Control the qubit-qubit coupling in the superconducting circuit with double-resonator couplers

Figure 1(a) describes the superconducting circuit consisting of two Xmon qubits coupling to two common fixed frequency resonators, the frequencies of qubits are tunable. Besides the direct interactions, the qubit-resonator couplings can also induce indirect qubit-qubit and resonator-resonator interactions. The resonant frequencies of resonators a and b are respective $\omega_{a}$ and $\omega_{b}$, while $\omega_{x}$ and $\omega_{y}$ label the transition frequencies of qubits x and y, respectively. In the case of zero magnetic fluxes, the frequencies of qubits and resonators satisfy $\omega_{a}<\omega_{x}<\omega_{y}<\omega_{b}$. We expect a large distance between two resonators and neglect their inductive coupling, then the interactions are all regarded as capacitive type as shown in Fig. 1(b). The capacitances of resonators should be distributed type and proportional to their lengths, but in the article we simply label the total capacitances of resonators a and b as $C_{a}$ and $C_{b}$, respectively.

Inspired by previous work[17, 20], the kinetic energy of superconducting circuit with double-resonator couplers can be written as $T=\sum_{\eta=a,b,x,y}C_{\eta}\dot{\phi}^{2}_{\eta}/2+\sum_{\eta,\eta^{\prime}=a,b,x,y\atop\eta\neq\eta^{\prime}}C_{\eta\eta^{\prime}}\left(\dot{\phi}_{\eta}-\dot{\phi}_{\eta^{\prime}}\right)^{2}/4$, where $C_{\eta}$ is the capacitance of qubit or resonator, and $C_{\eta\eta^{\prime}}$ ($C_{\eta\eta^{\prime}}=C_{\eta^{\prime}\eta}$) is relative capacitance between arbitrary two devices among qubits and resonators, with $\eta,\eta^{\prime}=a,b,x,y$ and $\eta\neq\eta^{\prime}$. The $\phi_{a}$ and $\phi_{b}$ are the respective magnetic fluxes of circuit nodes for resonators a and b, while $\phi_{x}$ and $\phi_{y}$ are the respective node fluxes of qubits x and y, and they can be tuned by the external magnetic fluxes $\Phi_{e,x}$ and $\Phi_{e,y}$[29, 30, 31]. If we label the $L_{a}$ and $L_{b}$ as the respective inductances of resonators a and b, thus the potential energy of superconducting circuit can be written as $U=\sum_{\lambda=a,b}\phi^{2}_{\lambda}/(2L_{\lambda})+\sum_{\beta=x,y}E_{J_{\beta}}\left[1-\cos\left(2\pi\phi_{\beta}/\Phi_{0}\right)\right]$, the subscript $\lambda=a,b$ label the respective variables of resonators a and b, while $\beta=x,y$ describe the variables of qubit x and qubit y, respectively. The $E_{J_{\beta}}=I_{c\beta}\Phi_{0}/(2\pi)$ is the Josephson energy of qubit $\beta$, where the $I_{c\beta}$ is the corresponding critical current, and $\Phi_{0}=h/2e$ is the flux quantum with the planck constant h and an electron charge e.

With the kinetic energy $T$ and potential energy $U$, the Lagrangian of the superconducting circuit in Fig. 1 can be formally written as $L=T-U$. If we define the generalized momentum operators as $q_{\eta}=\partial L/\partial\dot{\phi}_{\eta}=C\dot{\phi}_{\eta}$ (with $\eta=a,b,x,y$), under the conditions $C_{ab}\ll C_{xy}\ll C_{ax},C_{ay},C_{bx},C_{by}\ll C_{x},C_{y}\ll C_{a},C_{b}$, we obtain the expression of Hamiltonian (see Appendix B)

where $n_{\eta}=q_{\eta}/2e$ is the Cooper-pair number operator of a qubit or resonator, and the corresponding charging energy is $E_{C_{\eta}}=e^{2}/2C_{\eta}$. The transition frequencies of resonators and qubits are respectively defined as $\omega_{\lambda}=1/\sqrt{C_{\lambda}L_{\lambda}}$ and $\omega_{\beta}=(\sqrt{8E_{J_{\beta}}E_{C_{\beta}}}-E_{C_{\beta}})/\hbar$, while $\alpha_{\beta}=-E_{C_{\beta}}/\hbar$ labels the anharmonicity of qubit $\beta$. As shown in Appendix B, the two-body coupling strengths among qubits and resonators can be defined as

