Universal singlet-triplet qubits implemented near the transverse sweet spot

Here, we focus on introducing how to design the non-cyclic geometric gate using the piecewise continuous Hamiltonian Liu et al. (2020) (the other types of pulse sequences is introduced in Appendix. B), and then we will discuss the noise effect caused by the charge noise in the DQD.

The quantum gate considered here is using two-pieces of Hamiltonian in Eq. 4. In each of the piece, the Hamiltonian is with the form as

$$H_{n}(t)=\left\{\begin{array}[]{ll}\frac{\Omega_{0}}{2}(\cos(\phi_{0}+\frac{\pi}{2})\ \sigma_{x}+\sin(\phi_{0}+\frac{\pi}{2})\ \sigma_{y}),&T_{A}\leqslant t<T_{B}\\ \frac{\Omega_{0}}{2}(\cos(\phi_{0}+\phi_{1}+\frac{\pi}{2})\ \sigma_{x}+\sin(\phi_{0}+\phi_{1}+\frac{\pi}{2})\ \sigma_{y}),&T_{B}<t\leqslant T_{C}\end{array}\right.$$

(5)

where $T_{B}-T_{A}=\chi_{0}/\Omega_{0}$ and $T_{C}-T_{B}=\beta_{0}/\Omega_{0}$. Then, this two-connecting evolution together at the final time forms a desired quantum gate as

	$\displaystyle{U_{n}}\left({{\chi_{0}},{\phi_{0}},{\phi_{\rm{1}}},{\beta_{0}}}\right)$	$\displaystyle={U_{CB}}\left({{T_{C}},{T_{B}}}\right){U_{BA}}\left({{T_{B}},T_{A}}\right)$		(6)
		$\displaystyle=\left(\begin{array}[]{ccc}\cos\frac{\chi_{0}}{2}\cos\frac{\beta_{0}}{2}-\sin\frac{\chi_{0}}{2}\sin\frac{\beta_{0}}{2}e^{-i\phi_{1}}&-\cos\frac{\chi_{0}}{2}\sin\frac{\beta_{0}}{2}e^{-i\left(\phi_{0}+\phi_{1}\right)}-\cos\frac{\beta_{0}}{2}\sin\frac{\chi_{0}}{2}e^{-i\phi_{0}}\\ \cos\frac{\chi_{0}}{2}\sin\frac{\beta_{0}}{2}e^{i\left(\phi_{0}+\phi_{1}\right)}+\cos\frac{\beta_{0}}{2}\sin\frac{\chi_{0}}{2}e^{i\phi_{0}}&\cos\frac{\chi_{0}}{2}\cos\frac{\beta_{0}}{2}-\sin\frac{\chi_{0}}{2}\sin\frac{\beta_{0}}{2}e^{i\phi_{1}}\end{array}\right)$

Here, $\chi_{0}$, $\phi_{0}$, $\phi_{1}$ and $\beta_{0}$ are determined by the chosen evolution operator. In our simulation, the specific parameters are present in Appendix. D. Actually, ${U_{n}}\left({{\chi_{0}},{\phi_{\rm{0}}},{\phi_{\rm{1}}},{\beta_{0}}}\right)$ can enable arbitrary rotation on the Bloch sphere. For example, ${U_{n}}({\chi_{0}},\frac{\pi}{2},\pi,{\beta_{0}})$ and ${U_{n}}({\chi_{0}},-\pi,\pi,{\beta_{0}})$ can implement a rotation around $x$ axis and $y$ axis, respectively, the rotation angle for both of which have the same form as $\beta_{0}-\chi_{0}$. Note that, for a desired rotation around the $x$ or $y$ axis, even though one has many choices to pick up $\chi_{0}$ and $\beta_{0}$, the conditions for them are clear: $\chi_{0}>0$ and $\beta_{0}>0$. In this way, the evolution time is positive and it ensures the gate will not degenerate to the naive pulse sequence.

