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

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 $99.9\%$.

Figure 1

Schematic confinement potential and energy levels of a quantum-dot hybrid qubit. (a), (b) A hybrid qubit is formed in a double quantum dot containing three electrons, as depicted here in the (1,2) charge configuration. In this arrangement, the low-energy basis states, shown in the right-hand dots, comprise singlet-like ($|·S\rangle$) or triplet-like ($|·T\rangle$) spin states, where $S$ and $T$ refer to the doubly occupied dot [11, 12]. The (2,1) charge configuration has only one relevant low-energy basis state ($|S·\rangle$), depicted in the left-hand dots. Panel (a) illustrates the Hubbard Hamiltonian parameters of the undriven system: the energy detuning between the left and right dots, $\varepsilon$, and the tunnel couplings, ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, between $|S·\rangle$, and $|·S\rangle$ or $|·T\rangle$, respectively. The parameters ${\mathrm{\Delta }}_1$, ${\mathrm{\Delta }}_2$, and $\varepsilon$ are all controlled by voltages applied to the device top gates [5]. Detuning fluctuations caused by charge noise, $\delta \varepsilon \left(t\right)$, are the dominant decoherence mechanism for this system [6]. Panel (b) depicts the AC control of the detuning parameter ${\varepsilon }_{\text{ac}}$ and the tunnel couplings ${\mathrm{\Delta }}_{\text{ac}}$, which are used to implement resonant gates. (c) A typical energy-level diagram for a hybrid qubit, as a function of $\varepsilon$, showing the asymptotic energy splitting of the qubit states, $E_{\text{ST}}$. Here, the lowest two levels (red and blue) correspond to the qubit subspace, while the highest level (green) corresponds to a leakage state. Panel (c) is obtained by diagonalizing Eq. (2), assuming a realistic value of $E_{\text{ST}}/h=12$ GHz [13], and ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0.7E_{\text{ST}}$.

Figure 2

Dynamics of a strongly driven quantum-dot hybrid qubit in the presence of detuning fluctuations. For all calculations, we assume the DC control parameters $\{\varepsilon ,E_{\text{ST}},{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2\}/h=\{80,12,8.4,8.4\}\mathrm{GHz}$. Qubit evolutions are computed, taking $|0\rangle$ as the initial state, and applying a resonant AC signal of amplitude $A_{\mathrm{\Delta }}/h=3.5\mathrm{GHz}$ to both tunnel couplings, with a ratio of $r=1$ between the two driving amplitudes. The resulting density matrix is averaged over many realizations of $1/f$ charge noise, Eq. (5), assuming noise parameters $\sqrt{2\pi }{c}_{\varepsilon }=2.38\mu \mathrm{eV}$, ${\omega }_l/2\pi =1\mathrm{Hz}$, and ${\omega }_h/2\pi =256\mathrm{GHz}$. In all panels, the noise-averaged numerical simulations are plotted as solid lines, while corresponding analytical results, obtained up to second order in the cumulant expansion, Eq. (6), are plotted as white dashed lines. In panels (a) and (b), the differences between numerical and analytical results are also plotted as dotted lines (bottom panels), indicating errors $<0.1\%$. (a) The average occupancy of the initial state, $\langle {\rho }_{00}\rangle$ (blue), and the off-diagonal element, $|\langle {\rho }_{01}\rangle |$ (green), computed in the laboratory frame. Here, the smooth sinusoidal envelope reflects Rabi oscillations, as consistent with the rotating wave approximation (RWA), while the fast modulations are caused by strong driving. (b) The average occupancy of the leakage state, $\langle {\rho }_{LL}\rangle$ (red), in the laboratory frame. The inset shows the accumulated leakage. In panels (a) and (b), the analytical results are seen to capture all the significant features of the simulations, including the fast oscillations and the asymptotic decay. (c) The asymptotic decay of the density matrix, $\langle {\rho }_{00}^I\rangle$, in the interaction frame. The inset shows a blown-up view at short times. Here, the analytical results correspond to the full solution of Eq. (6) (dashed white line, inset), and its asymptotic form, Eq. (10) (dashed cyan line). Note that the fast oscillations, observed in the inset, are not captured in Eq. (10), but are accurately described in Eq. (6). As indicated, the Rabi decay time, $T_{\text{Rabi}}\simeq 24.0\mathrm{ns}$, is determined according to the definition $\langle {\rho }_{00}^I\left(T_{\text{Rabi}}\right)\rangle =(1+2e^{-1})/3$.