The two-body interactions are mainly decided by their relative capacitances $C_{\eta\eta^{\prime}}$ with $\eta,\eta^{\prime}=a,b,x,y$ and $\eta\neq\eta^{\prime}$. The qubit-resonator coupling strength $g_{\lambda\beta}$ in Eq.(2) could induce indirect interaction between two qubits, so the Eq. (4) can not describe the complete interaction between two qubits. In the single-coupler superconducting quantum chip, there are many restrictions on the capacitances and frequencies of qubits (or couplers), but these limitations might be unfrozen in the double-coupler circuit as will be discussed in the follow sections.

To get the effective qubit-qubit coupling, we try to decouple the qubit-resonator interactions in this section. For the Xmon qubit, the Josephson energy is much larger than its charging energy, $E_{J_{\beta}}/E_{C_{\beta}}\gg 1$, and then the $\phi_{\beta}$ should be very small and we can use the approximate equation: $\cos(\phi_{\beta})=1-\phi^{2}_{\beta}/2+\phi^{4}_{\beta}/24-...$. If we introduce the creation and annihilation operators by the definitions $\phi_{\beta}=\sqrt[4]{2E_{C}/E_{J_{\beta}}}(a^{\dagger}_{\beta}+a_{\beta})$, and $n_{\beta}=(i/2)\sqrt[4]{2E_{C}/E_{J_{\beta}}}(a^{\dagger}_{\beta}-a_{\beta})$, the second-quantization Hamiltonian can be obtained as $H=\sum_{\lambda=a,b}{H_{\lambda}}+\sum_{\beta=x,y}{H_{\beta}}+\sum_{\lambda=a,b\atop\beta=x,y}H_{\lambda\beta}+H_{ab}+H_{xy}$, with

We define $\alpha_{\beta}=-E_{C_{\beta}}/\hbar$ to describe anharmonicity of qubit $\beta$, and the nonlinear term $(\alpha_{\beta}/2)a^{\dagger}_{\beta}a^{\dagger}_{\beta}a_{\beta}a_{\beta}$ reflects the effects of high-excited states of superconducting artificial atom. We define the $\Delta_{\lambda\beta}=\omega_{\beta}-\omega_{\lambda}$ to describe the frequency detuning between the qubit $\beta$ and resonator $\lambda$, while $\Sigma_{\lambda\beta}=\omega_{\beta}+\omega_{\lambda}$ is the frequency summation of qubit $\beta$ and resonator $\lambda$. $\Delta_{xy}=\omega_{y}-\omega_{x}$ describes the frequency detuning between two qubits, and $\Delta_{ab}=\omega_{b}-\omega_{a}$ labels the frequency detuning between two resonators.

