Parametric-Resonance Entangling Gates with a Tunable Coupler

High-fidelity parametric gates have been demonstrated with superconducting qubits via rf flux modulation of the qubit frequency. The modulation however leads to renormalization of the bare qubit-qubit coupling, thereby reducing the gate speed. Here, we realize a parametric-resonance gate, which is activated by bringing the average frequency of the modulated qubit in resonance with a static-frequency qubit while approximately retaining the bare qubit-qubit coupling. The activation of parametric-resonance gates does not depend on the frequency of modulation, allowing us to choose the modulation frequencies and avoid frequency collisions. Moreover, we show that this approach is compatible with tunable coupler architectures, which reduce always-on residual couplings. Using these techniques, we demonstrate iswap and controlled-$Z$ gates between two qubits coupled via a tunable coupler with average process fidelities as high as $99.3\mathrm{\%}$ and $97.9\mathrm{\%}$, respectively. The flexibility in activating parametric-resonance gates combined with a tunable coupler architecture provides a pathway for building large-scale quantum computers.

Figure 1

Schematic of the device used in the experiment. The floating qubits (fuchsia and teal) are coupled to rf (blue) and flux (brown) control lines, while the tunable coupler (yellow) is connected to just a flux line. Both qubits and coupler are connected to individual readout resonators that in turn are coupled to a feed line (gray).

Figure 2

(a) Pulse sequence to measure qubit-qubit coupling under modulation as a function of the coupler flux bias. (b) Measured population of $|01⟩$ versus pulse duration $\tau$ and coupler flux bias ${\mathrm{\Phi }}_c$. (c) Bare qubit-qubit coupling $g_{01}/2\pi$ (dotted fuchsia) [measured by replacing the rf flux pulse in (a) by a fast flux pulse] and effective qubit-qubit coupling ${\tilde{g}}_{01}/2\pi$ (star teal) versus coupler flux bias.

Figure 3

(a) Average frequency of qubit 2 as a function of the flux modulation amplitude. The iswap gate is activated at resonance with ${\omega }_1/2\pi$ (solid line) and cz20 at $({\omega }_1-{\eta }_1)/2\pi$ (dashed line). (b) Qubit 2 flux modulation frequency ${\omega }_p$ versus flux modulation amplitude for $n=-2$ sideband [${\omega }_{p,\mathrm{i}\mathrm{swap}}=({\overline{\omega }}_2-{\omega }_1)/2$, ${\omega }_{p,\mathrm{cz}02}=({\overline{\omega }}_2-{\omega }_1-{\eta }_2)/2$, ${\omega }_{p,\mathrm{cz}20}=({\overline{\omega }}_2-{\omega }_1+{\eta }_1)/2$]. The plots are generated using the measured parameters of the qubits and the coupler. (c) Measured energy exchange (chevron) between $|10⟩$ and $|01⟩$ for activating the parametric-resonance iswap interaction once a $\pi$ pulse has been applied to qubit 1.

Figure 4

(a) The iswap quantum process tomography with an iswap fidelity of $99.3\mathrm{\%}$. (b) Reconstructed quantum process tomography of $\mathrm{fSim}(\theta ,\varphi )$ at angles $\theta =-1.57$, $\varphi =0.076$, yielding a maximum possible fidelity against an arbitrary fSim of $99.4\mathrm{\%}$.

Figure 5

(a) Quantum process tomography of a cz gate with a fidelity of $97.9\mathrm{\%}$. (b) Reconstructed quantum process tomography of $\mathrm{fSim}(\theta ,\varphi )$ at angles $\theta =-0.006$, $\varphi =2.8$, yielding a maximum possible fidelity against an arbitrary fSim of $98.4\mathrm{\%}$.

Figure 6

(a) Schematics of two floating transmon qubits (fuchsia and teal) coupled to a grounded tunable coupler (yellow). (b) Lumped-element circuit representation of the qubit-coupler-qubit system with the numbers showing the nodes that define the node fluxes used to derive the Hamiltonian.

Figure 7

Overview of the experimental setup. This diagram details the control electronics, wiring, and filtering for the two qubits and the coupler. The readout signal, rf signal ($XY$ lines), and dc and ac signals for flux delivery ($Z$ lines) are generated by custom Rigetti arbitrary waveform generators. All control lines go through various stages of attenuation and filtering in the dilution refrigerator.

Figure 8

Coupler frequency. Tunable coupler $|0⟩\to |1⟩$ transition frequency ${\omega }_c/2\pi$ versus tunable coupler fast flux amplitude ${\mathrm{\Phi }}_c$. The circles are measured data points and the solid fuchsia line is the fit to a transmon model. The black solid lines are the qubits’ maximum frequencies.

Figure 9

Qubit-qubit couplings. Bare net qubit-qubit coupling strength $g_{01}/2\pi$ (dotted fuchsia) and renormalized net qubit-qubit coupling ${\tilde{g}}_{01}/2\pi$ (star teal) versus coupler flux bias. This shows that the measured bare and the renormalized coupling rates are the same within the accuracy of the measurement.

Figure 10

Flux pulse transfer function. Achieved flux modulation amplitude for a given requested amplitude (0.3 V) versus modulation frequency.

Figure 11

Quantum process tomography fidelity of the iswap gate presented in the main text versus time. After each tomography measurement, the virtual $Z$ gates are updated.