High-Fidelity, High-Scalability Two-Qubit Gate Scheme for Superconducting Qubits

High-quality two-qubit gate operations are crucial for scalable quantum information processing. Often, the gate fidelity is compromised when the system becomes more integrated. Therefore, a low-error-rate, easy-to-scale two-qubit gate scheme is highly desirable. Here, we experimentally demonstrate a new two-qubit gate scheme that exploits fixed-frequency qubits and a tunable coupler in a superconducting quantum circuit. The scheme requires less control lines, reduces cross talk effect, and simplifies calibration procedures, yet produces a controlled-$Z$ gate in 30 ns with a high fidelity of 99.5%, derived from the interleaved randomized benchmarking method. Error analysis shows that gate errors are mostly coherence limited. Our demonstration paves the way for large-scale implementation of high-fidelity quantum operations.

Figure 1

Device schematic and concept of the adiabatic CZ gate. (a) Simplified circuit schematic of the experimental sample. The two qubits have an Xmon design, and the central tunable coupler is also a transmon-type qubit. The all-capacitive-coupling architecture allows convenient engineering of the nearest-neighbor and the weaker next-nearest-neighbor couplings. Single-qubit operations are implemented with local $XY$ control lines. Two-qubit gates are implemented by modulating the magnetic flux threading the coupler’s superconducting quantum interference device loop ${\mathrm{\Phi }}_C$ with the local $Z$ control line. The first qubit is treated as an equivalent of a fixed-frequency qubit by setting its loop flux ${\mathrm{\Phi }}_1$ to zero throughout the experiment. Two $\lambda /4$ resonators, coupled to the same transmission line, are used for reading out the qubit states simultaneously. (b) System eigenenergies as a function of the coupler frequency. Only states in the two-excitation manifold are shown. The adiabatic CZ gate is realized by a 30-ns flux pulse applied to the coupler. The pulse assumes a simple half-period cosine shape. The gray dashed line indicates the adiabatic trajectory of an initial $|101⟩$ state, which follows the pink level. The smallest gap between the pink state and the other states is about 238 MHz.

Figure 2

Tunability of the longitudinal coupling. (a) In the top panel is the control sequence that performs a Ramsey-like sequence on $Q_2$ conditioning on the state of $Q_1$ in order to extract the longitudinal coupling. Between the two $\pi /2$ pulses applied to $Q_2$, a squarelike flux pulse with amplitude $V_b$ and duration $\tau$ is applied to the coupler. The phase shift induced by this pulse is captured by $Q_2$ with the Ramsey-like sequence. In our experiment, the phase of the last $\pi /2$ pulse is a varying parameter (indicated by the circling arrow) so that a full oscillation can be resolved, as shown in the bottom panel. Data (markers) are fitted (solid lines) by a sinusoidal function to extract the differential phase ($\mathrm{\Delta }\varphi$) between the cases of $Q_1$ being at the ground (green) or excited (yellow) state. (b) Effective longitudinal coupling strength ${\chi }_{12}$ as a function of the coupler frequency. ${\chi }_{12}$ can be derived from the previous results in (a) by the relation $\mathrm{\Delta }\varphi ={\chi }_{12}\tau$. The experiment is repeated with different pulse amplitudes $V_b$ and fixed $\tau =50\mathrm{ns}$ (500 ns) for larger (smaller) $V_b$. Together with separately measured coupler spectrum [25], we obtain ${\chi }_{12}({\omega }_c)$ (blue dots), which is in good agreement with numerical results (red line). We choose ${\omega }_c/2\pi =6.74\mathrm{GHz}$ as the idling point in subsequent two-qubit gate experiments, due to the small residual coupling (20 kHz). Note that the negative of ${\chi }_{12}$ is plotted here.

Figure 3

Fidelity analysis of the adiabatic CZ gate. (a) Normalized sequence fidelity measured as a function of the number of Cliffords for both the reference (blue) and interleaved (red) RB experiments, with the sequences shown in the inset. Error bars on the data points are the standard deviations from the mean. We first obtain the decay constants, $p_{\mathrm{ref}}$ and $p_{\mathrm{int}}$, from an exponential fit, $F=A{p}^m+B$ (solid lines), and then derive the error rate per Clifford, $r_{\mathrm{ref}}$ and $r_{\mathrm{int}}$ from $r=\frac{3}{4}(1-p)$. We also extract the CZ gate error from $r_{\mathrm{CZ}}=\frac{3}{4}(1-p_{\mathrm{int}}/p_{\mathrm{ref}})$ and finally the CZ gate fidelity $F_{\mathrm{CZ}}=1-r_{\mathrm{CZ}}$. The uncertainty of the fidelity is determined by bootstrapping. (b) Effective energy relaxation time $T_1^{\mathrm{eff}}$ (green bar) and pure dephasing (Gaussian decay) time $T_{\varphi }^{\mathrm{eff}}$ (orange bar) during the adiabatic CZ gate. The results are calculated by averaging over ${\omega }_c$, weighted by the actual pulse shape (see Supplemental Material [25] for details). The blank outlines indicate measured characteristic times when the coupler is at the idling point. (c) Pulse-induced transitional errors by the CZ gate for different joint states (orange bars). Inset: control sequence. The qubits are prepared at the four different joint states with conditional $\pi$ pulses. In each case, a varying number ($m$) of identical CZ gates are repeatedly applied. The final decay curves (versus $m$) are compared with a reference case, in which CZ gates are replaced by identity gates of the same length, so that additional transition rates or errors due to the pulsing can be extracted (see [25] for details). The additional errors from the shortening of energy relaxation times during pulse are calculated by comparing the effective and idling $T_1$ times in (b) and shown for comparison (purple bars).

Figure 4

Considerations on cross talk and calibration. (a) Layout of a surface-code-compatible qubit array implementing our two-qubit gate scheme. Blue and gray squares denote qubits with different frequencies. The single-qubit ($XY$) control signals of two neighboring qubits ($Q_1$ and $Q_2$) have different frequency components, so their cross talk has little influence on qubits. Although the two-qubit ($Z$) control signals applied to the tunable couplers (purple squares) share the same bandwidth, the flux cross talk between neighboring couplers ($C_1$ and $C_2$) does not necessarily lead to adverse effect, because when the maximum frequency of the coupler is designed at the idling point (minimal residual coupling), an idling coupler becomes insensitive to flux, and also the longitudinal coupling is insensitive to the coupler frequency. Note that neighboring couplers can only be operated in turn in regular quantum circuits. (b) Comparison of the calibration flow. In our adiabatic CZ gate scheme, the qubit-$XY$ signals (for single-qubit gate) are first calibrated before calibrating the coupler-$Z$ signal (for two-qubit gate). In comparison, the iSWAP-like gate scheme [2], which requires stringent resonance condition between qubits demands iterative calibration procedures. For example, the qubit and coupler idling biases and control pulses have to be recalibrated once the other one is changed. The process may be further complicated by cross talk and signal distortion.