Separating the Hamiltonian as $H=H_{0}+H_{int}$, the free term is defined as $H_{0}=\sum_{\lambda=a,b}H_{\lambda}+\sum_{\beta=x,y}H_{\beta}$, while the interaction term is $H_{int}=H_{ab}+H_{xy}+\sum_{\lambda=a,b\atop\beta=x,y}H_{\lambda\beta}$. In the qubit-resonator dispersive coupling regimes $g_{\lambda\beta}/|\Delta_{\lambda\beta}|\ll 1$ and $g_{\lambda\beta}/\Sigma_{\lambda\beta}\ll 1$, we define $S=\sum_{\lambda=a,b\atop\beta=x,y}[(g_{\lambda\beta}/\Delta_{\lambda\beta})(c^{\dagger}_{\lambda}a_{\beta}-c_{\lambda}a^{\dagger}_{\beta})-(g_{\lambda\beta}/\Sigma_{\lambda\beta})(c^{\dagger}_{\lambda}a^{\dagger}_{\beta}-c_{\lambda}a_{\beta})]$. Under the Schrieffer–Wolff transformation, if we choose $H^{(d)}=\exp{(S)}H\exp{(-S)}$ and $H_{int}+[S,H_{0}]=0$, thus the decoupled Hamiltonian becomes $H^{(d)}=H_{0}-(1/2)[H_{int},S]+O(H_{int}^{3})$ (see Appendix C), that is

Since $g_{ab},g_{xy}\ll g_{\lambda\beta}$, the contributions of $H_{ab}$ and $H_{xy}$ have been neglected. Following the method of previous work[7], we assumed $\tilde{\alpha}_{\beta}\approx\alpha_{\beta}$ during the derivations of Eq.(10), thus the contributions of superconducting artificial atoms’ high-excited states are neglected.

The decoupled frequencies of qubits and resonators can be respectively defined as $\omega^{(d)}_{\beta}=\omega_{\beta}+\sum_{\lambda=a,b}(g^{2}_{\lambda\beta}/\Delta_{\lambda\beta}-g^{2}_{\lambda\beta}/\Sigma_{\lambda\beta})$ and $\omega^{(d)}_{\lambda}=\omega_{\lambda}-\sum_{\beta=x,y}(g^{2}_{\lambda\beta}/\Delta_{\lambda\beta}-g^{2}_{\lambda x}/\Sigma_{\lambda\beta})$ (as shown in Appendix C,). And the decoupled qubit-qubit coupling strength can be obtained as

Since $\Delta_{\lambda\beta}$ and $\Sigma_{\lambda\beta}$ depend the frequency of qubit $\beta$, so the induced qubit-qubit coupling $g^{(in)}_{\lambda,xy}=(1/2)\sum_{\atop\beta=x,y}(g_{\lambda x}g_{\lambda y}/\Delta_{\lambda\beta}-g_{\lambda x}g_{\lambda y}/\Sigma_{\lambda\beta})$ can be tuned by the external magnetic fluxes $\Phi_{e,x}$ and $\Phi_{e,y}$. To switch off the qubit-qubit coupling ($g^{(d)}_{xy}/(2\pi)=0$ Hz), we should find parameters to satisfy $-g^{(in)}_{xy}=g_{xy}$.

From the expression of $g^{(in)}_{\lambda,xy}$, both two resonators make contributions to the effective qubit-qubit coupling, and their contributions might cancel each other ($g^{(in)}_{a,xy}+g^{(in)}_{b,xy}=0$) if the qubit frequency of qubits satisfy certain conditions. Thus the direct qubit-qubit coupling might be not necessary for the switching off in the double-resonator couplers circuit. The qubits could also induce indirect interactions between the two resonators, and the decoupled resonator-resonator coupling strength can be defined as $g^{(d)}_{ab}=(1/2)\sum_{\lambda=a,b\atop\beta=x,y}\left(g_{a\beta}g_{b\beta}/\Delta_{\lambda\beta}-g_{a\beta}g_{b\beta}/\Sigma_{\lambda\beta}\right)+g_{ab}$ (see Appendix C). Because of the large frequency detuning between two resonators ($|\Delta_{ab}|\gg g^{(d)}_{ab}$), the effective resonator-resonator interaction makes little effect on the energy levels of qubits and resonators.

