Realization of High-Fidelity CZ and $ZZ$-Free iSWAP Gates with a Tunable Coupler

High-fidelity two-qubit gates at scale are a key requirement to realize the full promise of quantum computation and simulation. The advent and use of coupler elements to tunably control two-qubit interactions has improved operational fidelity in many-qubit systems by reducing parasitic coupling and frequency crowding issues. Nonetheless, two-qubit gate errors still limit the capability of near-term quantum applications. The reason, in part, is that the existing framework for tunable couplers based on the dispersive approximation does not fully incorporate three-body multilevel dynamics, which is essential for addressing coherent leakage to the coupler and parasitic longitudinal ($ZZ$) interactions during two-qubit gates. Here, we present a systematic approach that goes beyond the dispersive approximation to exploit the engineered level structure of the coupler and optimize its control. Using this approach, we experimentally demonstrate CZ and $ZZ$-free iSWAP gates with two-qubit interaction fidelities of $99.76\pm 0.07\%$ and $99.87\pm 0.23\%$, respectively, which are close to their $T_1$ limits.

Popular Summary

The full promise of quantum computation depends on the ability to perform computations with very low error rates. Despite tremendous progress toward this goal with superconducting quantum bits (qubits), a major bottleneck remains with errors in two-qubit gates, one of the fundamental building blocks of quantum computers. Here, we demonstrate a way to sharply reduce errors in two-qubit gates, demonstrating fidelities that are close to their coherence limits.

Operational fidelity in two-qubit gates has vastly improved in recent years thanks to the introduction of tunable couplers, an architectural add-on that can control interactions among neighboring qubits. However, improvements to tunable couplers have run into a roadblock—standard control and design techniques do not account for all the elements that introduce errors through parasitic interactions among qubits or leakage to the coupler itself.

We present a systematic approach to coupler control and design that goes beyond previous methods, allowing us to turn on and off the qubit-qubit coupling that we desire while eliminating unwanted interactions. Using this approach, we demonstrate two-qubit interaction fidelities of nearly 99.8%.

Leakage errors are detrimental to the implementation of quantum error-correcting codes, and gates free of parasitic interactions enable efficient circuit compilation with improved algorithmic accuracy. Taken together, the principles and demonstrations we present will help resolve major challenges facing quantum computing hardware for contemporary applications.

Figure 1

(a) Schematic diagram of a pairwise interacting three-body system. Each constituent body has anharmonic multiple energy level structure. (b),(c) Experimental realization of a three-body system in superconducting circuits. (d) Circuit schematic. In (c), false colors (blue, red, pink, green, and brown) are used to indicate the corresponding circuit components in (d).

Figure 2

The tunable coupling for the CZ and the iSWAP gates. (a) Illustrations of level crossings relevant to the CZ and the iSWAP gates. The energy splittings ($2{\tilde{g}}_{\mathrm{CZ}}$ and $2{\tilde{g}}_{\text{iSWAP}}$) are tunable by adjusting ${\omega }_c$. See the main text for details. (b),(c) Experimental data for the energy exchange between $|200⟩$ and $|101⟩$, and $|100⟩$ and $|001⟩$, as a function of the coupler frequency ${\omega }_c$, respectively. The pulse sequences are illustrated at the top. (d) By fitting the oscillations with sinusoidal curves, we extract the swap rates $|2{\tilde{g}}_{\mathrm{CZ}}|/2\pi$ and $|2{\tilde{g}}_{\text{iSWAP}}|/2\pi$ (circles). The top $x$ axis shows the corresponding perturbation parameter $g_{1c}/({\omega }_c-{\omega }_1)$ at each ${\omega }_c$.

Figure 3

(a),(b) Energy level diagrams of the single- and double-excitation manifolds. The dashed boxes indicate subspaces spanned by energy levels that are relevant to coherent leakage during the CZ gate. The red double-headed arrows denote exchange interactions between the energy levels. (c) Bloch-sphere representation of the relevant subspace in the single-excitation manifold. (d) Bloch-sphere representation of the two-level approximation for the relevant subspace in the double-excitation manifold. When $g_{1c}\gg g_{12}$, and because the transition between $|200⟩$ and $|011⟩$ is dipole forbidden, the state $|101⟩$ primarily interacts with a bright state $|B⟩\equiv \cos \mathrm{\Theta }|011⟩+\sin \mathrm{\Theta }|200⟩$, where $\mathrm{\Theta }\equiv {\tan }^{-1}(\sqrt{2}{g}_{12}/g_{1c})$.

