High-fidelity single-qubit gates in a strongly driven quantum dot hybrid qubit with $1/f$ charge noise

TL;DR

This paper demonstrates that strong microwave driving combined with pulse shaping and simultaneous detuning and tunneling modulation can significantly improve the fidelity of single-qubit gates in semiconductor hybrid qubits affected by charge noise.

Contribution

The study introduces a theoretical and numerical analysis showing how pulse shaping and AC sweet spots enhance gate fidelity in the presence of charge and phonon noise.

Findings

Achievable $X_{}$ gate fidelities exceed 99.9%.

Pulse shaping suppresses strong-driving effects.

AC sweet spots mitigate charge noise impact.

Abstract

Semiconductor double quantum dot hybrid qubits are promising candidates for high-fidelity quantum computing. However, their performance is limited by charge noise, which is ubiquitous in solid-state devices, and phonon-induced dephasing. Here we explore methods for improving the quantum operations of a hybrid qubit, using strong microwave driving to enable gate operations that are much faster than decoherence processes. Using numerical simulations and a theoretical method based on a cumulant expansion, we analyze qubit dynamics in the presence of $1/f$ charge noise, which forms the dominant decoherence mechanism in many solid-state devices. We show that, while strong-driving effects and charge noise both reduce the quantum gate fidelity, simple pulse-shaping techniques effectively suppress the strong-driving effects. Moreover, a broad AC sweet spot emerges when the detuning parameter and the tunneling coupling are driven simultaneously. Taking into account phonon-mediated noise, we find that it should be possible to achieve $X_{\pi}$ gates with fidelities higher than $99.9\%$.

Equations

$$\mathcal{H}={\mathcal{H}}_q+{\mathcal{H}}_{\text{ac}}+{\mathcal{H}}_n,$$

$${\mathcal{H}}_q=\left(\begin{array}{ccc}-\frac{\epsilon }{2}& 0& {\Delta }_1\\ 0& -\frac{\epsilon }{2}+E_{\text{ST}}& -{\Delta }_2\\ {\Delta }_1& -{\Delta }_2& \frac{\epsilon }{2}\end{array}\right),$$

$${\mathcal{H}}_{\text{ac}}=V\cos ({\omega }_{d}t)=\left(\begin{array}{ccc}-\frac{A_{\epsilon }}{2}& 0& A_{\Delta }\\ 0& -\frac{A_{\epsilon }}{2}& -r{A}_{\Delta }\\ A_{\Delta }& -r{A}_{\Delta }& \frac{A_{\epsilon }}{2}\end{array}\right)\cos ({\omega }_{d}t),$$

$${\mathcal{H}}_n=h_n\delta \epsilon (t)=\left(\begin{array}{ccc}-\frac{1}{2}& 0& 0\\ 0& -\frac{1}{2}& 0\\ 0& 0& \frac{1}{2}\end{array}\right)\delta \epsilon (t),$$

$$\tilde{S}(\omega)=\left\{\begin{array}[]{cl}c_{\varepsilon}^{2}\frac{2\pi}{|\omega|}&(\omega_{l}\leq|\omega|\leq\omega_{h})\\ 0&(\text{otherwise})\end{array}\right.,$$

$$⟨{\rho }^I(t)⟩=e^{-\frac{1}{{\hslash }^2}\underset{0}{\overset{t}{\int }}d{t}_1\underset{0}{\overset{t_1}{\int }}d{t}_2\mathcal{L}(t_1)\mathcal{L}(t_2)S(t_1-t_2)}{\rho }^I(0)$$

$${\vec{r}}^I(t)=\exp [K(t)]{\vec{r}}^I(0),$$

$$\begin{gathered} [K(t){]}_{ij} \\ =-\frac{1}{{\hslash }^2}\sum _{{\omega }_1,{\omega }_2}\sum _{k,l,m=1}^8{\alpha }_{k,{\omega }_1}{\alpha }_{l,{\omega }_2}{T}_{km}^{(i)}{T}_{lj}^{(m)}I(t,{\omega }_1,{\omega }_2), \end{gathered}$$

$$I(t,{\omega }_1,{\omega }_2)\equiv {\int }_0^{t}d{t}_1{\int }_0^{t_1}d{t}_2{e}^{i{\omega }_1{t}_1}{e}^{i{\omega }_2{t}_2}S(t_1-t_2).$$

$$⟨{\rho }_{00}^I⟩(t)=\frac{1}{3}+\left(\frac{1}{2}-\frac{a}{6}\right)e^{-{\Gamma }_{y}t-2{\Gamma }_{\phi }^2\phi (t)}+\frac{1}{6}(1-a)e^{-{\Gamma }_{L}t},$$

$${\Gamma }_y={\left(\frac{1}{\hslash }\right)}^2\{\frac{{\bar{h}}_{n,1}^2}{4}\left[\tilde{S}(\omega +\Omega )+\tilde{S}(\omega -\Omega )+4\tilde{S}(\omega )\right]+\frac{{\bar{h}}_{n,5}^2}{4}\left[\tilde{S}\left(\frac{E_L-E_0}{\hslash }+\frac{\Omega }{2}\right)+\tilde{S}\left(\frac{E_L-E_0}{\hslash }-\frac{\Omega }{2}\right)\right]$$

$$+\frac{{\bar{h}}_{n,7}^2}{4}\left[\tilde{S}\left(\frac{E_L-E_1}{\hslash }+\frac{\Omega }{2}\right)+\tilde{S}\left(\frac{E_L-E_1}{\hslash }-\frac{\Omega }{2}\right)\right]\},$$

$${\Gamma }_L={\left(\frac{1}{\hslash }\right)}^2\{\frac{3{\bar{h}}_{n,5}^2}{4}\left[\tilde{S}\left(\frac{E_L-E_0}{\hslash }+\frac{\Omega }{2}\right)+\tilde{S}\left(\frac{E_L-E_0}{\hslash }-\frac{\Omega }{2}\right)\right]$$

$$+\frac{3{\bar{h}}_{n,7}^2}{4}\left[\tilde{S}\left(\frac{E_L-E_1}{\hslash }+\frac{\Omega }{2}\right)+\tilde{S}\left(\frac{E_L-E_1}{\hslash }-\frac{\Omega }{2}\right)\right]\},$$

$${\Gamma }_{\phi }^2={\left(\frac{c_{\epsilon }}{\hslash }\right)}^2{\left(\frac{d{f}_{\text{Rabi}}}{d\epsilon }\right)}^2,$$

$$\phi (t)=t^2\left[\log (1/{\omega }_{l}t))-{\gamma }_E+3/2\right],$$

$$\text{vspace}1ina=\frac{3(E_L-E_0{)}^2{\bar{V}}_{0L}^2}{(E_L-E_0-hf{)}^2(E_L-E_0+hf{)}^2}+\frac{3(E_L-E_1{)}^2{\bar{V}}_{1L}^2}{(E_L-E_1-hf{)}^2(E_L+E_1+hf{)}^2}.$$

$$F^I=1-\frac{1}{12}[3{\Gamma }_x+6{\Gamma }_y+{\Gamma }_L]t-\frac{1}{2}{\Gamma }_{\phi }^2\phi (t)-\frac{1}{3}a,$$