With the parameters in Fig. 2, we can get $g_{by}/\min{(|\Delta_{by}|)}\sim 1/3$ and $g_{ax}/\min{(|\Delta_{ax}|)}\sim 1/15$ in the idling states of qubits (without external magnetic field). This means that the qubits and resonators are in the dispersive or weak-dispersive coupling regimes, thus the perturbation method can be used to calculate the effective qubit-qubit coupling . If we choose $\omega_{a}<\omega_{x}<\omega_{y}<\omega_{b}$, the signs of $\Delta_{a\beta}$ and $\Delta_{b\beta}$ are opposite. As indicated by $g^{(in)}_{\lambda,xy}$, the resonator b will induce negative indirect qubit-qubit coupling, and the contributions of resonators a is positive. With Eq. (11), the curved surfaces of $g^{(d)}_{xy}$ are plotted in Figs. 3(a) and 3(c), and the corresponding decoupled frequencies of qubits are shown in Figs. 3(b) and 3(d). The curved surface of $g^{(d)}_{xy}$ has many crossing points with the zero value plane ($g^{(d)}_{xy}/(2\pi)=0$ Hz) in Fig. 3(a) ($g_{xy}/(2\pi)=3$ MHz), which correspond to switch off positions for the qubit-qubit coupling. The multiple switching off points can be used to optimize the quantum operation parameters and reduce the effects of adjoint qubits. For the case of nonzero direct qubit-qubit coupling ($g_{xy}\neq 0$), the distribution of crossing points forms an approximately elliptical curve in $\phi_{x}$-$\phi_{y}$ plane as shown in Fig. 3(a), and this means that the contributions of resonator b to induced qubit-qubit coupling (in amplitudes) is larger than the contributions of resonator a.

In the case of $g_{xy}/(2\pi)=0$ Hz, the effective qubit-qubit can also be zero if the indirect qubit-qubit couplings induced by resonators a and b are the same in amplitudes but opposite in signs ($g^{(in)}_{a,xy}=-g^{(in)}_{b,xy}$). As shown in Fig. 3(c), the curved surface of $g^{(d)}_{xy}$ can also cross with the zero values plane ($g^{(d)}_{xy}/(2\pi)=0$ Hz) in the case of $g_{xy}/(2\pi)=0$ Hz, and this means that the switching off can be realized without the direct qubit-qubit interaction in the double-resonator couplers circuit. The distribution of switching off points approximately forms a circle in $\phi_{x}$-$\phi_{y}$ plane in Fig. 3(c), which indicates the approximate equal contributions (in amplitudes) of two resonators to the effective qubit-qubit couplings. The decoupled frequencies of qubits are plotted in Fig. 3(b) ($g_{xy}/(2\pi)=3$ MHz) and Fig. 3(d)($g_{xy}/(2\pi)=0$ Hz), and the effects of direct qubit-qubit coupling to the transition frequencies of qubits seems not very large.

The switching off positions are not totally decided by the direct qubit-qubit coupling in the double-resonator coupler circuit, thus we can take arbitrary small or even zero direct qubit-qubit coupling strength in principally, which might be helpful to suppress the state leakages and crosstalks on the superconducting quantum chips. And the restrictions on the direct qubit-qubit coupling strength and coupler’s frequency can be unfreezed on the double-resonator couplers superconducting quantum chip). If we choose the switching off positions close to two-qubit gate regimes, thus the maximal frequencies of couplers can be smaller, and this might create wider available frequency ranges for the readout resonators and relieve the frequency crowding on the superconducting quantum chip.

In current theoretical model for tunable coupler circuit, the nonlinear term $H_{nl,\beta}=(\alpha_{\beta}/2)a^{\dagger}_{\beta}a^{\dagger}_{\beta}a_{\beta}a_{\beta}$ term is regarded as invariant during dynamical decoupling processes for qubit-resonator interactions[7, 16]. This approximation in fact neglects the effects of superconducting artificial atoms’ high-excited states, so the $g^{(d)}_{xy}$ in Eq. (11) does not contain the information of anharmonicity $\alpha_{\beta}$. Because of the small anharmonicity for Xmon qubit (between 200 MHz and 400 MHz), the interactions between the resonators and high-excited states of superconducting artificial atoms should make corrections to the qubits’ energy levels and effective qubit-qubit coupling.

