Tunable Coupler for Realizing a Controlled-Phase Gate with Dynamically Decoupled Regime in a Superconducting Circuit

Controllable interaction between superconducting qubits is desirable for large-scale quantum computation and simulation. Here, based on a theoretical proposal by Yan et al. [Phys. Rev. Appl. 10, 054061 (2018)] we experimentally demonstrate a simply designed and flux-controlled tunable coupler with a continuous tunability by adjusting the coupler frequency, which can completely turn off adjacent superconducting qubit coupling. Utilizing the tunable interaction between two qubits via the coupler, we implement a different type of controlled-phase (cz) gate with “dynamically decoupled regime,” which allows the qubit-qubit coupling to be only “on” at the usual operating point while dynamically “off” during the tuning process of one qubit frequency into and out of the operating point. This scheme not only efficiently suppresses the leakage out of the computational subspace, but also allows for the acquired two-qubit phase being geometric at the operating point. We achieve an average cz gate fidelity of $98.3\pm 0.6\mathrm{\%}$, which is dominantly limited by qubit decoherence. The demonstrated tunable coupler provides a desirable tool to suppress adjacent qubit coupling and is suitable for large-scale quantum computation and simulation.

Figure 1

(a) Optical micrograph of three Xmon qubits with the middle one $C$ serving as a tunable coupler for the two computational qubits ($Q_1$ and $Q_2$). Each computational qubit has independent $XY$ and $Z$ control, and is coupled to a separate $\lambda /4$ resonator for simultaneous and individual readout. The coupler also has an individual flux-bias line for a frequency tunability. The combination of direct capacitive coupling and indirect tunable coupling via the coupler constitutes the total coupling between the two computational qubits. (b) Schematic electrical circuit of the device.

Figure 2

(a) Pulse sequence to characterize the tunability of the coupler. The two qubits are initialized in the ground state at their sweet spots with a detuning of 35 MHz. The coupler is originally set at ${\omega }_c/2\pi =5.905$ GHz where the coupling between the two qubits is nearly off. A $\pi$ pulse is first to excite $Q_2$, followed by two simultaneous fast flux pulses: $f_1$ brings $Q_1$ into resonance with $Q_2$; $f_c$ on the coupler to turn on the coupling. After the two qubits interact and evolve for time $t$, $Q_1$ and the coupler are pulsed back to the original points for measurements of qubit populations. (b) Population of $Q_2$ as a function of the amplitude of $f_c$ and $t$ clearly reveals the tunability of the coupling strength. The two dark dashed lines indicate the situation where the coupling is off. (c) Population of $Q_1$ as a function of time with the coupling on (orange dots) or off (black dots). (d) The effective qubit-qubit coupling strength $\tilde{g}/2\pi$ (black circles) extracted by fitting the oscillation of the qubit excitation in (b) as a function of the flux-bias amplitude on the coupler. The red line is a fit to the extracted $\tilde{g}$ according to Eq. (3). The coupler frequency (blue dots) can be measured independently by probing the dispersive shift of the qubit frequency when pulsing the coupler into the excited state. Inset: the $ZZ$ crosstalk coupling ${\xi }_{ZZ}/2\pi$ measured in a Ramsey-type experiment when the two qubits are detuned and at their sweet spots. The red arrow indicates where the coupling is off. (e) Individual and simultaneous RB for $Q_1$ and $Q_2$ with ${\xi }_{ZZ}/2\pi \approx 0$ and $-0.45$ MHz, respectively.

Figure 3

(a),(b) Bar charts of the measured ${\chi }_{\exp }$ from a QPT for $i$swap and $\sqrt{i\mathrm{swap}}$ with a gate fidelity of 96.8% and 95.0%, respectively. The solid black outlines are for the ideal gate.