$$\begin{gathered} {\Gamma }_x={\left(\frac{1}{\hslash }\right)}^2\{\frac{{\bar{h}}_{n,1}^2}{2}\left[\tilde{S}(\omega +\Omega )+\tilde{S}(\omega -\Omega )\right] \\ +\frac{{\bar{h}}_{n,5}^2}{4}\left[\tilde{S}\left(\frac{E_L-E_0}{\hslash }+\frac{\Omega }{2}\right)+\tilde{S}\left(\frac{E_L-E_0}{\hslash }-\frac{\Omega }{2}\right)\right] \\ +\frac{{\bar{h}}_{n,7}^2}{4}\left[\tilde{S}\left(\frac{E_L-E_1}{\hslash }+\frac{\Omega }{2}\right)+\tilde{S}\left(\frac{E_L-E_1}{\hslash }-\frac{\Omega }{2}\right)\right]\} \end{gathered}$$

$$p(t)=\left\{\begin{array}[]{cc}\vspace{0.03in}\frac{t_{g}[1-\cos(\pi t/t_{r})]}{2(t_{g}-t_{r})}&(0\leq t\leq t_{r}),\\ \vspace{0.03in}\frac{t_{g}}{t_{g}-t_{r}}&(t_{r}<t<t_{g}-t_{r}),\\ \frac{t_{g}[1+\cos(\pi[t-t_{g}+t_{r}]/t_{r})]}{2(t_{g}-t_{r})}&(t_{g}-t_{r}\leq t\leq t_{g}),\end{array}\right.$$

$$F_{\text{ch+ph}}\approx F-t_g/T_{\text{ph}},$$

$$i\hslash \frac{d}{dt}{U}_0=[{\bar{\mathcal{H}}}_q+{\bar{\mathcal{H}}}_{\text{ac}}]U_0.$$

$$\bar{V}\approx$$

$$A_{\Delta }\left(\begin{array}{ccc}-\frac{{\Delta }_1}{\epsilon }+\frac{{\Delta }_{2}r}{\epsilon -E_{\text{ST}}}& \frac{{\Delta }_{1}r}{\epsilon }+\frac{{\Delta }_2}{\epsilon -E_{\text{ST}}}& 1\\ \frac{{\Delta }_{1}r}{\epsilon }+\frac{{\Delta }_2}{\epsilon -E_{\text{ST}}}& \frac{{\Delta }_1}{\epsilon }-\frac{{\Delta }_{2}r}{\epsilon -E_{\text{ST}}}& -r\\ 1& -r& \frac{3{\Delta }_1}{\epsilon }+\frac{3{\Delta }_{2}r}{\epsilon -E_{\text{ST}}}\end{array}\right)+A_{\epsilon }\left(\begin{array}{ccc}\frac{{\Delta }_1^2}{2{\epsilon }^2}-\frac{{\Delta }_2^2}{2(\epsilon -E_{\text{ST}}{)}^2}& -\frac{{\Delta }_1{\Delta }_2}{\epsilon (\epsilon -E_{\text{ST}})}& -\frac{{\Delta }_1}{\epsilon }\\ -\frac{{\Delta }_1{\Delta }_2}{\epsilon (\epsilon -E_{\text{ST}})}& -\frac{{\Delta }_1^2}{2{\epsilon }^2}+\frac{{\Delta }_2^2}{2(\epsilon -E_{\text{ST}}{)}^2}& \frac{{\Delta }_2}{\epsilon -E_{\text{ST}}}\\ -\frac{{\Delta }_1}{\epsilon }& \frac{{\Delta }_2}{\epsilon -E_{\text{ST}}}& 1-\frac{3{\Delta }_1^2}{2{\epsilon }^2}-\frac{3{\Delta }_2^2}{2(\epsilon -E_{\text{ST}}{)}^2}\end{array}\right).$$

$$⟨\alpha |{\mathcal{H}}_{\text{QM}}|\alpha ⟩={\mathcal{H}}_{\text{semi}}+N\hslash {\omega }_d,$$

$$⟨i,n|{\mathcal{H}}_{\text{dot}}|j,m⟩=E_i{\delta }_{i,j}{\delta }_{n,m},$$

$$⟨i,n|{\mathcal{H}}_{\text{ph}}|j,m⟩=n\hslash {\omega }_d{\delta }_{i,j}{\delta }_{n,m},$$

$$⟨i,n|V_{\text{int}}|j,m⟩=\frac{{\bar{V}}_{ij}}{2}({\delta }_{n,m+1}+{\delta }_{n,m-1}).$$

$$i\hslash \frac{d}{dt}{\rho }^I=\delta \epsilon (t)\mathcal{L}{\rho }^I,$$

$$⟨{\rho }^I(t)⟩=\exp \left\{\sum _{n=1}^{\infty }\frac{(-i{)}^n}{{\hslash }^n}\underset{0}{\overset{t}{\int }}d{t}_1\cdots \underset{0}{\overset{t_{n-1}}{\int }}d{t}_n⟨\mathcal{L}(t_1)\delta \epsilon (t_1)\cdots \mathcal{L}(t_n)\delta \epsilon (t_n){⟩}_c\right\}{\rho }^I(0),$$

$$[K(t){]}_{ij}$$

$$\begin{array}[]{llll}\lambda_{1}=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&0\end{pmatrix},&\lambda_{2}=\begin{pmatrix}0&-i&0\\ i&0&0\\ 0&0&0\end{pmatrix},&\lambda_{3}=\begin{pmatrix}1&0&0\\ 0&-1&0\\ 0&0&0\end{pmatrix},&\lambda_{4}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&-2\end{pmatrix},\\ \\ \lambda_{5}=\begin{pmatrix}0&0&1\\ 0&0&0\\ 1&0&0\end{pmatrix},&\lambda_{6}=\begin{pmatrix}0&0&-i\\ 0&0&0\\ i&0&0\end{pmatrix},&\lambda_{7}=\begin{pmatrix}0&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix},&\lambda_{8}=\begin{pmatrix}0&0&0\\ 0&0&-i\\ 0&i&0\end{pmatrix}.\end{array}$$

$$i\hslash \frac{d}{dt}{\rho }^I=\delta \epsilon \mathcal{L}{\rho }^I\equiv \delta \epsilon [h_n^I,{\rho }^I].$$

$${\bar{h}}_n\simeq \left(\begin{array}{ccc}\frac{{\Delta }_1^2}{2{\epsilon }^2}-\frac{{\Delta }_2^2}{2(\epsilon -E_{\text{ST}}{)}^2}& -\frac{{\Delta }_1{\Delta }_2}{\epsilon (\epsilon -E_{\text{ST}})}& -\frac{{\Delta }_1}{\epsilon }\\ -\frac{{\Delta }_1{\Delta }_2}{\epsilon (\epsilon -E_{\text{ST}})}& -\frac{{\Delta }_1^2}{2{\epsilon }^2}+\frac{{\Delta }_2^2}{2(\epsilon -E_{\text{ST}}{)}^2}& \frac{{\Delta }_2}{\epsilon -E_{\text{ST}}}\\ -\frac{{\Delta }_1}{\epsilon }& \frac{{\Delta }_2}{\epsilon -E_{\text{ST}}}& 1-\frac{3{\Delta }_1^2}{2{\epsilon }^2}-\frac{3{\Delta }_2^2}{2(\epsilon -E_{\text{ST}}{)}^2}\end{array}\right),$$

$$S(t)=2c_{\epsilon }^2\left[\mathrm{Ci}({\omega }_h|t|)-\mathrm{Ci}({\omega }_l|t|)\right],$$

Full text

I Introduction

Semiconductor quantum dot qubits are promising platforms for quantum information processing. These qubits are controlled by manipulating either electric voltages or magnetic fields, using DC pulses or microwave drives. Microwave driving has several benefits for high-fidelity gates, including phase control of the rotation axis, amplitude control of the gate speed, protection against low-frequency noise, and the ability to perform the operations while keeping the qubit centered at a sweet spot. Recently, high-fidelity resonantly driven single-qubit gates have been realized in several quantum dot architectures, including single-electron spin qubits Zajac et al. (2017); Watson et al. (2018); Yoneda et al. (2018), singlet-triplet qubits Nichol et al. (2017), and quantum dot hybrid qubits Kim et al. (2015a).