The Bogoliubov transformation has been used to analyze the variations of nonlinear term $H_{nl,\beta}$ during the decoupling processes[40, 39], and the derived self-kerr and cross-kerr resonant terms under the unitary transformation reflect the contributions of superconducting artificial atoms’ high-excited states. To maintain consistency, in this section we continue to use the Schrieffer–Wolff transformation to calculate the effects of the nonlinear term $H_{nl,\beta}$ (see appendix D). In the qubit-resonator dispersive coupling regimes, $g_{\lambda\beta}/|\Delta_{\lambda\beta}|\ll 1$ and $g_{\lambda\beta}/|\Sigma_{\lambda\beta}|\ll 1$, we define $S_{\lambda\beta}=(g_{\lambda\beta}/\Delta_{\lambda\beta})(c^{\dagger}_{\lambda}a_{\beta}-c_{\lambda}a^{\dagger}_{\beta})-(g_{\lambda\beta}/\Sigma_{\lambda\beta})(c^{\dagger}_{\lambda}a^{\dagger}_{\beta}-c_{\lambda}a_{\beta})$ with $S=\sum_{\lambda=a,b\atop\beta=x,y}S_{\lambda\beta}$. Since $H_{nl,\beta}$ is a small quantity, we will separately conduct the Unitary transform $H^{\prime}_{nl,\beta}=\exp(S)H_{nl,\beta}\exp(-S)$ to study its contributions to the high-order effects, such as cross-kerr resonance, self-kerr resonance, and so on[40]. With tedious calculations (see Appendix D), up to the second-order perturbation expansion terms, we get

Since $g_{xy},g_{ab}\ll g_{\lambda\beta}$, the indirect interaction induced by the weakly direct qubit-qubit and resonator-resonator interactions have been neglected. The first and second lines in the right side of Eq.(12) respectively describe the self-kerr and cross-kerr resonance terms, and the complete calculation results can be seen in Appendix D. There are no external pump fields for resonator couplers, so the cavity photon numbers should be very small ($n_{\lambda}=a^{\dagger}_{\lambda}a_{\lambda}\ll 1$). Thus we can get the approximate frequency shift for qubit ${\beta}$ induced by the nonlinear terms,

We can see that the frequency shift $\Delta\omega_{\beta}$ for qubit $\beta$ is proportional to the qubit’s anharmonicity $\alpha_{\beta}$, which reflects the effects of the second-excited state of superconducting artificial atoms. Adding the frequency shift induced by the nonlinear term $H_{nl,\beta}$, we can approximately get the corrected frequency $\omega^{(cr)}_{\beta}$ of qubit $\beta$ in decoupled coordinate frame

The transition frequency of qubit x in decoupled coordinate is plotted in Fig. 4(a), the deviation between $\omega^{(cr)}_{x}$ and $\omega^{(d)}_{x}$ is larger at the regimes far from the zero magnetic flux points, which coincides with the curve of $\Delta\omega_{x}$ in insert figure. On the contrary, the maximal deviation between $\omega^{(cr)}_{y}$ and $\omega^{(d)}_{y}$ appears at the regime close to the zero magnetic flux ($\Phi_{e,y}\rightarrow 0$ or $\phi_{y}\rightarrow 0$) in Fig. 4(b), this also coincides with the curve of $\Delta\omega_{y}$ in the insert figure.