Figure 4

Suppressing leakage to the coupler by optimizing the coupler control. (a),(e) Square-shaped and Slepian-based optimal control waveforms for 60-ns-long CZ gates, respectively. (b),(f) State population of $|101⟩$ after applying repeated CZ gates versus the coupler pulse amplitude ($f_c^{\min }-f_1$) and the number of CZ gates $N_{\mathrm{CZ}}$. (c),(g) Leakage population to $|011⟩$ after applying repeated CZ gates. The square-shaped pulse shows periodic leakage to $|011⟩$, which is suppressed down to the background noise limit by optimizing the pulse shape. (d),(h) Interleaved randomized benchmarking (RB) results of the CZ gates. The pulse sequence is illustrated at the top; we apply $N_{\text{Clifford}}$ random two-qubit Clifford gates ($C_2$) and the recovery Clifford gate $C_2^{-1}$, which makes the total sequence equal to the identity operation. Errors bars represent $\pm 1$ standard deviations. We measure 30 random sequences for each sequence length $N_{\text{Clifford}}$. To ensure accurate uncertainties of the error rates ($r_{\text{Clifford}}$ and $r_{\mathrm{int}}$), we perform a weighted least-squares fit using the inverse of variance as the weights.

Figure 5

Canceling out residual $ZZ$ interaction of the iSWAP by exploiting the engineered coupler level structure. (a) The residual $ZZ$ interaction during iSWAP originates from the level repulsion between $|101⟩$ and the second excited level of the qubits (red arrows). This level repulsion is counteracted by utilizing the level repulsion from the second excited state of the coupler (blue arrow). (b) Residual $ZZ$ strength $\zeta$ as a function of the coupler frequency ${\omega }_c$, when the two qubits are on resonance. The top $x$ axis shows the corresponding perturbation parameter $g_{1c}/({\omega }_c-{\omega }_1)$. The solid curves correspond to numerical simulation assuming either ${\eta }_c=\infty$ (yellow) or ${\eta }_c/2\pi =-90\mathrm{MHz}$ (green). (c) $ZZ$ angle of the iSWAP gate ${\varphi }_{ZZ}/N_{\text{iSWAP}}$ as a function of the gate length $t_G$. We cancel out the $ZZ$ angle by exploiting the tunability of $\zeta$ from positive to negative values. The inset shows the dynamic change of $\zeta$ during the excursion of ${\omega }_c$ for a 30-ns-long iSWAP gate. Each data point is obtained by fitting the accumulated $ZZ$ angle ${\varphi }_{ZZ}$ of $N_{\text{iSWAP}}$-times repeated iSWAP gates with a linear function as shown in (d). (e) The results of interleaved RB for the $ZZ$-free 30-ns-long iSWAP gate (see the main text for details). We measure 30 random sequences for each sequence length ($N_{\text{Clifford}}$). Error bars represent $\pm 1$ standard deviations. The error rates ($r_{\text{Clifford}}$ and $r_{\mathrm{int}}$) and their uncertainties are extracted by performing a weighted least-squares fit using the inverse of variance as the weights.

Figure 6

A schematic diagram of the experimental setup.

Figure 7

$|0⟩\to |1⟩$ transition frequencies of the qubits (red and blue) and the coupler (black). Circles correspond to experimental data. Solid curves correspond to simulations based on the fitted circuit parameters: QB1 ($E_J/h=12.2\mathrm{GHz}$ and $E_c/h=0.195\mathrm{GHz}$), CPLR ($E_J^1/h=46\mathrm{GHz}$, $E_J^2/h=25\mathrm{GHz}$, and $E_c/h=0.085\mathrm{GHz}$), and QB2 ($E_J^1/h=13\mathrm{GHz}$, $E_J^2/h=2.8\mathrm{GHz}$, and $E_c/h=0.19\mathrm{GHz}$), where $E_J$ and $E_c$ denote the corresponding Josephson energy and the charging energy, respectively [28].

Figure 8

Coherence times of QB1 (a), QB2 (b), and CPLR as a function of time.

Figure 9

(a) Energy relaxation times of QB2 as a function of its frequency ${\omega }_2$. Its $T_1$ drops at ${\omega }_2/2\pi =3.75$ and 4.07 GHz due to TLSs. (b) Energy relaxation times of CPLR as a function of its frequency ${\omega }_c$. Its $T_1$ drops at ${\omega }_2/2\pi =5.1$ and 5.9 GHz due to TLSs.

Figure 10

(a) Experimental data of coupler spectroscopy measurement when TLSs appear in the operating frequency range of CPLR. (b) We displace the TLSs from the operating frequency range of the coupler by thermal cycling to room temperature.