Quantum dot hybrid qubits can be operated fully electrically, allowing for fast microwave manipulations, while maintaining insensitivity to charge noise by operating at an extended sweet spot. However, gate fidelities are still limited by detuning charge noise Thorgrimsson et al. (2017); Yang et al. (2017). An obvious strategy for outpacing noise effects is to increase gate speeds by applying a strong microwave drive, although this can also cause control errors. In a previous paper, we showed that, in the absence of noise, high fidelities can be achieved by accounting for the strong-driving-induced Bloch-Siegert shift of the qubit frequency and employing smoothed pulse envelopes to suppress fast oscillations and leakage Yang et al. (2017). In this paper, we include the effects of charge noise with a $1/f$ spectrum. We find that, while AC driving reduces the direct effects of low-frequency noise  Yan et al. (2013); Ithier et al. (2005); Wong (2016), the interplay between strong driving and charge noise can also suppress the fidelity. Indeed, when crosstalk effects dominate, increasing the driving amplitude is typically harmful for the gate performance. Clarifying the effects of strong driving on decoherence is therefore critical for achieving high-fidelity gates.

In this work, we use numerical and analytical methods to investigate the crosstalk between strong driving and charge noise. We show that it is possible to achieve high-fidelity single-qubit rotations in quantum dot hybrid qubits under these conditions. We also find that the infidelity caused by decoherence can be greatly reduced by driving the detuning and tunnel coupling simultaneously and coherently. In this way, we show that $X_{\pi}$ gates with fidelities higher than $99.9\%$ can be achieved over a broad range of control parameters. We further include phonon decoherence phenomenologically and show that $X_{\pi}$ gate fidelities can still be larger than $99.9\%$, provided that the phonon decoherence time scale is larger than $2\;\mathrm{\mu s}$.

The paper is organized as follows. Section II briefly introduces the double quantum dot hybrid qubit. In Sec. III, we provide an analytical description of the hybrid qubit dynamics, subject to charge noise in the detuning parameter, and compare these results to numerical simulations. In Sec. IV, we propose methods for improving the fidelity and show that $X_{\pi}$ gates with fidelities $>99.9\%$ are feasible, even in the presence of realistic levels of $1/f$ charge noise and phonon dephasing. We conclude in Sec. V. Additional technical details of the calculations are provided in the Appendices.

II Theoretical Model

We now present the full theoretical model used in this work for the quantum dot hybrid qubit. The Hamiltonian is given by

[TABLE]

where the three components of the Hamiltonian are described below.

The Qubit Hamiltonian, $\mathcal{H}_{q}$. The quantum dot hybrid qubit is formed of three electrons in a double quantum dot, with total spin quantum numbers $S=1/2$ and $S_{z}=-1/2$ Shi et al. (2012); Koh et al. (2012). For the operating regime of interest, we consider the three-dimensional (3D) basis composed of $|{\cdot S}\rangle\equiv|{\downarrow\!S}\rangle$, $|{\cdot T}\rangle\equiv\sqrt{1/3}|{\downarrow\!T_{0}}\rangle-\sqrt{2/3}|{\uparrow\!T_{-}}\rangle$, and $|{S\cdot}\rangle\equiv|{S\!\downarrow}\rangle$, where $|{\cdot}\rangle$ denotes a dot with one electron, the singlet state $|S\rangle=(|{\uparrow\downarrow}\rangle-|{\downarrow\uparrow}\rangle)/\sqrt{2}$, and the triplet states $|T_{0}\rangle=(|{\uparrow\downarrow}\rangle+|{\downarrow\uparrow}\rangle)/\sqrt{2}$ and $|T_{-}\rangle=|{\downarrow\downarrow}\rangle$ denote the spin states of dots with two electrons. In this basis, the qubit Hamiltonian is given by

[TABLE]

where $\varepsilon$ is the detuning parameter, corresponding to the energy difference between the dots, $E_{\text{ST}}$ approximately corresponds to the singlet-triplet energy splitting in the doubly occupied dot, and $\Delta_{1}$ ($\Delta_{2}$) are the tunnel couplings between the states $|{S\cdot}\rangle$ and $|{\cdot S}\rangle$ ($|{\cdot T}\rangle$). The various parameters are labelled in the schematic diagram shown in Fig. 1(a), and a typical energy diagram is shown in Fig. 1(c), where the two low-energy states $|{0}\rangle$ and $|{1}\rangle$ comprise the qubit, while the high-energy state $|{L}\rangle$ is a leakage state. To simplify our analysis later, we transform Eq. (1) to the energy basis, $\{|0\rangle,|1\rangle,|L\rangle\}$, yielding $\bar{\mathcal{H}}_{q}=\text{diag}[E_{0},E_{1},E_{L}]$, where the bar indicates the energy basis, and $\{E_{i}\}$ are the corresponding energy eigenvalues.

The AC Drive, $\mathcal{H}_{\text{ac}}$. We consider two different schemes for AC driving Koh et al. (2013), as shown in Fig. 1(b). In the first case, we modulate the tunnel couplings as $\Delta_{i}=\Delta_{i0}+r_{i}\Delta_{\text{ac}}(t)$, where $i=1,2$ [purple arrow in Fig. 1(b)]. The AC drive $\Delta_{\text{ac}}$ is achieved by applying a microwave voltage signal to one of the device top-gates Kim et al. (2015a). It is reasonable to assume that the same modulation drives both $\Delta_{1}$ and $\Delta_{2}$, although they may be affected differently, which we take into account through the variable $r_{i}$. In the second case, we modulate the detuning as $\varepsilon=\varepsilon_{0}+\varepsilon_{\text{ac}}(t)$ [orange arrow in Fig. 1(b)]. We first consider rectangular pulses, $\Delta_{\text{ac}}(t)=A_{\Delta}\cos(\omega_{d}t)$ and $\varepsilon_{\text{ac}}(t)=A_{\varepsilon}\cos(\omega_{d}t)$, for which the pulse amplitudes $A_{\Delta}$ and $A_{\varepsilon}$ are piecewise constant in time, and $\omega_{d}$ is the driving angular frequency. The driving Hamiltonian expressed in the $\{|\cdot S\rangle,|\cdot T\rangle,|S\cdot\rangle\}$ basis can then be written as

[TABLE]

where we refer to $V$ as the driving matrix, and we simplify the expression by defining $r_{1}=1$ and $r_{2}=r$. Note that we keep $r$ as a variable here for generality; however in our simulations, we choose $r=1$ for simplicity.