If we replace the $\omega^{(d)}_{\beta}$ with $\omega^{(cr)}_{\beta}$ in Eq.(11), we can get the corrected effective qubit-qubit coupling $g^{(cr)}_{xy}$. The resonators’ resonant frequencies $\omega_{a}$ and $\omega_{b}$ are fixed in this article, so the effective qubit-qubit coupling are mainly tuned by the qubits’ transition frequencies $\omega_{x}$ and $\omega_{y}$. By setting $\omega_{x}/(2\pi)=4.56$ GHz, we plot the curves of effective qubit-qubit coupling $g^{(cr)}_{xy}$ and $g^{(d)}_{xy}$ on the respective $\phi_{y}$ and $\omega_{y}$ in Figs. 4(c) and  4(d), the points satisfying $g^{(cr)}_{xy}/(2\pi)=0$ Hz or $g^{(d)}_{xy}/(2\pi)=0$ Hz corresponds to the switching off position for the qubit-qubit interaction. The zero value points of $g^{(cr)}_{xy}$ and $g^{(cr)}_{xy}$ are different, which reflects the effects of the nonlinear term $H_{nl}$ ( also the second-excited state of superconducting artificial atom) on the switching off positions. The calculation results of the corrections to qubits’ frequencies and effective qubit-qubit coupling strength can help to accurately design the superconducting quantum chip.

In this section, we study the effects of direct qubit-qubit coupling $g_{xy}$ and qubit’s anharmonicitiy $\alpha_{\beta}$ on the effective qubit-qubit coupling $g^{(cr)}_{xy}$ and the switching off position. If we take the parameters of Fig. 2, we can get $g_{by}/\min{(|\Delta_{by}|)}\sim 1/3$ and $g_{ax}/\min{(|\Delta_{ax}|)}\sim 1/15$ in the idling states of qubits (without external magnetic field), so the qubits and resonators are in the dispersive or weak-dispersive coupling regimes. To see clearer the working mechanism of switching-off processes in the double-resonator couplers circuit, we plot the one-dimensional curves of effective qubit-qubit coupling $g^{(cr)}_{xy}$ with the variation of qubit transition frequency $\omega_{y}$ in Fig. 5(a). By fixing $\omega_{x}/(2\pi)=4.56$ GHz, the three curves without markers in Fig. 5(a) correspond to different direct qubit-qubit coupling strengths: $g_{xy}/(2\pi)=0$ Hz in the blue-solid curve, $g_{xy}/(2\pi)=0.5$ MHz in black-dashed curve, and $g_{xy}/(2\pi)=1$ MHz in the red-dotted curve. The crossing points of three curves with zero value line ( $g^{(cr)}_{xy}/(2\pi)=0$ Hz) are different, this means that the direct qubit-qubit coupling could affect the switching off positions. If we set $\omega_{x}/(2\pi)=4.53$ GHz, the switching off points in the three marked curves show considerable shifts relative to the corresponding same color curves without markers.

In Fig. 5(b), we plot the curves of effective qubit-qubit coupling $g^{(cr)}_{xy}$ on the node phase $\phi_{y}$ with $\omega_{x}/(2\pi)=4.56$ GHz. For the same ranges of node phase $\phi_{y}$, the qubit’s maximal frequencies $\omega^{(max)}_{y}$ are not the same for different anharmonicities $\alpha_{y}$ as shown in the insert figure. For different anhamonicities $\alpha_{y}$, the shifts of switching off positions on three curves reflect the effects of superconducting artificial atom’s second-excited states.

The frequency of qubit y should be tuned close to $\omega_{y}\approx\omega_{x}$ for the iSWAP gate and $\omega_{y}+\alpha_{y}\approx\omega_{x}$ for the Controlled-Z gate. For the single-path coupler circuit, the switching off point is usually close to the idling coupler frequency (usually about 6.0 GHz) which is far from the two-qubit gate regimes (usually below 5.0 GHz). In the double-resonator couplers circuit, the switching off positions can be very close to the two-qubit gate regimes as shown in Fig. 5(a). So the maximal frequencies of couplers can be smaller in the double-resonator couplers superconducting circuit, this leaves wider available ranges for readout resonators (or qubits) and might relieve the frequency crowding on superconducting quantum chip.