The evolution of the quantum gate can be visualized by using the orthogonal dressed states

	$\displaystyle\left|\psi_{+}(t)\right\rangle=\cos\frac{\chi}{2}e^{-\frac{i}{2}\eta}|0\rangle+\sin\frac{\chi}{2}e^{\frac{i}{2}\eta}|1\rangle$		(7)
	$\displaystyle\left|\psi_{-}(t)\right\rangle=\sin\frac{\chi}{2}e^{-\frac{i}{2}\eta}|0\rangle-\cos\frac{\chi}{2}e^{\frac{i}{2}\eta}|1\rangle$

As shown in Fig. 1(a), the whole evolution path of the quantum gate is divided into two proper parts, which are denoted as path AB and BC. In the first part of the evolution, $\left|\psi_{+}\right\rangle$ starts from a given point A, and evolves along the geodesic line up to the north pole B. Then, in the second part, it goes down to a given point C along another geodesic line. Using this dressed states basis, the evolution operator ${U_{n}}\left({{\chi_{0}},{\phi_{\rm{0}}},{\phi_{\rm{1}}},{\beta_{0}}}\right)$ can also be written as

$\displaystyle{U_{n}}={e^{i{\gamma}}}\left|{{\psi_{+}}({T_{C}})}\right\rangle\left\langle{{\psi_{+}}({T_{A}})}\right|+{e^{-i{\gamma}}}\left|{{\psi_{-}}({T_{C}})}\right\rangle\left\langle{{\psi_{-}}({T_{A}})}\right|$

(8)

where $\gamma$ is the related phase that is obtained. This two-piece path in the dressed state basis is not closed. On the other hand, the evolution path in the dressed state basis can alternatively be cyclic, as shown in Fig. 1(b) which can form a geometric gate. The detail of the geometric gate is as shown in Ref. Zhao et al. (2017). For this geometric gate, its whole evolution time is fixed to be $2\pi/\Omega_{0}$ Zhang et al. (2020c). By carefully selecting proper $\chi_{0}$ and $\beta_{0}$ so as to satisfy $\chi_{0}+\beta_{0}<2\pi$, the evolution time for the non-cyclic geometric gate can be always shorter than the geometric ones. Note that, in Ref.Liu et al. (2020), the authors have pointed out that the non-cyclic geometric gate introduced here is non-cyclic and non-Abelian. However, for the case $m\neq n$ with $m,n\in+,-$, one finds $\int_{0}^{\tau}\left\langle\psi_{m}(t)|\mathcal{H}(t)|\psi_{n}(t)\right\rangle dt\neq 0$, namely, the dynamical phase in this case is present. Therefore, it does not satisfy the parallel transport condition and cannot be the non-Abelian geometric gate Kult et al. (2006).

Then, we study the noise effect of this AC-driven system to the non-cyclic geometric gate. Considering the charge noise effect, the detuning value turns to be $\varepsilon(t)=\varepsilon_{\rm{SS}}+\varepsilon_{\rm{AC}}\cos(\omega t+\phi(t))+\delta\varepsilon(t)$, where $\delta\varepsilon(t)$ is the time-dependent charge noise. $\delta\varepsilon(t)$ results in the error for the chosen $\varepsilon_{\rm{SS}}$, i.e. $\varepsilon_{\rm{SS}}\rightarrow\varepsilon_{\rm{SS}}+\delta\varepsilon(t)$ and further induce the drift for the energy-level structure, namely, $E_{n}\rightarrow E_{n}+\delta E_{n}$. Assuming that $\varepsilon_{\rm{AC}}$, $\delta\varepsilon(t)\ll\varepsilon_{\rm{SS}}$, we can expand the qubit energy as

$\displaystyle\omega_{q}\approx\omega_{q}\left(\varepsilon_{\rm{SS}}\right)+\delta\omega_{q},$

(9)

where we have $\delta\omega_{q}\simeq(\partial{\omega_{q}}/\partial{\varepsilon})\ \delta\varepsilon$ when operating near the TSS point. This in turn leads to error for $H_{\rm{AC}}$ in the rotating frame (see Appendix. A):

$$H_{\rm{AC}}^{\prime}=\frac{\Omega_{0}}{2}\left(\cos\phi(t)\ \sigma_{x}+\sin\phi(t)\ \sigma_{y}\right)+\delta\omega_{q}/2\ \sigma_{z}.$$

(10)