Figure 11

Fluctuation in the QB2 frequency as a function of time, measured via repeated Ramsey experiments. We compute the PSD of the frequency fluctuation by using the scipy.signal.welch function from the open-source python library SciPy.

Figure 12

The estimated flux noise PSD affecting QB2 from repeated Ramsey measurements (blue circles and the corresponding fit, orange dashed curve) and spin-echo experiments (green dashed curve).

Figure 13

Single-shot measurements in the $I\text{−}Q$ plane for the qubits (a),(b) and the coupler (c). For the same color points, we repetitively prepare at the corresponding state and measure the $I\text{−}Q$ outcomes (number of repetitions, 10 000). Black markers denote the median points.

Figure 14

Static $ZZ$ interaction strength $\zeta$ as a function of the coupler frequency ${\omega }_c$. QB1 and QB2 are biased at ${\omega }_1/2\pi =4.16\mathrm{GHz}$ and ${\omega }_2/2\pi =4.00\mathrm{GHz}$. The solid black curve corresponds to $\zeta$ obtained by the perturbation theory up to the fourth order without rotating-wave approximation [Eq. (e3)]. The blue dashed curve corresponds to $\zeta$ obtained by numerically diagonalizing the system Hamiltonian [Eq. (1); see Appendix pp17 for the parameters used].

Figure 15

(a) Measurement of a turn-off transient of a $5\text{−}\mu \mathrm{s}$-long $\mathrm{QB}2Z$ pulse (${\tau }_{\text{pulse}}=5\mu \mathrm{s}$) without predistortion (orange crosses) and with predistortion (blue circles). (b) Measurement of a turn-off transient for a $1\text{−}\mu \mathrm{s}$-long CPLR $Z$ pulse (${\tau }_{\text{pulse}}=1\mu \mathrm{s}$). (c) Measurement of a turn-off transient of flux cross talk from the CPLR’s flux line to QB2’s SQUID. A $3\text{−}\mu \mathrm{s}$-long CPLR $Z$ pulse (${\tau }_{\text{pulse}}=3\mu \mathrm{s}$) is applied. The pulse sequences are illustrated in the insets.

Figure 16

Energy level diagrams of the single-excitation manifold (a) and the double-excitation manifold (b) when performing the CZ gate. The dashed boxes indicate subspaces spanned by energy levels that are relevant to coherent leakage during the CZ gate. The red double-headed arrows denote exchange interactions between the energy levels.

Figure 17

Energy level diagrams of the single-excitation manifold (a) and the double-excitation manifold (b) when performing the iSWAP gate. The dashed boxes indicate subspaces spanned by energy levels that are relevant to coherent leakage during the iSWAP gate. The red double-headed arrows denote exchange interactions between the energy levels.

Figure 18

Numerical simulation of coherent leakage of CZ gates. (a) Square-shaped and optimal control waveforms for 60-ns-long CZ gates, respectively. (b) Coherent leakage in the double-excitation manifold. We prepare $|101⟩$, apply a control pulse, and then measure the state populations. By using the optimal pulse shaping, we suppress population of the leakage state $|011⟩$ (orange curve) below ${10}^{-7}$ for pulses longer than 60 ns. (c) Coherent leakage in the single-excitation manifold. We prepare $|100⟩$, apply a control pulse, and then measure the state populations. By using optimal pulse shaping, we suppress population of the leakage state $|010⟩$ (blue curve) below ${10}^{-7}$ for pulses longer than 60 ns. The leakage to $|001⟩$ is not suppressed as much, since the optimal control relies on the effective Hamiltonian $H_{\mathrm{eff}}^{\mathrm{CZ}}(t)$ that addresses only leakage from $|100⟩$ to $|010⟩$ in the single-excitation manifold. The data points in (b),(c) are obtained every 1 ns.

Figure 19

Numerical simulation of coherent leakage of iSWAP gates. (a) Square-shaped and optimal control waveforms for 30-ns-long iSWAP gates, respectively. (b) Coherent leakage in the double-excitation manifold. We prepare $|101⟩$, apply a control pulse, and then measure the state populations. By using optimal pulse shaping, we suppress population of the leakage states $|110⟩$ and $|011⟩$ (blue and orange curves) below ${10}^{-7}$ for pulses longer than 30 ns (black dashed line). The leakage to $|200⟩$ and $|002⟩$ is not suppressed as much, since the optimal control relies on the effective Hamiltonian $H_{\mathrm{eff}}^{\text{iSWAP}}(t)$ that addresses only the leakage from $|101⟩$ to $|110⟩$ and $|011⟩$ in the double-excitation manifold. (c) Coherent leakage in the single-excitation manifold. We prepare $|101⟩$, apply a control pulse, and then measure the state populations. By using the optimal pulse shaping, we suppress population of the leakage state $|010⟩$ (blue curve) below ${10}^{-7}$ for pulses longer than 30 ns (black dashed line). The data points in (b),(c) are obtained every 1 ns.

