Current address: ]Nord Quantique, Sherbrooke, Québec, J1K 0A5, Canada

Quantum information processing with superconducting qubits has been intensively studied recently. High-fidelity quantum manipulations and projective measurements have been achieved in multi-qubit systems martinis2014 ; ibm2016 ; rigetti ; martinis2019 ; johnson2020 , and basic quantum error-correction protocols have been demonstrated schoelkopf2012 ; martinis2015 ; schoelkopf2016 ; correct2019 ; wallraf2019 . For fault-tolerant quantum computing, however, the gate and readout fidelity should be further improved by a few orders of magnitude fowler ; terhal .

To this end, a variety of two-qubit gates have been proposed and demonstrated. These can be classified into two groups, based on their use of either a coupling between (near-)degenerate qubits martinis2014 ; martinis2019 or a microwave-induced parametric coupling nakamura2007 ; ibm2011 ; ibm2016 ; rigetti ; johnson2020 ; screview . For the gate operation, the former usually requires fast frequency tuning of the qubits and/or a coupler through a flux bias, while the latter only uses microwave pulses for the dynamical control. For the parametric gates, qubits are usually far off-resonant from each other in order to suppress residual couplings between them. On the other hand, a large detuning slows down the parametric gate, causing a trade-off that hinders the improvement of the gate fidelity. Recent works have addressed this issue by introducing various types of coupler circuits to eliminate the residual coupling without sacrificing the gate speed significantly DiVincenzo2017 ; steele2018 ; houck2019 ; duan2020 . A simpler scheme combining two qubits with opposite signs of anharmonicity also allows a residual-coupling-free two-qubit cross-resonance gate plourde2020 .

In this letter, we propose and demonstrate fast parametric two-qubit gates using sideband transitions between an ordinary transmon qubit and a parity-violated superconducting qubit, which we call a cubic transmon. The parity violation originates in a cubic component of the inductive potential of a Josephson-junction circuit under a finite magnetic flux bias. This circuit, known as a superconducting nonlinear asymmetric inductive element (SNAIL), was recently proposed SNAIL and utilized in parametric amplifiers SNAIL2 , bosonic-mode qubits SNAIL3 , and hybrid quantum systems noguchi2018 . The parity symmetry breaking is essential for a physical system to acquire a second-order nonlinearity, which allows three-wave-mixing-type first-order sideband transitions and thus the parametric interactions with a neighboring qubit nakamura2015 ; sideband ; DiVincenzo2016 . The large capacitive coupling strength between the qubits introduces strong parametric interactions, but also a large residual static interaction. We eliminate the latter by using ac Stark shifts induced by a continuous near-resonant drive of the sideband transitions and solve the trade-off. This approach of microwave-assisted elimination of the static interactions brings in more tuning knobs, i.e. amplitudes and frequencies of multiple drives, which can be applied to the cases with multiple qubits and higher-order residual couplings.

Figures 1(a) and 1(b) present optical micrographs and a circuit diagram of the device, which contains two superconducting qubits and two resonators for the dispersive readout of each qubit. The qubit on the right-hand side is a conventional transmon, which is composed of a capacitively-shunted single Josephson junction koch2007 . The other qubit is a cubic transmon, which is a capacitively-shunted SNAIL circuit. The SNAIL is a Josephson-junction loop formed by a parallel circuit of a single small Josephson-junction and two large Josephson junctions. The SNAIL loop is threaded by a flux $\Phi$. The Hamiltonian of the two-qubit system reads

where $\hbar=h/2\pi$ is the reduced Planck constant, $\omega_{\mathrm{c0}}$ and $\omega_{\mathrm{t0}}$ are the bare eigenmode frequencies, and $\hat{a}$ and $\hat{b}$ are the annihilation operators for the cubic transmon and conventional transmon, respectively. The coefficient $\beta_{\mathrm{c0}}$ is the second-order nonlinearity of the cubic transmon, $\alpha_{\mathrm{c0}}$ and $\alpha_{\mathrm{t0}}$ are the third-order nonlinearities of each qubit, and $g_{0}$ is the capacitive coupling strength between the two qubits.

In the dispersive coupling regime $|\Delta_{0}|\equiv|\omega_{\mathrm{c0}}-\omega_{\mathrm{t0}}|\gg g_{0}$, the effective Hamiltonian can be written as

where $g$($\propto\beta_{\mathrm{c0}}$) is the effective coupling strength, and $\omega_{\mathrm{c}}$, $\omega_{\mathrm{t}}$, $\alpha_{\mathrm{c}}$, and $\alpha_{\mathrm{t}}$ are the eigenmode frequencies and self-Kerr nonlinearities of the qubits after the perturbative treatment of the coupling term in Eq. (1), respectively. The term with a coefficient $g$ arises from the second-order nonlinearity in the parity-violated cubic transmon and gives the interaction in the same form as the radiation pressure in optomechanics optomechanics ; noguchi2018 and the state-dependent force in trapped ions blatt2008 ; ion . There is also a static longitudinal (ZZ) interaction between the qubits, whose amplitude is $J_{\mathrm{ZZ}}$. The detailed derivations and expressions of the parameters in Eqs. (1) and (2) are presented in the Supplementary Material supple .