Figure 4

(a) Schematic to realize the cz gate using the usual $\left|11\right.⟩$ and $\left|20\right.⟩$ resonance. (b) Flux sequence to realize the cz gate with DDR. The $Q_1$ frequency is unchanged during the process. A gradual flux pulse (for a cosine-type frequency adjustment) on $Q_2$ is to tune its frequency to the operating point, while a flux pulse on the coupler with an appropriate pulse shape to dynamically keep the coupling “off” (the shaded regions). The coupling is then turned on by applying another gradual flux pulse on the coupler (for a cosine-type coupler frequency adjustment) for proper two-qubit interaction time to acquire the required $\pi$-phase shift on $\left|11\right.⟩$. (c),(d) Confirmation of the “off” state of the coupling in the DDR and the corresponding measurement sequences. Top: the population of $\left|11\right.⟩$ state shows purely exponential decay without any oscillation. Middle: phase accumulation of $\left|01\right.⟩$ with respect to $\left|00\right.⟩$. Bottom: phase accumulation of $\left|11\right.⟩$ with respect to $\left|10\right.⟩$. The difference between the lower two panels shows zero phase accumulation of $\left|11\right.⟩$, confirming zero coupling. (e) Population of $\left|11\right.⟩$ in three different schemes to realize the cz gate. Upper: rectangular pulses to have a positive coupling strength. $t_R=222$ ns is required for realizing the standard two-qubit cz gate. Middle: rectangular pulses to have a negative coupling regime. Obvious high-frequency oscillations occur and indicate the leakage out of $\left|11\right.⟩$. Lower: DDR pulses to have a large negative coupling but with a smooth oscillation. A cz gate can be accomplished with a gate time $t_D$=119 ns. (f) Bar chart of the measured real part of ${\chi }_{\exp }$ from a QPT shows a gate fidelity of $98.3\pm 0.6$%. The solid black outlines are for the ideal gate.

Figure 5

$ZZ$ crosstalk coupling ${\xi }_{ZZ}$ versus coupler frequency when the two qubits are detuned and at their sweet spots as in the inset of Fig. 2 of the main text. Inset: the experimental sequence for measuring $ZZ$ crosstalk coupling ${\xi }_{ZZ}$ at a specific coupler flux bias.

Figure 6

Schematic electrical circuit of the device.

Figure 7

Measurement setup. Our measurement circuit contains three AWGs (two Tek5014C and one Tek70002A), three signal generators, and other microwave components. The $XY$ control pulses are generated from a signal generator modulated by the AWG. Flux pulses are generated directly from Tek5014C (for $Q_1$) and Tek70002A (for $Q_2$ and the coupler). Readout signal is amplified by a JPA at the base, a HEMT amplifier at 4 K and two room-temperature amplifiers, and finally down-converted and digitized by an analog-to-digital converter (ADC).

Figure 8

Measured coupler spectrum. We drive the coupler through the $XY$ control line of $Q_1$ by applying a microwave pulse at variable frequency with a rectangular envelope of a duration of 500 ns, followed by a population measurement of $Q_2$ with a wide selective Gaussian pulse. Inset: schematic of the measurement. The two computational qubits ($Q_1$ and $Q_2$) are shown in blue and green respectively, and the coupler is shown in purple.

Figure 9

Qubit-coupler anticrossing. At the resonance point, anticrossing of the energy level is observed and the separation characterizes the coupling strength.

Figure 10

Lower threshold for the coupler frequency. (a) Pulse sequences to measure the operation range with negative coupling strength. (b) An enlarged part of Fig. 2 in the main text. The swap interaction is completely turned off at the marked dark dashed line, while the red dashed line indicates the approximate threshold for non-negligible excitation of the coupler. (c) Direct leakage experiment for finding the approximate threshold of the coupler frequency. The two qubits are initialized in $\left|11\right.⟩$, then are tuned into resonance while varying the coupler frequency, and finally the population of $\left|11\right.⟩$ state is measured. The red dashed line shows the threshold of having leakage out of the computational space.

Figure 11

Qubit coherence times versus coupler frequency. When the coupler frequency is above the threshold for non-negligible excitation of the coupler, the qubit coherence times remain nearly unaffected. Three specific coupling strengths are marked by the black arrows.

Figure 12

cz gate calibration. (a) Experimental sequences to measure ${\varphi }_{10}$, phase accumulation of $\left|01\right.⟩$ with respect to $\left|00\right.⟩$, and ${\varphi }_{11}$, phase accumulation of $\left|11\right.⟩$ with respect to $\left|10\right.⟩$. (b) Probabilities of measuring $\left|01\right.⟩$ and $\left|11\right.⟩$ versus the phase of the second $\pi /2$ pulse. The sinusoidal oscillations reveal the phases acquired in the Ramsey measurements for the two-qubit states $\left|01\right.⟩\to e^{i{\varphi }_{01}}\left|01\right.⟩$ and $\left|11\right.⟩\to e^{i{\varphi }_{11}}\left|11\right.⟩$, respectively. The solid lines are fits to sinusoidal oscillations. With a standard cz gate time of $t_R=222$ ns, a $\pi$-phase shift between the orange and green curves is observed.

Figure 13

Simulation of the population evolution of each state during the cz gate with an initial joint qubit-coupler state $\left|Q_{1}C{Q}_2\right.⟩=\left|+0+\right.⟩$. (a) cz gate with rectangular flux pulses. (b) cz gate with DDR.