Note that here we have assumed $\delta\varepsilon$ is a constant value for simplicity and its time-dependent effect will be considered later. At the TSS point the qubit is first-order insensitive to the charge noise because of $\partial{\omega_{q}}/\partial{\varepsilon}=0$. However, when it is away from the TSS point, this first-order charge noise effect cannot be ignored anymore. In Fig. 3, we show the derivative $\partial{\omega_{q}}/\partial{\varepsilon}$ as a function of the detuning value $\varepsilon$ around the TSS point. We find that the derivative in Fig. 3(a), which corresponds to $\Delta B<\tau$, is much smaller than that one when $\Delta B>\tau$, as shown in Fig. 3(b). This means the qubit is more insensitive to the charge noise when operating in this region. However, as stated above, the leakage effect in this region is substantially large regardless of this superiority to the charge noise. Meanwhile, when $\Delta B>\tau$, $\partial{\omega_{q}}/\partial{\varepsilon}$ grows almost linearly with $\varepsilon-\varepsilon_{\rm{SS}}$. This implies that small AC drive is more appropriate in order to avoid large charge noise. As stated above, we have taken $\varepsilon_{\rm{AC}}/2\pi=0.1\ \rm{GHz}$. Considering all the operating region of the detuning value $\varepsilon$ we have $\partial{\omega_{q}}/\partial{\varepsilon}\leq\partial{\omega_{q}}/\partial{\varepsilon}|_{\varepsilon=\varepsilon_{\rm{AC}}}\simeq 0.02$. According to Ref. Mielke et al. (2020), the standard deviation related to the detuning charge noise is with the order of $\sigma_{\varepsilon}=1\ \mu\rm{eV}$. Thus, the deviation with respect to the qubit energy can be $0<\sigma_{\omega q}\leq(\partial{\omega_{q}}/\partial{\varepsilon})\ \sigma_{\varepsilon}\simeq 0.02\ \mu\rm{eV}$. In our simulation we have used $\sigma_{\omega q}=0.02\ \mu\rm{eV}$.

Note that, for the ST qubits in semiconductor quantum dot, there are several extra noise channels that should be carefully treated. For example, the charge noise can also bring fluctuation for the tunneling value $\tau$, leading to tunneling noise. However, the magnitude of tunneling noise is much weaker than that of the detuning noise Koski et al. (2020). Here, the error in the AC-driven field $\varepsilon_{\rm{AC}}$ is neglected, which is also much weaker than the detuning noise Abadillo-Uriel et al. (2019). In addition, the fluctuation in the Overhauser field (nuclear noise) in GaAs can lead to unwanted error in $\Delta B$ and also introduce relaxation Reilly et al. (2008). To deal with the nuclear noise, the silicon-based semiconductor quantum dot with isotopic purification is appreciated, where the nuclear noise is strongly suppressed Huang et al. (2019). Meanwhile, the valley in Si/SiGe platforms may introduce extra valley-spin coupling leading to relaxation Zhang et al. (2019). Fortunately, recent experiment indicated that by using the silicon metal-oxide-semiconductor (Si-MOS) platform the valley splitting can be as high as 0.8 meV Kawakami et al. (2014); Rančić and Burkard (2016). In this way, the valley effect can also be safely neglected. According to the latest experiment, the relaxation time based on silicon platform has reached $T_{1}=9\ \rm{s}$ Ciriano-Tejel et al. (2021). In this work, we focus on the suppression on $\delta\omega_{q}$ due to the detuning noise, and we simulate the single-qubit gate fidelity by using the two-level structure as shown in Eq. 10.

In the real evolution process, the noise is time-dependent and is always described by the so-called $1/f^{\alpha}$ noise model. The power spectral density of the $1/f^{\alpha}$-type noise has the form as $S(\omega)=A/(\omega t_{0})^{\alpha}$ Yang and Wang (2016). Here, $A$ denotes the noise amplitude, $t_{0}$ the time unit, and the exponent $\alpha$ the noise correlation. For the semiconductor quantum-dot environment, the charge noise is typically about $\alpha\simeq 1$. The noise amplitude can be determined by Zhang et al. (2017)

$\displaystyle\int_{\omega_{\mathrm{ir}}}^{\omega_{\mathrm{uv}}}\frac{A_{\varepsilon}}{\left(\omega t_{0}\right)^{\alpha}}d\omega=\pi\sigma_{\omega q}^{2}.$

(11)