The Noise Hamiltonian, $\mathcal{H}_{n}$. The parameter $\varepsilon$ in Eq. (2) represents the desired value of the detuning, and is controlled by voltages applied to the device top-gates. However, charge noise within the device causes the detuning to fluctuate by $\delta\varepsilon(t)$, which represents the dominant source of decoherence for hybrid qubits Thorgrimsson et al. (2017). The resulting noise Hamiltonian in the $\{|\cdot S\rangle,|\cdot T\rangle,|S\cdot\rangle\}$ basis is given by

[TABLE]

where $h_{n}$ is the dimensionless noise matrix. The noise is characterized by the time correlation function $S(t_{1}-t_{2})=\langle\delta\varepsilon(t_{1})\delta\varepsilon(t_{2})\rangle$, where the brackets denote an average over noise realizations and the noise is assumed to be stationary with zero mean ($\langle\delta\varepsilon\rangle=0$). The corresponding noise power spectrum is given by $\tilde{S}(\omega)=\int_{-\infty}^{\infty}dt\,e^{i\omega t}S(t)$ Clerk et al. (2010). In solid-state devices, the charge noise typically exhibits a $1/f$ power spectrum density. In this work, we adopt the following model for $1/f$ charge noise Kuhlmann et al. (2013); Paladino et al. (2014):

[TABLE]

where $c_{\varepsilon}$ is the noise amplitude, and $\omega_{l}$ ($\omega_{h}$) are the low (high) angular frequency cutoffs. In the simulations described below, time sequences for $\delta\varepsilon(t)$ are obtained by first generating a white noise sequence, then scaling its Fourier transform by the spectral density function given in Eq. (5), following the procedure described in Refs. Kawakami et al. (2016); Yang et al. (2019), and Appendix A.

III Theoretical Methods

In this section, we perform simulations and analytical calculations of a strongly driven hybrid qubit in the presence of charge noise, obtaining excellent agreement between the two methods. We also obtain simple analytical expressions for the decoherence rates of the density matrix in the long-time limit.

Figure 2 shows the results of numerical simulations of the density matrix, using the methods described in Appendix A, when the qubit is initialized into the $|{0}\rangle$ state. Here and throughout this work, we choose fixed values of $E_{\text{ST}}$ and $\varepsilon$, as indicated in the figure caption, which are representative of quantum dot hybrid qubits Thorgrimsson et al. (2017). The tunnel couplings are chosen such that $\Delta_{1}=\Delta_{2}=0.7E_{\text{ST}}$, which approximately corresponds to a second-order DC sweet spot for this system, in the limit of large $\varepsilon$ Yang et al. (2017). The AC driving parameter $A_{\Delta}$ is also chosen to be representative, experimentally; in later calculations, the AC parameters are allowed to vary. The noise parameters are also specified in the figure caption, and the resulting density matrix is averaged over 10,000 realizations of $\delta\varepsilon(t)$. For the results shown in Fig. 2(a), the qubit exhibits Rabi oscillations, as well as small-amplitude fast oscillations caused by leakage and the counter-rotating terms in the qubit evolution, as previously explained in Ref. Yang et al. (2017). The leakage component of the density matrix also exhibits fast oscillations, as shown in Fig. 2(b). However, distinct from Ref. Yang et al. (2017), the Rabi oscillations here decay due to charge noise. Moreover, the leakage component accumulates on longer time scales, as seen in the inset of Fig. 2(b).

To differentiate between the effects of decoherence and strong driving, we recompute the noise-averaged density matrix after first transforming to the interaction frame, defined as $\rho^{I}=U_{0}^{\dagger}\rho U_{0}$, where $\rho$ is expressed in the energy basis, and $U_{0}$ is the time-evolution operator for the energy frame, including strong-driving dynamics but not charge noise, as derived in Appendix B. In this interaction frame, noise-free dynamics simply correspond to $\rho^{I}\!(t)=\text{const.}$ However, the numerical results in Fig. 2(c) decay over time, due to the presence of charge noise. Here, short-time behavior is plotted in the inset, where a careful examination shows that small-amplitude, high-frequency oscillations still persist, as an example of the crosstalk between noise and strong driving. This particular effect can be classified as ‘non-Markovian,’ as discussed below.

We now obtain analytical expressions for the dynamics of strongly driven hybrid qubits in the presence of charge noise using a technique developed for quantum dot charge qubits Yang et al. (2019), which is based on a cumulant expansion. The method is summarized as follows, with further details provided in Appendix C. First, we calculate the qubit evolution in the interaction frame, as governed by the equation $i\hbar\,d\rho^{I}\!/dt=\delta\varepsilon(t)\mathcal{L}\rho^{I}$, where $\mathcal{L}\rho^{I}\equiv[h_{n}^{I},\rho^{I}]$, $h_{n}^{I}=U_{0}^{\dagger}\bar{h}_{n}U_{0}$, and $\bar{h}_{n}$ is expressed in the energy basis. To determine $h_{n}^{I}$, we follow Ref. Yang et al. (2017) in expanding $U_{0}$ in powers of the small parameter $\gamma\sim V/\hbar\omega_{d}$, obtaining $U_{0}=U_{0}^{(0)}+U_{0}^{(1)}+U_{0}^{(2)}+\cdots$, where $U_{0}^{(n)}\propto\gamma^{n}$. We then perform a noise average of $\rho^{I}$, similar to Ref. Yang et al. (2019), adopting $\delta\varepsilon/V$ as a small parameter in the cumulant expansion Kubo (1962), obtaining

[TABLE]

at $\mathcal{O}[(\delta\varepsilon/V)^{2}]$. To simplify the calculation, we expand our result in terms of Gell-Mann matrices $\{\lambda_{i}\}_{i=1,\cdots,8}$, as defined in Appendix D, obtaining $\langle\rho^{I}\rangle=I_{3}/3+\sum_{i=1}^{8}\vec{r}^{I}_{i}\lambda_{i}/2$, where $I_{3}$ is the $3\times 3$ identity matrix and $\vec{r}^{I}$ is a generalized, 8D Bloch vector. The noise matrix in the interaction frame can also be expressed as $h_{n}^{I}=\sum_{i=1}^{8}h_{n,i}^{I}\lambda_{i}$, and the Bloch vector can be rewritten as

[TABLE]

where $K(t)$ is an $8\times 8$ matrix. Expanding $h_{n,i}^{I}(t)$ in a Fourier series, $h_{n,i}^{I}(t)=\sum_{\omega}\alpha_{i,\omega}e^{i\omega t}$, we obtain

[TABLE]

where $T_{ij}^{(k)}$ is a structure constant defined as $[\lambda_{i},\lambda_{j}]\equiv\sum_{k=1}^{8}T_{ij}^{(k)}\lambda_{k}$, and

[TABLE]

The physics of the noisy qubit dynamics is encoded in $K(t)$. We can characterize different types of behavior by decomposing $K(t)$ into a sum of pure-dephasing ($K_{\varphi}$), Markovian ($K_{\text{M}}$), and non-Markovian-non-dephasing terms ($K_{\text{nMn}\varphi}$) Yang et al. (2019). Note that pure dephasing is defined, here, with respect to the interaction frame, and is associated with the integral $I(t,\omega_{1}\text{=}0,\omega_{2}$=$0)\sim t^{2}\ln(1/\omega_{l}t)$ Paladino et al. (2002); Astafiev et al. (2004); Ithier et al. (2005). The Markovian terms induce characteristic exponential decay ($e^{-\Gamma t}$), which is associated with the Markovian approximation and the integral $\text{Re}[I(t,\omega,-\omega)]$. The non-Markovian-non-dephasing terms describe any additional effects, including the small-amplitude, high-frequency oscillations observed in Fig. 2(c).

