Implementation of Conditional Phase Gates Based on Tunable $ZZ$ Interactions

High fidelity two-qubit gates exhibiting low cross talk are essential building blocks for gate-based quantum information processing. In superconducting circuits, two-qubit gates are typically based either on rf-controlled interactions or on the in situ tunability of qubit frequencies. Here, we present an alternative approach using a tunable cross-Kerr-type $ZZ$ interaction between two qubits, which we realize with a flux-tunable coupler element. We control the $ZZ$-coupling rate over 3 orders of magnitude to perform a rapid (38 ns), high-contrast, low leakage ($0.14\pm 0.24\%$) conditional phase $CZ$ gate with a fidelity of $97.9\pm 0.7\%$ as measured in interleaved randomized benchmarking without relying on the resonant interaction with a noncomputational state. Furthermore, by exploiting the direct nature of the $ZZ$ coupling, we easily access the entire conditional phase gate family by adjusting only a single control parameter.

Figure 1

(a) False-colored micrograph of the sample, featuring two transmons used as computational qubits ($Q_1$, blue; $Q_2$, red), a coupler transmon ($C$, purple), its flux line for frequency control (orange), the readout circuitry (green) and additional charge- and flux control lines of $Q_1$, $Q_2$. (b) Schematic illustrating the transversal interactions $g_{12}$, $g_{1\mathrm{c}}$, $g_{2\mathrm{c}}$ between the computational qubits and the coupler qubit. The $ZZ$ coupling (orange arrow) arises from the hybridization of all three nonlinear modes and can be tuned as a function of the coupler frequency.

Figure 2

(a) Measured $ZZ$ coupling strength ${\alpha }_{ZZ}$ vs coupler frequency ${\omega }_{\text{c}}$ (black) and values from a numerical eigenvalue analysis based on the electrical parameters of the device (orange). Data points represented with diamonds are extracted from a flux-pulsed measurement, see Fig. 3. (b) Combined level and pulse diagram used for the measurement of the conditional phase. The control pulse on the coupler is a flat-top Gaussian, parametrized by the total length $\tau$ and the maximal amplitude ${\mathrm{\Phi }}_{\mathrm{c}}$. The conditional phase acquired by $Q_2$ is obtained by determining the difference in phase when the $\pi$ pulse is or is not applied to $Q_1$.

Figure 3

(a) Measured conditional phase ${\phi }_{\text{c}}$ vs pulse length $\tau$ for various pulse amplitudes ${\mathrm{\Phi }}_{\mathrm{c}}$ [color coded, see (b)] and fit to linear model. Black crosses indicate the parameters for which we have performed quantum process tomography. (b) Measured (colored points) and simulated (dashed line) accumulated leakage $p_{\ell }$ vs pulse amplitude ${\mathrm{\Phi }}_{\text{c}}$.

Figure 4

(a) Measured $⟨{\sigma }_z^{(1)}{\sigma }_z^{(2)}⟩$ and (b) accumulated leakage population $p_{\ell }^{\mathrm{tot}}$ in randomized benchmarking (purple) and interleaved randomized benchmarking (orange) vs sequence length $N_{\text{c}}$.