Figure 3

$X_{\pi }$ gate fidelities of a strongly driven quantum-dot hybrid qubit in the presence of $1/f$ detuning noise. Plots show the dependence of the fidelity on the detuning driving amplitude $A_{\varepsilon }$ and tunnel coupling driving amplitude $A_{\mathrm{\Delta }}$ using simulation parameters that are the same as Fig. 2. Panels (a)–(d) are computed in the laboratory frame, while panels (e) and (f) are computed in the interaction frame, which represents an upper bound on the fidelity in the laboratory frame, since in this case, strong-driving effects are viewed as part of the coherent evolution in the interaction frame. Simulation results obtained for (a) rectangular pulse envelopes, or (b) smoothed-rectangular pulse envelopes, with rise and fall times of $0.83\mathrm{ns}$. For the rectangular pulse envelope, the fidelity exhibits fringes due to strong driving. The dashed white line corresponds to an AC sweet spot, where $d{f}_{\text{Rabi}}/d\varepsilon =0$; in a broad region near this line, we observe fidelities $>99.9\%$. For the smoothed-rectangular pulse envelope, the fidelity fringes are suppressed, and the quality of the AC sweet spot is improved. In panel (c), the inset shows simulation results, similar to panels (a) and (b), where we phenomenologically include the effects of phonon-induced dephasing according to Eq. (13), with $T_{\text{ph}}=3\mu \mathrm{s}$. Here, we observed broad regions with fidelities $>99.9\%$, and the location of the fidelity maximum is indicated with a star. In the main panel, we plot a series of fidelity maxima, obtained in the same manner, as a function of $T_{\text{ph}}$, where the starred point corresponds to the inset. (d) Analytical results obtained from Eq. (6), keeping expansion terms up to $O\left[{(V/\hslash {\omega }_d)}^2\right]$. The simulation results in panel (a) and analytical results in panel (d) are nearly identical, except in the lower-right portion of the plots, where higher-order terms in the expansion are non-negligible. (e), (f) Here, in the interaction frame, the main fringes due to strong driving are absent, and any suppression of the fidelity can be attributed to charge noise, or crosstalk between charge noise and strong-driving effects. In panel (e) we plot the full analytical results based on Eq. (6), expanding up to $O\left[{(V/\hslash {\omega }_d)}^2\right]$. In panel (f), we plot the asymptotic results based on Eq. (11). We see that the simpler asymptotic results accurately reproduce the main features of the full analytical results, allowing us to determine straightforwardly the location of the AC sweet spot.

Figure 4

Theoretical estimates for the figure of merit, $f_{\text{Rabi}}/{\mathrm{\Gamma }}_{\text{Rabi}}$, of a strongly driven quantum-dot hybrid qubit, as a function of the detuning driving amplitude $A_{\varepsilon }$ and the tunnel coupling driving amplitude $A_{\mathrm{\Delta }}$, for the same DC control parameters and charge noise parameters used in Figs. 2 and 3 of the main text. Here, ${\mathrm{\Gamma }}_{\text{Rabi}}=1/T_{\text{Rabi}}$ is the Rabi decay rate, estimated from Eq. (E5), and $f_{\text{Rabi}}$ is the Rabi frequency. The dashed white line corresponds to an AC sweet spot, $d{f}_{\text{Rabi}}/d\varepsilon =0$, for which the ${\mathrm{\Gamma }}_{\phi }$ dephasing term vanishes, up to $O\left[{(V/\hslash {\omega }_d)}^2\right]$, yielding an optimal figure of merit. This line accurately matches our theoretical estimates, and is the same white dashed line plotted in Fig. 3. At the AC sweet spot, the figure of merit can be $>1000$, consistent with $X_{\pi }$ fidelities approaching $99.9\%$.

Figure 5

Comparison of the infidelities of $X_{\pi }$ gates, for three different AC driving methods: detuning driving, tunnel coupling driving, and optimal driving at an AC sweet spot, in the presence of detuning noise and phenomenological phonon dephasing. The results are presented in the laboratory frame and assume rectangular pulse envelopes. We use the same DC control parameters and noise parameters as Figs. 2 and 3, with a variable detuning parameter $\varepsilon$. (a) Detuning driving. Here, the vertically oriented, low-fidelity feature is caused by leakage. The fidelity attains a maximum value of $F=0.9973$ at the point $\{\varepsilon ,A_{\varepsilon }\}/h=\{120,27\}\mathrm{GHz}$. (b) Tunnel-coupling driving. The fidelity attains a maximum value of $F=0.9989$ at the point $\{\varepsilon ,A_{\mathrm{\Delta }}\}/h=\{120,2\}\mathrm{GHz}$. (c) Optimal, simultaneous driving of the detuning and tunnel coupling parameters at an AC sweet spot. The fidelity is higher than in panels (a) and (b) for almost every parameter value and attains a maximum value $F=0.9993$ at the point $\{\varepsilon ,A_{\varepsilon },A_{\mathrm{\Delta }}\}/h=\{88,35,3.64\}\mathrm{GHz}$. Importantly, we observe a broad range of parameters where $F>0.999$, particularly when $\varepsilon \gtrsim 90$ GHz.

Figure 6

Dependence of the infidelity on the phase shift between the detuning and tunnel coupling drives. Here the hybrid qubit is driven at an AC sweet spot, with rectangular pulse envelopes. The DC control parameters are the same as in Figs. 2 and 3, while the AC parameters are $\{A_{\varepsilon },A_{\mathrm{\Delta }}\}/h=\{27,3.1\}\mathrm{GHz}$. For the range of phase shift shown here, the infidelity is found to vary by more than two orders of magnitude. To suppress this effect below $0.1\%$, the phase shift should be less than $0.02\mathrm{rad}\approx 1^{\circ }$.