The formalism described above allows us to compare and contrast various decoherence mechanisms within a common framework. We now describe the main results. In Fig. 2, our analytical results are plotted as white dashed lines, for comparison with the simulations. Figures 2(a) and 2(b) demonstrate that the analytical calculations can accurately describe the main features of the simulations, including decoherence, with deviations between $\langle\rho_{00}\rangle$, $\langle\rho_{01}\rangle$, and $\langle\rho_{LL}\rangle$ and their simulated values falling below $10^{-3}$ in all cases (bottom panels).

We can also obtain simple, approximate expressions for the long-time dynamics, as described in Appendix E. For the initial state $\rho(t=0)=|0\rangle\!\langle 0|$, we obtain the leading order behavior 111Note that we use slightly different notation here, as compared to Ref. Yang et al. (2019). In the latter, $\Gamma_{\varphi}$ had units of $\text{(sec)}^{-2}$. Here, $\Gamma_{\varphi}$ has units of $\text{(sec)}^{-1}$, and is a true rate.

[TABLE]

where

[TABLE]

Here, $\Gamma_{y}$ and $\Gamma_{L}$ represent decoherence rates associated with the Markovian approximation ($\Gamma_{L}$ specifically describes leakage processes), $\Gamma_{\varphi}$ is the pure-dephasing rate, $a$ describes the asymptotic occupation of the leakage state, and we define $\bar{h}_{n}=\sum_{i=1}^{8}\bar{h}_{n,i}\lambda_{i}$. Note that the terms $\varphi$ and $a$ are induced by strong driving and are therefore responsible for the crosstalk with decoherence processes. Equation (10) [plotted as a cyan dashed line in Fig. 2(c)] accurately describes the coarse features of the simulation, except at extremely short times (inset), where it incorrectly assigns a non-zero leakage. A more accurate description of the short-time behavior, and other fine-scale features, requires retaining the full analytical expressions, as shown in the inset of Fig. 2(c) (dashed white line). Since $\varphi(t)\sim t^{2}\log(t)$ dominates the decay of the density matrix for typical gate times (see below), it suggests that reducing $\Gamma_{\varphi}$ would significantly improve qubit performance.

IV High-fidelity $X_{\pi}$ Gates

In this section, we first obtain an analytical expression for the fidelity at the asymptotic time scales relevant for quantum gates. We then demonstrate that, by driving the detuning and tunnel coupling simultaneously, in a specific ratio, pure dephasing can be fully suppressed, $\Gamma_{\varphi}=0$, in principle enabling significant improvements in the fidelity. Finally we show, via simulations and analytical calculations, that in the presence of charge noise and strong driving effects, the fidelity of an $X_{\pi}$ gate can be higher than $99.9\%$.

IV.1 Asymptotic Fidelity Results

We define the asympotic time regime as $1/\omega_{l}\gg t/(2\pi)\sim 1/\Omega\gg 1/\omega\gg 1/\omega_{h}$, where $\omega$ represents any angular frequency associated with resonant driving, other than the Rabi frequency $\Omega$, which is the slowest frequency in the system. In this regime, Eq. (10) approximates the noise-averaged, driven density matrix in the interaction frame. As outlined in Appendices E and F, we can use this to compute the process fidelity as a function of time, obtaining the simple expression

[TABLE]

where

[TABLE]

is also associated with the Markovian approximation. For typical system parameters, we find that the $\Gamma_{\varphi}$ term in Eq. (11) dominates the infidelity. Suppressing this contribution requires suppressing $df_{\text{Rabi}}/d\varepsilon$, or equivalently, $d\bar{V}_{01}/d\varepsilon$, where $\bar{V}_{ij}$ refers to a particular component of the $V$ matrix in the energy basis. Remarkably, we find that it is possible to satisfy this condition exactly, $df_{\text{Rabi}}/d\varepsilon=d\bar{V}_{01}/d\varepsilon=0$, over a continuous range of driving parameters $A_{\varepsilon}$ and $A_{\Delta}$, as consistent with an AC sweet spot Didier et al. . In the present case, it is easy to show that $\bar{V}_{0L}$, $\bar{V}_{1L}$, and thus $a$ are suppressed at such tunings, further improving the fidelity.

We illustrate the effectiveness of the AC sweet spot technique by simulating the $X_{\pi}$ gate fidelity for a 2D sweep over the driving amplitudes $A_{\varepsilon}$ and $A_{\Delta}$, as shown in Fig 3(a). Here, we compute the process fidelity, as described in Appendix E, by comparing the noise-averaged simulation results to an ideal $X_{\pi}$ rotation. The location of the AC sweet spot is determined numerically and plotted as a white dashed line. Near this line, we observe a broad region with gate fidelities greater than $99.9\%$. Further away from the line, in the upper-left portion of the plot, we observe fast oscillations, which arise due to strong driving (e.g., counter-rotating terms; see the discussion below). In the lower-right portion of the plot, we observe a region of low fidelity; here, the two driving fields interfere to produce a slow gate that is strongly affected by charge noise. To demonstrate the effectiveness of our theoretical formalism, we also compute the $X_{\pi}$ gate fidelity theoretically, using Eq. (6). These results, shown in Fig. 3(d), quantitatively reproduce all the features of the simulations.

IV.2 Suppressing Strong-Driving Effects

In the previous section, we noted that the oscillations observed in the top-left corner of Figs. 3(a) and 3(d) are caused by strong driving. To substantiate this claim, we determine the $X_{\pi}$ gate fidelity in the interaction frame, where the ‘ideal’ density matrix, used to compute $F^{I}$, encompasses the full coherent evolution, including strong driving effects. The results, plotted in Fig. 3(e), exhibit strongly suppressed oscillations, indicating that the oscillations in Fig. 3(d) indeed arise from strong driving. Closer inspection of Fig. 3(e) reveals weak, residual oscillations, which arise at higher order in the strong-driving $\gamma$ expansion, and are a manifestation of the crosstalk between strong-driving effects and charge noise. In Fig. 3(f), we also plot the simplified form of $F^{I}$ obtained in Eq. (11), which contains strong-driving corrections, including as the Bloch-Siegert shift of the resonant frequency, but no strong-driving fast oscillations. The results reproduce the essential features in Fig. 3(e), explaining why this result can accurately predict the position of the AC sweet-spot line, as plotted in Fig. 3(a).