Here, we take the cutoffs as $\omega_{\mathrm{ir}}/2\pi=100\ \mathrm{kHz}$ and $\omega_{\mathrm{uv}}/2\pi=20\ \mathrm{GHz}$ Abadillo-Uriel et al. (2019). In our simulation, we take the time unit as $t_{0}=1/\Omega_{0}$. Therefore, by substituting $\alpha=1$, we estimate $A_{\varepsilon}t_{0}\simeq 10^{-3}$. Here, the $1/f$ noise in the simulation is generated by using the method as described in Yang and Wang (2016). On the other hand, for the piecewise continuous Hamiltonian (see Eq. 4), the time-dependent noise spectrum and the infidelity can be well characterized via the filter transfer function Green et al. (2012, 2013); Paz-Silva and Viola (2014). The detail on how to analyze the fidelity using this method can be seen in Ref. Green et al. (2013). Here, we briefly introduce this method and focus on the $z$-component noise. For the desired gate rotation $U(t)$ which satisfies the Schrodinger equation $i\dot{U}(t)=H(t)U(t)$, one can define a specific control matrix Green et al. (2013)

$\displaystyle R_{ij}(t)=[\boldsymbol{R}(t)]_{ij}=\operatorname{Tr}[U^{\dagger}(t)\sigma_{i}U(t)\sigma_{j}]/2,$

(12)

with its Fourier transform as

$\displaystyle R_{ij}(\omega)=-i\omega\int_{0}^{T}dtR_{ij}(t)e^{i\omega t},$

(13)

where $i,j\in\{x,y,z\}$. And then, the average gate fidelity in the noise environment that is determined by the specific cross-spectral density matrix $S_{ij}(\omega)$ can be Green et al. (2013)

$\displaystyle\mathcal{F}_{\mathrm{av}}\simeq$

$\displaystyle 1-\frac{1}{2\pi}\sum_{i,j,k=x,y,z}\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{\omega^{2}}S_{ij}(\omega)R_{jk}(\omega)R_{ik}^{*}(\omega).$

(14)

Since we only consider the $z$-component noise, the cross-spectral density is with the form of $S_{z}(\omega)$. In this way, Eq. 14 reduces to

$\displaystyle\mathcal{F}_{\mathrm{av}}=$

$\displaystyle 1-\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega S_{z}(\omega)\frac{F_{z}(\omega)}{\omega^{2}}$

(15)

where

$\displaystyle F_{z}(\omega)=\sum_{k=x,y,z}R_{zk}(\omega)R_{zk}^{*}(\omega)$

(16)

is the filter transfer function for the $z$-component detuning noise. In Appendix. C, we have given a step-by-step example to calculate the filter transfer function. One can then use it to calculate the fidelity.

Considering the $1/f$ noise spectrum, one can further derive the fidelity as

$\displaystyle\mathcal{F}_{\mathrm{av}}=1-\frac{1}{2\pi}\int_{\omega_{\mathrm{ir}}}^{\omega_{\mathrm{uv}}}\frac{A_{\epsilon}}{\omega t_{0}}\frac{F_{z}(\omega)}{\omega^{2}}$

(17)

Note that, to simplify the calculation one can use the unit of $\Omega_{0}$ and the corresponding cutoffs turn to be $\omega_{\mathrm{ir}}^{\prime}=\omega_{\mathrm{ir}}/\Omega_{0}\simeq 2.220\times 10^{-3}$ and $\omega_{\mathrm{uv}}^{\prime}=\omega_{\mathrm{uv}}/\Omega_{0}\simeq 4.439\times 10^{2}$.

From Eq. 15 it is clear that for a given noise spectrum the related filter transfer function i.e. $F_{z}(\omega)/\omega^{2}$ has positive relationship with the infidelity. Therefore, it can be used to characterize the noise effect. Fig. 4 shows the results of the filter transfer function for the naive, the CORPSE, the geometric and the non-cyclic geometric quantum gates. In the plot, we show four different kinds of gates ($R(\hat{x},\pi/2)$, $R(\hat{z},\pi/2)$, $R(\hat{x}+\hat{y}-\hat{z},4\pi/3)$ and $R(\hat{x}+\hat{z},\pi)$), which are the representatives for the rotation around different axis in the single-qubit Clifford group. The design of these gates can be seen in Table. 1. The performance of other gates in the Clifford group is similar and is thus not shown. We find that in the low-frequency region i.e. $\omega/\Omega_{0}<10^{-1}$, the filter transfer function for the CORPSE gate is normally lower than others. This means that the CORPSE gate is more robust to the low-frequency noise. However, it stands out in the high-frequency region $10^{-1}<\omega/\Omega_{0}<10^{0}$. When the frequency is high enough, the performance for all the gates are not distinguishable. Using these filter transfer function results, we further calculate the gate fidelity as shown in the plot. It is shown that whether the non-cyclic geometric gate can offer improvement over the naive gates depends on the specific noise spectrum up to the gates. For example, for the gate $R(\hat{x},\pi/2)$ in (a), it performs worse than the naive gate. Meanwhile, for the gate $R(\hat{x}+\hat{z},\pi)$ in (d), it performs the same as the naive gate. On the other hand, both $R(\hat{z},\pi/2)$ in (b) and $R(\hat{x}+\hat{y}-\hat{z},4\pi/3)$ in (c) can outperform the naive gate. On the other hand no improvement is offered by the CORPSE gate.