Figure 20

Numerical simulations of (a) coupling strength ${\tilde{g}}_{\text{iSWAP}}$ and (b) residual $ZZ$ strength ${\zeta }_{\text{iSWAP}}$ for an iSWAP interaction versus the coupler anharmonicity ${\eta }_c$ ($y$ axis) and the coupler frequency ${\omega }_c$ ($x$ axis), where ${\omega }_1/2\pi ={\omega }_2/2\pi =4.16\mathrm{GHz}$. Note that smaller ${\eta }_c$ enables $ZZ$-free iSWAP interaction (${\zeta }_{\text{iSWAP}}=0$, red dashed curves) with stronger ${\tilde{g}}_{\text{iSWAP}}$, thereby enabling faster $ZZ$-free iSWAP gates.

Figure 21

Numerical simulations of (a) coupling strength ${\tilde{g}}_{\mathrm{CZ}}$ for the CZ interaction and (b) the state overlap $|⟨020|\widetilde{101}⟩{|}^2+⟨020|\widetilde{200}⟩{|}^2$ versus CPLR anharmonicity ${\eta }_c$ ($y$ axis) and CPLR frequency ${\omega }_c$ ($x$ axis), where $({\omega }_1+{\eta }_1)/2\pi ={\omega }_2/2\pi =3.94\mathrm{GHz}$. Note that smaller ${\eta }_c$ prevents $|020⟩$ from being strongly hybridized with $|101⟩$ and $|200⟩$ during the CZ interaction while keeping ${\tilde{g}}_{\mathrm{CZ}}$ constant; this constraint enables us to simplify the multilevel leakage dynamics and use the Slepian-based optimal control.

Figure 22

Measurements and cancellation of a microwave cross talk. (a),(b) Rabi oscillations of QB1 (blue) and QB2 (orange) when driving through the QB2 local drive line. (c),(d) Rabi oscillations of QB1 (blue) and QB2 (orange) when driving through the QB1 local drive line. (e) Rabi oscillations of QB2 when driving through the QB1 local drive line.

Figure 23

Experimental results of single-qubit randomized benchmarking, when QB1 and QB2 are detuned by 150 MHz. (a) Single-qubit RB measurement data for QB1. (b) Single-qubit RB measurement data for QB2. “S” denotes the simultaneous application of single-qubit Cliffords, and “I” denotes the isolated application of single-qubit Cliffords. The pulse duration of $X$- and $Y$-rotation gates is 30 ns. QB1 and QB2 are biased at 4.16 and 4.00 GHz, respectively. The gate fidelities are degraded, when measured simultaneously, possibly due to microwave cross talk. The data are averaged over 20 random sequences for each sequence length.

Figure 24

Experimental results of single-qubit randomized benchmarking, when QB1 and QB2 are in resonance. (a) Single-qubit RB measurement data for QB1. (b) Single-qubit RB measurement data for QB2. S denotes the simultaneous application of single-qubit Cliffords, and I denotes the isolated application of single-qubit Cliffords. The pulse duration of $X$- and $Y$-rotation gates is 70 ns. Both QB1 and QB2 are biased at 4.16 GHz. We apply cancellation pulses to offset microwave cross talk (orange and green circles) and reduce the gate errors by more than a factor of 10. The data are averaged over 20 random sequences for each sequence length.

Figure 25

Two-qubit Clifford classes.

Figure 26

(a) Decomposition of the cnot gate into the iSWAP gates. (b) Decomposition of the SWAP gate into the iSWAP gates.

Figure 27

Two-qubit Clifford classes written in terms of the iSWAP gate and single-qubit gates. Since our native iSWAP gate accompanies additional unwanted single-qubit $Z$ rotations, we incorporate compensating $Z$ rotations into the existing single-qubit gates that are either preceded or followed by the iSWAP gate to undo the unwanted $Z$ rotations. The orange arrows denote which single-qubit gates are subject to be updated to undo the $Z$ rotations of the iSWAP.

Figure 28

(a) A diagram of the standard (or reference) two-qubit RB sequence. (b) A diagram of the two-qubit RB sequence interleaved by the iSWAP gate. The additional unwanted $Z$ rotations of the interleaved iSWAP gate are canceled out by the subsequent two-qubit Clifford (orange arrows).