In Figs. 2(a)-2(b), we have numerically calculated the energy level curves of states $|0100\rangle$, $|0010\rangle$, and $|0110\rangle$, in principally the static ZZ coupling can be easily calculated through the definition $\xi_{ZZ}=\omega_{|0110\rangle}-\omega_{|0100\rangle}-\omega_{|0010\rangle}+\omega_{|0000\rangle}$. In practically, it is difficult to accurately fit the energy level curves of qubits because the avoided crossing gaps are affected by multi-body interactions. By setting $\omega_{x}/(2\pi)=4.56$ GHz, if we tune the frequency of qubit y to be near resonant with qubit x ($\omega_{y}\approx\omega_{x}$), thus we can get $g_{by}/|\Delta_{by}|\approx g_{bx}/|\Delta_{bx}|\sim 1/20$ and $g_{ax}/|\Delta_{ax}|\approx g_{ay}/|\Delta_{ay}|\sim 1/15$. Thus the qubit-resonator are dispersive coupling regimes, and the perturbation method can be used to analyze the static ZZ coupling close to the two-qubit gate regimes.

For convenience and consistency, we still use $\omega_{\beta}$ to describe the energy levels of first-excited state of qubit $\beta$ in this section, and the energy level for second-excited state is $2\omega_{\beta}+\alpha_{\beta}$, where the $\alpha_{\beta}$ is the qubit’s anharmoncity . If we temporarily disregard the weak direct qubit-qubit and direct resonator-resonator interactions, up to the fourth-order perturbation theory the effective Hamiltonian on the qubits’ eigenstates space can be obtained as [40, 39]

The ket vector $|j_{\beta}\rangle$ describes the $j_{\beta}$-th excited state of qubit $\beta$, with $j_{\beta},j^{\prime}_{\beta}=0,1,2,...$. We define $g^{j_{\beta}j^{\prime}_{\beta}}_{\lambda}$ as the coupling strength between resonator $\lambda$ and the transition $|j_{\beta}\rangle\leftrightarrow|j^{\prime}_{\beta}\rangle$ of qubit $\beta$. Considering the selection rule, the resonator $\lambda$ can only interact with the neighbour quantum states of qubit $\beta$: $g^{j_{\beta}j^{\prime}_{\beta}}_{\lambda}=0$ for $j^{\prime}_{\beta}\neq j_{\beta}\pm 1$. In this section, we neglect the small differences for the coupling strengths between resonator $\lambda$ and different neighbour state transitions of qubit $\beta$ , then $g^{j_{\beta}-1,j_{\beta}}_{\lambda}=g^{j_{\beta},j_{\beta}+1}_{\lambda}=g_{\lambda\beta}$ ( $j_{\beta}=1,2,...$). Defines $\chi^{j_{\beta}-1,j_{\beta}}_{\lambda}=j_{\beta}g_{\lambda\beta}/[\Delta_{\lambda\beta}+(j_{\beta}-1)\alpha_{\beta}]$, the $\kappa_{\lambda,j_{\beta}}=\chi^{j_{\beta}-1,j_{\beta}}_{\lambda}$ describes the level shifts of Lamb type for the quantum state $|j_{\beta}\rangle$ which is induced by the interaction between resonator $\lambda$ and qubit $\beta$ ( $|j_{\beta}-1\rangle\leftrightarrow|j_{\beta}\rangle$ and $|j_{\beta}\rangle\leftrightarrow|j_{\beta}+1\rangle$), while $\chi_{\lambda,j_{\beta}}=\chi^{j_{\beta}-1,j_{\beta}}_{\lambda}-\chi^{j_{\beta},j_{\beta}+1}_{\lambda}$ describes the corresponding ac-stark type dispersive shifts for the quantum state $|j_{\beta}\rangle$ ($\chi_{\lambda,0_{\beta}}=-\chi^{0_{\beta},0_{\beta}+1}=-g^{2}_{\lambda\beta}/(2\Delta_{\lambda\beta})$)[39, 40]. If we add the contributions of second-excited states of superconducting artificial atoms, besides the self-kerr resonant term $\mu_{\lambda,j_{\beta}}(c^{\dagger}_{\lambda}c_{\lambda})^{2}$, the cross-kerr resonant terms $\nu_{ab,j_{\beta}}c^{\dagger}_{a}c_{a}c^{\dagger}_{b}c_{b}$ and $\nu_{ba,j_{\beta}}c^{\dagger}_{b}c_{b}c^{\dagger}_{a}c_{a}$ should also make contributions to the static ZZ coupling(as will be discussed in follow sections)[39, 40, 41].