In experiments, it is easy to account for the Bloch-Siegert shift by recalibrating the resonance frequency. However, avoiding the fidelity oscillations in Fig. 3(a) may require careful tuning. A simpler approach is to incorporate a smooth pulse envelope $p(t)$ into the driving term $\bar{\mathcal{H}}_{\text{ac}}=\bar{V}p(t)\cos(\omega_{d}t)$ Deng et al. (2015, 2016), as opposed to the rectangular pulse envelope used in Fig. 3(a). To investigate this possibility, we consider a “smoothed-rectangular” pulse envelope, defined as Yang et al. (2017)

[TABLE]

where $t_{g}$ is the pulse width, and we choose a rise time of $t_{r}=h/E_{ST}\sim 0.83\,\mathrm{ns}$, to give a pulse that is sufficiently adiabatic. Figure 3(b) shows the improved simulation results for the gate fidelity in the presence of charge noise, using this pulse form. In comparison with Fig. 3(a), we now observe a suppression of strong-driving effects and a broad regime with gate fidelities >$$99.9\%. In fact, the gate fidelities obtained with the smoothed rectangular pulse are nearly identical to those in Fig. 3(e), which do not include direct strong-driving effects, therefore representing an upper bound. This suggests that charge noise, rather than strong driving, limits the fidelity in this case.

IV.3 Phenomenological Treatment of Phonons

A full treatment of phonons is outside the scope of this work; however, the following considerations allow us to estimate their effect. Dephasing is known to be the greatest threat from phonons for hybrid qubits, occurring on time scales of order microseconds Gamble et al. (2012). We account for this process here by phenomenologically expressing the total fidelity as

[TABLE]

as appropriate for Markovian processes at short times. Here, $F$ is the fidelity obtained in previous sections, describing the effects of charge noise, $t_{g}$ is the gate time, and $T_{\text{ph}}$ is the phonon decoherence time. As apparent in this expression, phonon dephasing effects are less effective for shorter gates.

In the inset of Fig. 3(c), we plot the results of a typical $X_{\pi}$ gate fidelity calculation based on Eq. (13), including both charge noise and phonon effects, and assuming smoothed rectangular pulse envelopes. While $T_{\text{ph}}=3\;\mathrm{\mu s}$ is held constant throughout this plot, we note that $t_{g}$ is a function of both $A_{\varepsilon}$ and $A_{\Delta}$. Here we observe a relatively large region with fidelities >$$99.9\%, with the maximum fidelity occurring at the spot marked with a star. Repeating this calculation for a range of $T_{\text{ph}}$, we obtain the fidelity maxima shown in the main panel of Fig. 3(c). Generally, we conclude that fidelities >$$99.9\% can be achieved when $T_{\text{ph}}>2\,\mathrm{\mu s}$.

V Summary and Conclusions

We have studied the dynamics of a strongly driven double quantum dot hybrid qubit in the presence of $1/f$ detuning charge noise, both analytically and numerically. Our analytical results accurately reproduce the numerical simulations, and therefore provide insight into the dependence of the fidelity on the experimental parameters. In particular, the asymptotic fidelity in Eq. (11) is quite accurate for typical gate times, and can therefore be used to design high-fidelity gate protocols. Using these results, we have shown that high-fidelity $X_{\pi}$ gates can be achieved by simultaneously and coherently driving the detuning and tunnel coupling, and that unwanted fast oscillations caused by strong driving can be suppressed by using smoothed rectangular pulse envelopes. The predicted gate fidelities are above $99.9\%$, over a wide parameter regime, even in the presence of phonon-induced dephasing, which we treat phenomenologically here.

Moving forward, we note that our analytical formalism, based on a cumulant expansion, can be readily generalized to systems with multiple qubits. For example, quantum dot spin qubits Zajac et al. (2017); Watson et al. (2018); Yoneda et al. (2018), single-triplet qubits Nichol et al. (2017), charge qubits Hayashi et al. (2003); Gorman et al. (2005); Kim et al. (2015b); Yang et al. (2019), and hybrid qubits Kim et al. (2015a), as well as one and two-qubit gate operations, may all be investigated using the methods described here. In each case, the gate performance can be improved by identifying optimal working points or working strategies, such that the device is less susceptible to the dominant decoherence channel, or other control errors.

*Acknowledgments: * We thank M. A. Eriksson for enlightening conversations. This work was supported in part by ARO (W911NF-17-1-0274) and the Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through grant N00014-15-1-0029. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

Appendix A Details of the Numerical Simulations

For all the numerical simulations shown in the main text, we follow the method described in Ref. Yang et al. (2019), which we briefly summarize as the follows: We first generate noise realizations $\{\delta\varepsilon^{\alpha}(t)\}_{\alpha=1,\ldots,\alpha_{\text{max}}}$ whose power spectral density is $1/f$, as described in Eq. (5). We then solve the Schr$\ddot{\text{o}}$dinger equation $i\hbar\frac{d}{dt}|\psi(t)\rangle=[\mathcal{H}_{q}+\mathcal{H}_{\text{ac}}+\mathcal{H}_{n}]|\psi(t)\rangle$ numerically for each realization using the method we developed in Ref. Yang et al. (2019). The solutions are labeled as $\rho^{\alpha}=|\psi^{\alpha}\rangle\langle\psi^{\alpha}|$. Finally, we evaluate $\langle\rho\rangle$ by averaging over these solutions, $\langle\rho\rangle=\frac{1}{\alpha_{\text{max}}}\sum\limits_{\alpha=1}^{\alpha_{\text{max}}}\rho^{\alpha}$, where for this work, we take $\alpha_{\text{max}}=10,000$.

Appendix B Evolution of the Hybrid Qubit in the Absence of Charge Noise

In this Appendix, we sketch out the perturbative derivation of the time evolution of a strongly driven hybrid qubit in the absence of charge noise, which is in principle exact, when keeping all orders of the expansion. While portions of these results were first derived in Ref. Yang et al. (2017), we present them again here for completeness, because they will be used, below, in our cumulant expansion analysis.

In the absence of noise, the dynamics is governed by the Schrödinger equation

[TABLE]

Expressing Eq. (2) in its eigenbasis $\{|0\rangle,|1\rangle,|L\rangle\}$, we have $\bar{\mathcal{H}}_{q}=\text{diag}[E_{0},E_{1},E_{L}]$, where $\{E_{i}\}$ are the energy eigenvalues, and the AC driving Hamiltonian is given by $\bar{\mathcal{H}}_{\text{ac}}=\bar{V}\cos(\omega_{d}t)$. In the large-detuning regime, $\varepsilon\gg E_{\text{ST}},\Delta_{1},\Delta_{2}$, $\bar{V}$ is given by

[TABLE]

We derive $U_{0}$ using the dressed-state approach of Ref. Yang et al. (2017). We first extend the semiclassical Hamiltonian, $\mathcal{H}_{\text{semi}}=\bar{\mathcal{H}}_{q}+\bar{\mathcal{H}}_{\text{ac}}\cos(\omega_{d}t)$, into a quantum Hamiltonian $\mathcal{H}_{\text{QM}}$ by imposing the condition

