Optimization of a Controlled-$Z$ Gate with Data-Driven Gradient-Ascent Pulse Engineering in a Superconducting-Qubit System

The experimental optimization of a two-qubit controlled-$Z$ (cz) gate is realized following two different data-driven gradient-ascent pulse engineering (GRAPE) protocols with the aim of optimizing the gate operator and the output quantum state, respectively. For both GRAPE protocols, the key computation of gradients utilizes mixed information of the input $Z$-control pulse and the experimental measurement. With an imperfect initial pulse in a flattop waveform, our experimental implementation shows that the cz gate is quickly improved and the gate fidelities subject to the two optimized pulses are around 99%. Our experimental study confirms the applicability of the data-driven GRAPE protocols in the gate optimization problem.

Figure 1

Schematic diagrams of two data-driven GRAPE protocols implemented in our experiment. These two protocols are designed under the optimization of (a) the fitted gate operator $U$ and (b) an output density matrix $\rho (T)$, respectively.

Figure 2

(a) The blue and red solid lines present an avoided crossing for the states $|11⟩$ and $|20⟩$, while the two crossed dashed lines denote energy levels with a zero coupling. Inset is an optical micrograph composed of two coupled X-shaped transmon qubits. (b) The probability $P_{11}$ of the state $|11⟩$ versus the target qubit frequency and the swap time.

Figure 3

The experimental results of data-driven GRAPE protocol I. (a) The fidelities of the process matrix $F({\chi }^{(l)})$ (black squares) and the fitted gate operator $F(U_{\exp }^{(l)})$ (red circles) as functions of the iteration step $l$. (b) The pulse shape modification through the iteration procedure. The initial flattop waveform (blue) and the optimal pulse ${\mu }_A^{(5)}(t)$ (red) are shown. In the inset, the resonance region of six pulses [${\mu }_A^{(0)}(t)\to {\mu }_A^{(5)}(t)$ from top to bottom] are enlarged. (c) The QPT measurement of the $\chi$ matrix subject to the optimal cz gate. (d) The sequence fidelities (reference in blue and interleaved in red) versus the number of Clifford gates, where the cz gate is implemented by the optimal $Z$ pulse. For each result, an error bar of the standard deviation is shown together with the average value.

Figure 4

The experimental results of data-driven GRAPE protocol II. (a) The fidelities of an output density matrix $F[{\rho }_{\exp }^{(l)}(T)\mid {\phi }_1(0)]$ (red circles) and the process matrix $F({\chi }^{(l)})$ (black squares) as functions of the iteration step $l$. (b) The pulse shape modification through the iteration procedure. (c) The QPT measurement of the $\chi$ matrix subject to the optimal cz gate. (d) The sequence fidelities under the optimal $Z$ pulse in the RB measurement. The legends in (b)–(d) are the same as those in Figs. 3–3.