If we temporarily disregard the cross-kerr resonant terms, after adding the contributions of weak direct qubit-qubit coupling, and then the static ZZ coupling in qubit-resonator dispersive coupling regimes can be obtained as [8, 17, 16, 25],

The $\xi^{(2)}_{ZZ}$ is the second-order static ZZ coupling between two qubits, and $\xi^{(3)}_{ZZ,\lambda}$ describes the third-order static ZZ coupling between two qubits intermediated by the resonator $\lambda$. We use $\xi^{(4s)}_{ZZ,\lambda}$ to label fourth-order static ZZ coupling contributed by the self-kerr resonance intermediated by the resonator $\lambda$, and the $\xi^{(4c)}_{ZZ,\lambda}$ describe the static ZZ coupling induced by the cross-kerr resonance.

Even being listed together, the second-order $\xi^{(2)}_{ZZ}$, third-order $\xi^{(3)}_{ZZ}=\sum_{\lambda=a,b}\xi^{(3)}_{ZZ,\lambda}$, and fourth-order (self-kerr) $\xi^{(4s)}_{ZZ}=\sum_{\lambda=a,b}\xi^{(4s)}_{ZZ,\lambda}$ static ZZ coupling terms come from different sources. As shown in Eqs.(16) and (17), the second-order term $\xi^{(2)}_{ZZ}$ originates from the direct qubit-qubit coupling [6, 7, 8, 17], and the fourth-order term $\xi^{(4s)}_{ZZ}$ results from the perturbation expansion of qubit-resonator dispersive coupling[39, 41], while the third-order term $\xi^{(3)}_{ZZ}$ is joint effects of direct qubit-qubit coupling and qubit-resonator interaction. Here $\alpha_{\lambda}=\sum_{\beta=x,y}\alpha_{\beta}(g_{\lambda\beta}/\Delta_{\lambda\beta})^{4}$ describes the nonlinearity of resonator $\lambda$ induced by the qubit-resonator dispersive coupling[40]. Since $g_{xy},g_{ab}\ll g_{\lambda\beta}$, the contributions of direct qubit-qubit and resonator-resonator couplings to fourth-order static ZZ coupling are neglected.

In this section, we try to suppress the Static ZZ coupling with the direct qubit-qubit coupling which can be arbitrary small in the double-coupler superconducting circuit. By setting $\omega_{x}/(2\pi)=4.52$ GHz, the curves of static ZZ coupling $\xi_{ZZ}=\xi^{(2)}_{ZZ}+\xi^{(3)}_{ZZ}+\xi^{(4s)}_{ZZ}$ are plotted in Fig. 6(b) according to Eqs.(16)-(18). The four curves correspond to different direct qubit-qubit coupling strengths. In the regimes suitable for the two-qubit gates, the values of static ZZ coupling are apparently suppressed by the weaker direct qubit-qubit coupling as shown in the insert figure. And the values of $\xi^{(3)}_{ZZ}$ and $\xi^{(4s)}_{ZZ}$ are obviously suppressed by weaker direct qubit-qubit coupling in Fig. 6(c) ($g_{xy}/(2\pi)=0.5$ MHz) compared with the results in Fig. 6(d) ($g_{xy}/(2\pi)=1.0$ MHz). As shown in Fig. 6(b)-6(f), the static ZZ coupling can be suppressed below Sub-MHz in double-resonator coupler circuit, which is the similar level with transmon-based coupler circuit[7, 16].