Figure 29

Tune-up measurements for the CZ gate. (a),(b) Experimental data of tune-up measurements for a 60-ns-long CZ gate. We measure leakage from $|101⟩$ and conditional phase (CPHASE) angle ${\varphi }_{\mathrm{CZ}}$ as functions of QB2 $Z$-pulse amplitude ($x$ axis) and CPLR $Z$-pulse amplitude ($y$ axis). The control pulse and sequences to measure leakage and CPHASE angle are illustrated at the top. We find an optimal parameter set that minimizes both the leakage and CPHASE angle error (red circles). (c),(d) Numeric simulation reproducing the experimental data of tune-up measurements.

Figure 30

Experimental data of fine-tuning measurements for the CZ gate. (a) Measuring the leakage of multiple CZ pulses to finely adjust the CPLR $Z$ amplitude ($f_c^{\min }$). (b) Measuring the CPHASE angle error (${\varphi }_{\mathrm{CZ}}-N_{\mathrm{CZ}}\times 180°$) of multiple CZ pulses to finely adjust the QB2 $Z$ amplitude ($f_2^{\mathrm{peak}}$).

Figure 31

Tune-up measurement for the iSWAP gate. (a) Experimental data of tune-up measurements for a 30-ns-long iSWAP gate. We measure the iSWAP angle ${\theta }_{\text{iSWAP}}$ as functions of $\mathrm{QB}2Z$-pulse amplitude ($x$ axis) and CPLR $Z$-pulse amplitude ($y$ axis). The control sequence to measure the iSWAP angle is illustrated at the top. A red circle denotes an optimal parameter set that maximizes ${\theta }_{\text{iSWAP}}$. (b) Numeric simulation reproducing the experimental data of tune-up measurements.

Figure 32

Experimental data of fine-tuning measurements for the iSWAP gate. (a) Measurements of the swap angle ${\theta }_{\text{iSWAP}}$ for multiple iSWAP pulses to finely adjust the CPLR $Z$ amplitude ($f_c^{\min }$). (b) Measurements of the swap angle ${\theta }_{\text{iSWAP}}$ for multiple iSWAP pulses to finely adjust the QB2 $Z$ amplitude ($f_2^{\mathrm{peak}}$).

Figure 33

Characterization of the leakage rate of the CZ gate with an optimized pulse shape. Population in the computational subspace $p_{{\mathcal{X}}_1}=p_{|000⟩}+p_{|001⟩}+p_{|100⟩}+p_{|101⟩}$ for the reference and interleaved two-qubit randomized benchmarking sequences.

Figure 34

Implementation of arbitrary $Z$ gates by combining $X$ and $Y$ rotations. Experimental data of the Ramsey-type experiment to validate the Euler-$Z$ gate. We vary the angle ${\theta }_z$ of $Y$ rotation, which effectively adjusts the rotation angle of the Euler-$Z$ gate.

Figure 35

Numerical simulation results for the average gate errors of the CZ and the iSWAP gates. (a) The average gate infidelity ($1-F_g$) of the iSWAP gate as a function of its gate length $t_G$. For each gate length, the control pulse shape is optimized as detailed in Appendixes pp7 and pp8. Each data point is obtained by numerically simulating quantum process tomography and computing the corresponding gate infidelity. (b) The average gate infidelity ($1-F_g$) of the iSWAP gate as a function of its gate length $t_G$. The lowest gate error is achieved when $t_G\approx 25\mathrm{ns}$. At this point, the residual $ZZ$ interaction of the iSWAP is minimized, owing to the cancellation induced by the higher level of CPLR.

Figure 36

Contribution of $1/f^{\alpha }$ flux noise to error rates of CZ gates. Notably, the error contribution from CPLR flux noise increases as the gate duration decreases (orange circles), possibly due to stronger qubit-coupler hybridization.

Figure 37

Contribution of $1/f^{\alpha }$ flux noise to error rates of iSWAP gates. Notably, the error contribution from CPLR flux noise increases as the gate duration decreases (orange circles), possibly due to stronger qubit-coupler hybridization.

Figure 38

State population of (a) $|101⟩$, (b) $|200⟩$, (c) $|011⟩$, (d) $|110⟩$, (e) $|020⟩$, and (f) $|002⟩$ for the repeated square CZ pulses.

Figure 39

State population of (a) $|101⟩$, (b) $|200⟩$, (c) $|011⟩$, (d) $|110⟩$, (e) $|020⟩$, and (f) $|002⟩$ for the repeated optimal CZ pulses.