[TABLE]

where the coherent state $|\alpha\rangle$ is defined as $a|\alpha\rangle=e^{-i\omega_{d}\!t}\alpha_{0}|\alpha\rangle$, with $a$ the photon annihilation operator and $N=\langle\alpha|a^{\dagger}a|{\alpha}\rangle$ the number of photons. The quantum Hamiltonian can be written as $\mathcal{H}_{\text{QM}}=\mathcal{H}_{\text{dot}}+\mathcal{H}_{\text{ph}}+V_{\text{int}}$. Here, the uncoupled dot Hamiltonian is given by $\mathcal{H}_{\text{dot}}=\sum_{i=0,1,L}E_{i}|i\rangle\langle i|\otimes I_{\text{ph}}$, the uncoupled photon Hamiltonian is given by $\mathcal{H}_{\text{ph}}=I_{\text{dot}}\otimes\hbar\omega_{d}a^{\dagger}a$, and the interaction term is defined as $V_{\text{int}}=\bar{V}/2\sqrt{N}\otimes(a^{\dagger}+a)$. By evaluating the dynamics of the quantum Hamiltonian, $U_{\text{QM}}(t)=e^{-\frac{i}{\hbar}\mathcal{H}_{\text{QM}}t}$, the time evolution of the semiclassical Hamiltonian can be obtained as $U_{0}(t)=\langle\alpha(t)|U_{\text{QM}}(t)|\alpha(0)\rangle$.

To simplify the calculation, we note that predominant modes of coherent states occur in the range $n\in[N-\Delta N,N+\Delta N]$, where $\Delta N/N\ll 1$. We therefore express the quantum Hamiltonian in the basis of $|i,n\rangle$ for qubit state $|i\rangle$ and photon state $|n\rangle$ for $n\in[N-\Delta N,N+\Delta N]$. In this way, we obtain

[TABLE]

To increase the accuracy of $U_{0}$ when $A_{\varepsilon}$ is large [e.g. the rightmost part of Fig. 3(d) in the main text], we make the following modification in the perturbation: We first diagonalize the leakage subspace, i.e. the $\langle L,n|\mathcal{H}_{\text{QM}}|L,m\rangle$ part of the Hamiltonian. The new basis now is $\{|0,n\rangle,|1,n\rangle,|\tilde{L},n\rangle\}$ where $|\tilde{L},n\rangle=\sum_{m=-\infty}^{\infty}J_{m}(-[\bar{V}]_{LL}/\hbar\omega_{d})|L,n+m\rangle$, and $J_{m}(x)$ is a Bessel function of the first kind. Rewriting the Hamiltonian in this basis, the magnitude of the off-diagonal elements $\tilde{V}_{ij}$ is now at most ${\cal O}[A_{\varepsilon}\,J_{m}(-A_{\varepsilon}/\hbar\omega_{d})]$. We then proceed to evalulate $U_{0}$ perturbatively over $\bar{V}/\hbar\omega_{d}$ using the method of Ref. Yang et al. (2017). A similar diagonalization scheme has been applied to floquent treatment of strongly driven two-level system, for example in Ref. Deng et al. (2015). This is equivalent to performing the basis transformation $U_{L}=\text{diag}[1,1,e^{-\frac{i\bar{V}_{LL}}{\hbar\omega}\sin(\omega t)}]$ before going into the dressed space, similar to the basis transformation for the two-level system in Ref. Ashhab et al. (2007). We use this modification when we perform the analytical calculation of fidelity, plotted as Fig. 3(d) and (e) in the main text.

Appendix C Evolution in the Presence of Charge Noise

To incorporate the effects of detuning noise, we first move to the interaction frame. Here, the equation of motion for the density matrix $\rho^{I}$ is given by

[TABLE]

where $\rho^{I}=U_{0}^{\dagger}\rho U_{0}$, $\mathcal{L}\rho^{I}\equiv[h_{n}^{I},\rho^{I}]$, and $h_{n}^{I}=U_{0}^{\dagger}\bar{h}_{n}U_{0}$. The evolution can be evaluated in terms of the cumulant expansion Kubo (1962)

[TABLE]

where $\langle\cdots\rangle$ is the ensemble average over $\delta\varepsilon(t)$ and $\langle\cdots\rangle_{c}$ is the cumulant average. To second order in $\delta\varepsilon$, this gives Eq. (6) in the main text.

To further simplify the calculation, we express the density and noise matrices in terms of the 3$\times$3 Gellman matrices, $\{\lambda_{i}\}_{i=1,\cdots,8}$, which are presented in Appendix. D. The Gellman matrices can be categorized into two subsets: the first set $\Lambda_{q}=\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\}$ does not mix qubit and leakage states, while the second set $\Lambda_{\text{sup}}=\{\lambda_{5},\lambda_{6},\lambda_{7},\lambda_{8}\}$ mixes the two subspaces. Using these matrices, we can write $\rho^{I}=1/3I_{3}+1/2\sum\limits_{i=1}^{8}\vec{r}^{I}_{i}\lambda_{i}$ where $I_{3}$ is the 3$\times$3 identity matrix, and $\vec{r}^{I}$ is the generalized Bloch vector. Similarly, we can express the noise matrix in the interaction frame as $h_{n}^{I}=\sum\limits_{i}h_{n,i}^{I}\lambda_{i}$. As noted in the main text, Eq. (6) may then be replaced by Eq. (7), where

[TABLE]

Determining the cumulant $K(t)$ therefore reduces to evaluating $I(t,\omega_{1},\omega_{2})$ in Eq. (9). This integral can be solved analytically for a generic noise spectrum, leading to a complete analytical result for the dynamics that provides a rather accurate description. It can also be approximated in the asymptotic limit, as in Appendix E, yielding results that the capture key features of the decay profile and the process fidelity.

Appendix D Gellman Matrices

For completeness, we reproduce here the Gellman matrices, $\{\lambda\}_{i=1,\cdots,8}$, which are linearly independent, traceless, and Hermitian, as originally defined in Ref. Gell-Mann (1962):

[TABLE]

Appendix E Asymptotic Results in the Long-Time Limit

In the presence of noise, the dynamics of the density matrix in the interaction frame is governed by the Schr$\ddot{\text{o}}$dinger equation

[TABLE]

In the large detuning regime, the noise matrix can be approximated (by expanding in powers of the tunnel couplings) as

[TABLE]

similar to the second term of Eq. (15). We note that, by choosing $\Delta_{1}=\Delta_{2}\sim 0.7\,E_{\text{ST}}$, as noted in the main text, the main term controlling the dephasing is $\bar{h}_{n,3}\approx 0$, consistent with a higher-order DC sweet spot Wong (2016); Yang et al. (2017).

In this work, we focus on detuning charge noise with a $1/f$ power spectral density, as defined in Eq. (5). The corresponding time correlation function is given by

[TABLE]

where $\operatorname{Ci}(x)$ is the cosine integral. For this case, we derive the the asymptotic results, Eqs. (10) and (11), discussed in the main text, as follows.

