 0.936 ± 0.013. The \(\left| 2 \right\rangle\)-state occupation probability of each qubit averaged over the 16 output states is no higher than 0.015 in a separate measurement. We also perform the CZ gate with Q1 and Q3, and obtain a similar gate fidelity