The two-qubit entangling gate can be implemented by coupling two ST qubits via a cavity resonator as shown in Fig. 1(c). In the DQD eigenbasis, the total Hamiltonian of the hybrid system composed of the qubits and the resonator reads Abadillo-Uriel et al. (2019)

	$\displaystyle H_{\rm{tot}}=$	$\displaystyle\omega_{r}a^{\dagger}a+\sum_{k=1}^{2}\sum_{m=1}^{3}E_{m}^{(k)}\sigma_{mm}^{(k)}$		(18)
		$\displaystyle+\sum_{k=1}^{2}\sum_{m,n=1}^{3}g^{(k)}d_{mn}^{(k)}\left(a+a^{\dagger}\right)\sigma_{mn}^{(k)},$

where $a^{\dagger}$ ($a$) is the photon creation (annihilation) operator for the resonator and $\omega_{r}$ is the intrinsic frequency of the resonator, while $g^{(k)}$ is the coupling strength between the $k$th DQD and the resonator. $E_{m}^{(k)}$ and $\sigma_{mn}^{(k)}$ are the eigenvalue and the Pauli matrix for the $k$th DQD. By moving to the rotating frame defined by $U^{\prime}=e^{-iH_{0}t}$ where

$\displaystyle H_{0}=\omega_{r}a^{\dagger}a+\sum_{k=1}^{2}\sum_{m=1}^{3}E_{m}^{(k)}\sigma_{mm}^{(k)},$

(19)

one finds

$\displaystyle H_{\rm{int}}^{\prime}=$

$\displaystyle\sum_{k=1}^{2}\sum_{m=1}^{3}g^{(k)}d_{mm}^{(k)}e^{i\omega_{r}t}\sigma_{mm}^{(k)}a^{\dagger}+\sum_{k=1}^{2}\sum_{m<n}g^{(k)}d_{mn}^{(k)}(\sigma_{mn}a^{\dagger}e^{i(E_{m}^{(k)}-E_{n}^{(k)}+\omega_{r})t}+\sigma_{mn}ae^{i(E_{m}^{(k)}-E_{n}^{(k)}-\omega_{r})t})+h.c.$

(20)

In the experiments, the resonator frequency can be as high as several GHz, while the coupling strength is typically less than 100 MHz Abadillo-Uriel et al. (2019), indicating $|g^{(k)}d_{mn}^{(k)}|/2\pi\leq 50$ MHz and therefore we have $|g^{(k)}d_{mm}^{(k)}|\ll\omega_{r}$. Also, we can set the proper parameters in the Hamiltonian Eq. 1 to ensure $|g^{(k)}d_{mn}^{(k)}|\ll|E_{m}^{(k)}-E_{n}^{(k)}-\omega_{r}|$. In this way, the leakage to the excited states $|f\rangle$ (related to the terms $d_{gf}$ and $d_{ef}$) and the longitudinal coupling between the qubits and the resonator (related to the terms $d_{gg}$, $d_{ee}$ and $d_{ff}$) are strongly suppressed. This is owing to the fact that they can be regarded as the fast-oscillating terms. Further, if we set $\omega_{q}^{(k)}=\omega_{r}$, namely, both of the qubits are in resonance with the resonator, Eq. 20 reduces to

$\displaystyle H_{\rm{int}}^{\prime}\approx$

$\displaystyle\sum_{k=1}^{2}\Omega^{(k)}\sigma_{ge}^{(k)}a^{\dagger}+h.c.$

(21)

where $\Omega^{(k)}=g^{(k)}d_{ge}^{(k)}$. In the single-excitation subspace spanned by $\{|eg0\rangle,|ge0\rangle,|gg1\rangle\}$, Eq. 21 can establish an effective resonant three-level $\Lambda$ system. Here, $|abc\rangle$ represents the first and the second qubit and the resonator. When the evolution time satisfies $\int_{0}^{t}\Omega(t)\mathrm{d}t^{\prime}=\pi$, a two-qubit entangling gate is obtained Zhou et al. (2017); Egger et al. (2019); Li et al. (2020); Zhang et al. (2020b). In the computational subspace, which corresponds to the zero-photon subspace $\{|gg0\rangle,|ge0\rangle,|eg0\rangle,|ee0\rangle\}$ one finds