1. (1)

First, calculate the Fourier components $\alpha_{i,\omega}$, as defined in $h_{n}^{I}(t)=U_{0}^{\dagger}\bar{h}_{n}U_{0}=\sum\limits_{i,\omega}\alpha_{i,\omega}e^{i\omega t}\lambda_{i}$, and insert these results into Eq. (8). 2. (2)

Next, approximate $I(t,\omega_{1},\omega_{2})$ [Eq. (9)] in the regime of interest and only retain non-oscillatory terms. (a) For the decay profile, Eq. (10), we consider the long-time limit, $1/\omega_{l}\gg t/(2\pi)\gg 1/\omega\gg 1/\omega_{h}$, where $\omega$ is any angular frequency relevant to the hybrid qubit. (b) For the gate infidelity in the rotating frame, Eq. (11), we consider the limit $1/\omega_{l}\gg t/(2\pi)\sim 1/\Omega\gg 1/\omega\gg 1/\omega_{h}$, where $\Omega$ is the Rabi angular frequency. 3. (3)

Finally, evaluate $K(t)$ in Eq. (8). In the long-time limit, the dominant contribution to $K$ arises from pure dephasing in the rotating frame, $K_{\varphi}=D_{\varphi}I(t,\omega_{1}=0,\omega_{2}=0)$, where $D_{\varphi}$ is an 8$\times$8 matrix of coefficients. To evaluate $\exp[K(t)]$, we first perform a Schrieffer-Wolff decomposition $D_{\varphi}$, up to ${\cal O}[\gamma^{2}]$, to decouple the qubit and leakage subspaces. The decoupled subspaces exhibit distinct behaviors: the qubit subspace exhibits slow dephasing, with $[D_{\varphi}]_{ij}\sim{\cal O}[\gamma^{2}]$, while the leakage subspace exhibits fast dephasing with $[D_{\varphi}]_{ij}\sim{\cal O}[\gamma^{0}]$. Consequently, the elements in $e^{K_{\varphi}}$ all vanish at this order, except in the space spanned by $\Lambda_{q}$. Similarly, we evaluate the leading order terms in $K_{\text{M}}$ and $K_{\text{nMn}\varphi}$, which occur at ${\cal O}[\gamma^{0}]$. For pedagogical purposes, we retain the leading order terms in each of these categories, and evaluate the exponential $e^{K(t)}=e^{K_{\varphi}(t)+K_{\text{M}}(t)+K_{\text{nMn}\varphi}(t)}$.

The final asymptotic results are summarized as the follows.

1. (a)

For the time limit appropriate for the decay profile, we obtain

[TABLE]

where $\gamma_{E}\approx 0.5772$ is Euler’s constant, and

[TABLE]

where $[e^{K(t)}]_{ij}=0$ otherwise. Here, $\varphi(t)$, $\Gamma_{x}$, $\Gamma_{y}$, $\Gamma_{L}$, $\Gamma_{\varphi}$, and $a$ are defined in the main text, and we further define

[TABLE]

In Fig. 4, we plot the resulting figure of merit, $f_{\text{Rabi}}/\Gamma_{\text{Rabi}}$, where the Rabi decay rate $\Gamma_{\text{Rabi}}$ is identified from Eq. (26), where we define $\langle\rho^{I}_{00}\left(t=1/\Gamma_{\text{Rabi}}\right)\rangle\equiv(1+2e^{-1})/3$, for the initial state $\rho^{I}(t=0)=|0\rangle\langle 0|$. The white dashed line in the figure is obtained by maximizing the figure of merit, or equivalently, by setting $\Gamma_{\varphi}\sim(df_{\text{Rabi}}/d\varepsilon)^{2}=0$, as consistent with an AC sweet spot Didier et al. . This condition determines the optimal combination of driving amplitudes, as described in the main text. 2. (b)

For the calculation of the gate infidelity in the rotating frame, we note that the condition $t/(2\pi)\sim 1/\Omega$ makes the calculation somewhat more complicated. Fortunately, the DC sweet-spot condition, $\bar{h}_{n,3}=0$, causes $\alpha_{i,\pm\Omega}=0$, at $O[\gamma^{2}]$, which simplifies the calculation. After lengthy manipulations, we finally arrive at Eq. (11) in the main text.

Appendix F Process Fidelity

Following Ref. Nielsen and Chuang (2010), a generic quantum process $\mathcal{E}$ on a three-dimensional Hilbert space may be expressed as

[TABLE]

where $\{E_{m}\}$ is a basis for the vector space of $3\times 3$ matrices, $\chi$ is the process matrix, and $\rho_{0}$ represents any initial density matrix. The process fidelity is then defined as $F=\text{Tr}[\chi_{\text{sys}}\chi_{\text{ideal}}]$, where $\chi_{\text{sys}}$ is the actual process matrix, describing the system evolution, including non-ideal contributions from strong driving and decoherence, and $\chi_{\text{ideal}}$ describes the ideal evolution. Since $\chi_{\text{ideal}}$ does not involve the leakage channel, we simplify the calculation of $F$ by projecting $\rho_{0}$ and $\mathcal{E}$ onto the 2D logical subspace and solving for the corresponding 4$\times$4 $\chi$ matrices Yang et al. (2017). In this case, we choose $E_{m}$ from the Pauli basis $\{I,\sigma_{x},-i\sigma_{y},\sigma_{z}\}$ and follow the standard procedure for computing $F$ Nielsen and Chuang (2010). For an $X_{\pi}$ gate, the ideal operation is defined as

[TABLE]

where $\tilde{E}_{0}$ and $\tilde{E}_{1}$ are the qubit energies renormalized by the Bloch-Siegert shifts.

By moving to the interaction frame, defined by $U_{0}$, we can focus specifically on the effects of decoherence on the fidelity, $F^{I}$. Here, we compare the $\chi$ matrix of the actual process, $\chi^{I}_{\text{sys}}$, with the ideal process, defined as $\chi^{I}_{\text{ideal}}=\text{diag}[1,0,0,0]$:

[TABLE]

As noted in the main text, $\chi^{I}_{\text{sys}}$ is not affected by strong driving effects, which are already included in $U_{0}$. $F^{I}$ therefore describes the fidelity due to decoherence (only), and serves as an upper-bound on the fidelity in the laboratory frame, where $F$ is suppressed by strong driving effects as well as decoherence.

Appendix G Comparison Of Different Drives

In this section, we compare the performance of three different methods of AC driving: detuning driving, tunnel coupling driving, and simultaneous driving of the detuning and the tunnel coupling, along the AC sweet spot (dashed line) in Fig. 4. In each case, we numerically simulate the $X_{\pi}$ gate fidelity in the presence of $1/f$ detuning charge noise and phonon effects, introduced as in Eq. (13) of the main text. Here, we take $T_{\text{ph}}=5\;\mathrm{\mu s}$ as the phonon decay rate. In Fig. 5, we plot the resulting infidelities as a function of detuning parameter, $\varepsilon$, and the appropriate amplitude for each type of drive. We find that the optimal, combined drive is superior to the other options over the entire parameter space. Moreover, the optimal drive exhibits a broad regime of parameters for which the fidelities are $>$99.9%.

Appendix H Infidelity Caused by Phase Errors

For the optimal driving method, we now show that the infidelity is sensitive to the relative phase between the two driving terms, even when the driving amplitudes are tuned to an AC sweet spot. In our simulations, we include a variable phase shift $\phi$ to the tunnel coupling driving term in the modified driving Hamiltonian, given by

[TABLE]

[Compare to Eq. (3).] Figure 6 shows the simulated dependence of the infidelity on $\phi$. As a benchmark, for this example, we find that achieving an infidelity below $0.1\%$ requires phases errors below $0.02\,\mathrm{rad.}\approx 1^{\circ}$.