Figure 1(c) illustrates the eigenstates $\widetilde{|ij\rangle}$ $(i,j\in\{g,e,f\})$ and their frequencies. The coupling between qubits hybridizes the bare qubit states, $|i\rangle_{\mathrm{c}}|j\rangle_{\mathrm{t}}$, and forms the eigenstates $\widetilde{|ij\rangle}$ supple . When a drive field at frequency $\omega_{\mathrm{d}}=\Delta\equiv\omega_{\mathrm{t}}-\omega_{\mathrm{c}}$ is applied to the cubic transmon, the two qubits resonate with each other in the rotating frame. Under the rotating-wave approximation, the parametric coupling follows

where $\Omega$ and $\theta$ are the amplitude and phase of the drive field, respectively. Note that $\eta$ is proportional to the second-order nonlinearity $\beta_{\mathrm{c0}}$ for the three-wave-mixing process occurring under the Hamiltonian $\hat{H}_{\mathrm{p}}$ in Eq. (3).

Under the resonant condition $\omega_{\mathrm{d}}=\Delta$, the drive exchanges the excitation of the two qubits, and thus the iSWAP and SWAP gates can be implemented. Another type of two-qubit gate, controlled-phase (CZ) gate, is similarly achieved with a parametric drive. When the drive frequency is equal to $\Delta+\alpha_{\mathrm{t}}$, the transition $\widetilde{|ee\rangle}\leftrightarrow\widetilde{|gf\rangle}$ takes place. A $2\pi$-rotation of the transition induces a geometric phase factor of $-1$ only to the $\widetilde{|ee\rangle}$ state in the computational subspace.

In parallel with the dynamically-induced coupling, there remains the spurious static ZZ interaction, the last term in Eq. (2), between the capacitively coupled qubits with higher energy levels ibm2019 . Remarkably, the residual interaction can be eliminated also with a parametric drive. We irradiate the cubic transmon with a continuous-wave (CW) microwave field, whose frequency is slightly detuned from the transition of $\widetilde{|ee\rangle}\leftrightarrow\widetilde{|fg\rangle}$. The ac Stark effect by the CW drive shifts the eigenfrequencies in the two-qubit subspace. These shifts give rise to a tunable ZZ interaction and allow compensation for the unwanted interaction.

In the experiment, we use the device shown in Fig. 1. The parameters at the operating flux bias, $\Phi=0.34\Phi_{0}$, where $\Phi_{0}\equiv h/2e$, are the followings: The eigenfrequencies of the cubic transmon and the transmon are $\omega_{\mathrm{c}}/2\pi=3.633\mathrm{~{}GHz}$ and $\omega_{\mathrm{t}}/2\pi=4.479\mathrm{~{}GHz}$, respectively. The third-order nonlinearities of the qubits are $\alpha_{\mathrm{c}}/2\pi=-132~{}\mathrm{MHz}$ and $\alpha_{\mathrm{t}}/2\pi=-168~{}\mathrm{MHz}$, and the bare coupling strength between the qubits is $g_{0}/2\pi=75\mathrm{~{}MHz}$, which are determined by spectroscopic measurements. The details of the sample characterization are described in the Supplementary Material supple . Using these values, we estimate the second-order nonlinearity $\beta_{\mathrm{c}}/2\pi=-195\mathrm{~{}MHz}$, the effective coupling strength $g/2\pi=-14\mathrm{~{}MHz}$, and the coupling coefficient of the parametric drive, $\eta=0.022$. The energy-relaxation and Ramsey-dephasing times of the qubits are $T_{1}=3.9\mathrm{~{}\mu s}$ and $T_{2}^{*}=0.6\mathrm{~{}\mu s}$ for the cubic transmon, and $T_{1}=4.0\mathrm{~{}\mu s}$ and $T_{2}^{*}=2.3\mathrm{~{}\mu s}$ for the transmon, respectively. The dephasing time of the cubic transmon is improved to $T_{2}^{\mathrm{E}}=1.5\mathrm{~{}\mu s}$ with an echo pulse, while no change is seen for the transmon.

We first eliminate the residual ZZ interaction by the CW drive (Fig. 2). The drive frequency is 930 MHz, and the detuning from the $\widetilde{|ge\rangle}\leftrightarrow\widetilde{|eg\rangle}$ transition is 84 MHz. The inset of Fig. 2 shows the shifts of eigenstates induced by the CW drive. The CW drive is red detuned from the $\widetilde{|ee\rangle}\leftrightarrow\widetilde{|fg\rangle}$ transition and blue detuned from the $\widetilde{|ge\rangle}\leftrightarrow\widetilde{|eg\rangle}$ and $\widetilde{|ee\rangle}\leftrightarrow\widetilde{|gf\rangle}$ transitions. Thus, the sign of the frequency shift is different from each other. The amplitude of the ZZ interaction corresponds to the frequency difference between the $\widetilde{|gg\rangle}\leftrightarrow\widetilde{|ge\rangle}$ and $\widetilde{|eg\rangle}\leftrightarrow\widetilde{|ee\rangle}$ transitions, which amounts to $5$ MHz in the absence of the CW drive. The frequency difference vanishes at a certain power of the drive. It is also found that the CW drive does not degrade the coherence of the qubits (data not shown).