$\displaystyle U_{\rm{ent}}\left(\xi\right)=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&\cos\xi&\sin\xi&0\\ 0&\sin\xi&-\cos\xi&0\\ 0&0&0&-1\end{array}\right),$

(22)

where we have assumed $\Omega=\sqrt{\left(\Omega^{(1)}\right)^{2}+\left(\Omega^{(2)}\right)^{2}}$ and $\tan\xi/2=-\Omega^{(1)}/\Omega^{(2)}$.

The performance of the two-qubit gate can be evaluated via numerically solving the master equation Blais et al. (2007)

$\displaystyle\dot{\rho}=-i\left[H_{\rm{int}}^{\prime},\rho\right]+\Gamma_{a}\mathcal{D}[a]+\Gamma_{2}\mathcal{D}[\sigma_{z}]+\Gamma_{1}\mathcal{D}[\sigma],$

(23)

where

$\displaystyle\mathcal{D}[L]=\left(2L\rho L^{\dagger}-L^{\dagger}L\rho-\rho L^{\dagger}L\right)/2.$

(24)

Here, $\Gamma_{a}$ denotes the internal decay effect of the resonator, while $\Gamma_{1}$ and $\Gamma_{2}$ denote the relaxation and pure dephasing rate, respectively. Meanwhile, we assume the pure dephasing is mainly owing to the charge noise and the fast-oscillating terms, both of which have been considered in the Hamiltonian $H_{\rm{int}}^{\prime}$, we therefore set $\Gamma_{2}=0$. Note that, the relaxation of the ST qubits is complicated. However, as stated above, the relaxation rate in the silicon-based platform can be substantially low, since the relaxation time in the experiment has reached $T_{1}=9\ \rm{s}$ Ciriano-Tejel et al. (2021). Therefore, the relaxation of the coupled system can be mainly owing to the leakage to the excited state $|f\rangle$ and the resonator decay.

Under the action of the entangling gate $U_{\rm{ent}}\left(\xi=-\pi/2\right)$ which corresponds to $g^{(1)}=g^{(2)}=g^{\prime}$, the initial state $|ge0\rangle$ will transform into state $|eg0\rangle$ at the final evolution time. In Fig. 5(a) and (b) we show the population of the states $|ge0\rangle$ and $|eg0\rangle$, which corresponds to the cases $\Delta B<\tau$ and $\Delta B>\tau$, respectively. Here, we set $g^{\prime}=2\pi\times 100\ \rm{MHz}$. The resonator decay is taken as $\Gamma_{a}/2\pi=0.028\ \rm{MHz}$ according to the recent experiment, which corresponds to the resonator quality factor of about $Q=10^{5}$ Samkharadze et al. (2016). The population of $|eg0\rangle$ in Fig. 5(a) is obviously smaller than the case in Fig. 5(b) due to the strong leakage effect. To deeply study the robustness of the two-qubit gate, we further plot the fidelity as a function of $\sigma_{\omega q}/g^{\prime}$, as shown in Fig. 5 (c) and (d). To ensure convergence, we have averaged the fidelities over 1000 implementations for each $\sigma_{\omega q}/g^{\prime}$. When the noise is small, i.e. $\sigma_{\omega q}/g^{\prime}\simeq 0$, the fidelities for both cases are closed to 1. As $\sigma_{\omega q}/g^{\prime}$ increases, the fidelity in Fig. 5(c) drops quickly, and it is down to 92% for the large noise of $\sigma_{\omega q}/g^{\prime}\simeq 0.2$. For comparison, the fidelity in Fig. 5(d) falls more slowly. When $\sigma_{\omega q}/g^{\prime}\simeq 0.2$, the fidelity is still surpassing 97%. Thus, to achieve high-fidelity entangling gate, one is wise to operate it in the region $\Delta B>\tau$, which is consistent with the single-qubit case. Assuming that the charge noise here is with the same level as that of the single-qubit case, namely $\sigma_{\omega q}/g^{\prime}\simeq 0.05$, the fidelity in Fig. 5(d) can be as high as 99.63%.