In the presence of the CW drive, we implement the two-qubit Clifford gate set, i.e. CZ, iSWAP and SWAP gates, using parametric couplings induced by additional microwave pulses. These gates are within the family of the Fermionic Simulation gate set, characterized by two parameters, the swap angle $\theta_{\mathrm{sw}}$ and the conditional phase $\theta_{\mathrm{cp}}$ fermi2018 ; martinis2020 . The CZ, iSWAP, and SWAP gates have the parameters $(\theta_{\mathrm{sw}},\theta_{\mathrm{cp}})=(0,\pi)$, $(\pi,0)$ and $(\pi,\pi)$, respectively. In our setup, $\theta_{\mathrm{sw}}$ and $\theta_{\mathrm{cp}}$ are independently and simultaneously controlled via parametric couplings.

Figure 3(a) illustrates the waveforms of the synthesized two-tone pulses for these gates. The total gate time is 50 ns for each. The swap pulse (blue) is resonant to the $\widetilde{|ge\rangle}\leftrightarrow\widetilde{|eg\rangle}$ transition and is used to control $\theta_{\mathrm{sw}}$. The control-phase pulse (orange) is resonant to the $\widetilde{|ee\rangle}\leftrightarrow\widetilde{|gf\rangle}$ transition and controls $\theta_{\mathrm{cp}}$ through the relative phase between two serial segments, applying the conditional phase $\theta_{\mathrm{cp}}$ as a geometrical phase only to the $\widetilde{|ee\rangle}$ state. Because the swap pulses also generate a small conditional phase due to the Stark shift, we simultaneously apply a control-phase pulse for iSWAP and SWAP gate to eliminate the unwanted phase.

Figure 3(b) [3(c)] shows the pulse sequence of the Ramsey interferometry for characterizing the identity and CZ gates [iSWAP and SWAP gates]. For the iSWAP and SWAP gates, which exchange an excitation between the qubits, we apply the second $\pi$/2-pulse to the cubic transmon instead of the transmon to form an interferometric sequence [Fig. 3(c)], in contrast to the standard Ramsey experiments. Figures 3(d) and 3(e) show the experimental data of Ramsey oscillations, conditioned on the state of the cubic transmon, revealing the amount of the conditional phase $\theta_{\mathrm{cp}}$ of each two-qubit gate. The phase difference between the Ramsey oscillations, with and without an initial $\pi$-rotation of the cubic transmon, corresponds to the conditional phase shift. The experimental data have good agreements with the ideal behaviors in Figs. 3(d) and 3(e).

Finally, we characterize the average two-qubit gate fidelities with the randomized-benchmarking (RB) protocols RBth1 ; RBth2 ; RB1 . The gate time is uniformly set to 50 ns for the CZ, iSWAP, SWAP gates and all the single-qubit gates. Figure 4 shows the experimental results of the two-qubit RB. From the standard RB, the average gate fidelity of the two-qubit Clifford gates is determined to be $0.950\pm 0.001$. Using this value and those from the interleaved RB, we estimate the average gate fidelity of each two-qubit gate: $0.971\pm 0.002$, $0.958\pm 0.001$ and $0.962\pm 0.001$ for CZ, iSWAP and SWAP gates, respectively. The achieved fidelities of the two-qubit gates are comparable to those of the single-qubit gates supple and mostly limited by the energy relaxation time of the qubits. The coherence limits are approximately 0.97 according to the gate pulse widths, which are close to the observed fidelities. As the cubic transmon has basically the same layout as conventional transmons, we expect improvement of the relaxation time through optimizations of the design and fabrication.

In conclusion, we have demonstrated a parity-violated qubit called a cubic transmon, and realized microwave-controlled fast two-qubit gates between a cubic transmon and a conventional transmon. As the gates originate from the second-order nonlinearity of the circuit, the coupling strength scales inversely proportional to the detuning between the qubits, not to the square of it, and is thus sufficiently large for a wide detuning range of the qubits. This is advantageous for a multi-qubit system, which often suffers from a frequency-crowding problem. The residual static ZZ interaction is eliminated by applying a continuous microwave field, which will allow us to increase the bare coupling strength further and make the two-qubit gates as fast as 20 ns with optimal device parameters. This scheme for the suppression of the residual coupling can also be extended to multi-qubit systems as well as to higher-order interactions by using multiple drives.

The authors acknowledge Y. Sunada, K. Nittoh and K. Kusuyama for the help in sample fabrication and W. Oliver for providing a TWPA. This work was partly supported by JSPS KAKENHI (Grant Number 26220601, 18K03486), JST PRESTO (Grant Number JPMJPR1429), JST ERATO (Grant Number JPMJER1601), and Q-LEAP (Grant Number JPMXS0118068682).

Supplementary Material for
“Fast parametric two-qubit gates with suppressed residual interaction
using a parity-violated superconducting qubit”