Achieving high-fidelity single-qubit gates in a strongly driven silicon-quantum-dot hybrid qubit

Performing qubit gate operations as quickly as possible can be important to minimize the effects of decoherence. For resonant gates, this requires applying a strong ac drive. However, strong driving can present control challenges by causing leakage to levels that lie outside the qubit subspace. Strong driving can also present theoretical challenges because preferred tools such as the rotating wave approximation can break down, resulting in complex dynamics that are difficult to control. Here we analyze resonant $X$ rotations of a silicon-quantum-double-dot hybrid qubit within a dressed-state formalism, obtaining results beyond the rotating wave approximation. We obtain analytic formulas for the optimum driving frequency and the Rabi frequency, which are both affected by strong driving. While the qubit states exhibit fast oscillations due to counter-rotating terms and leakage, we show that they can be suppressed to the point that gate fidelities above $99.99\%$ are possible, in the absence of decoherence. Hence decoherence mechanisms, rather than strong-driving effects, should represent the limiting factor for resonant-gate fidelities in quantum dot hybrid qubits.

Figure 1

Control of a quantum dot hybrid qubit. (a), (b) Double quantum dots with three electrons, depicted here in their $(1,2)$ charge configurations. In this arrangement, the low-lying energy levels, depicted in the right-hand dots, correspond to singletlike ($|·S\rangle$) and tripletlike ($|·T\rangle$) spin states, where $S$ and $T$ refer to the two-electron dots [23, 24]. The $(2,1)$ charge configuration has only one low-lying energy level ($|S·\rangle$), as depicted in the left-hand dots. Panel (a) illustrates control of the tunnel couplings ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, between $|S·\rangle$, and $|·S\rangle$ or $|·T\rangle$, respectively. Panel (b) illustrates control of the detuning parameter $\varepsilon$, corresponding to the energy difference between the left and right quantum dots. The parameters ${\mathrm{\Delta }}_1, {\mathrm{\Delta }}_2$, and $\varepsilon$ are all controlled by voltages applied to the double-dot top gates [11]. Resonant gates are implemented by adding an ac drive to either the tunnel couplings or the detuning. (c) A typical energy-level diagram for the quantum dot hybrid qubit, as a function of $\varepsilon$, obtained by diagonalizing Eq. (1), using parameter values $E_{\text{ST}}/h=12$ GHz and ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0.7E_{\text{ST}}$ [25]. Here, the lowest two levels (red and blue) correspond to the qubit subspace, while the highest level (green) corresponds to a leakage state. The effective two-level system of the qubit, derived in Eq. (A6), is indicated by white dashed lines that overlay the qubit levels.

Figure 2

A cartoon depiction of the manifold structure of bare states of a quantum dot hybrid qubit coupled to a photon field with frequency $f$. The 3D manifold labeled $g_n$ contains two qubit levels, $|0,n+1\rangle$ and $|1,n\rangle$, which are nearly degenerate near the resonance condition $hf\simeq E_1-E_0$, and a leakage level $|L,n-k\rangle$. Here, $n$ and $k$ indicate photon numbers. If $E_{\text{intra}}=hf-E_{\text{inter}}$ represents the energy width of a given manifold, as indicated in the diagram, then $k$ is defined such that $E_{\text{intra}}<hf/2$.

Figure 3

Dynamics of a strongly driven quantum dot hybrid qubit. (a), (b) Numerical solutions of Eq. (2) are plotted as solid lines. Here, the simulation parameters are given by $E_{\text{ST}}/h=12$ GHz, $\{\varepsilon ,{\mathrm{\Delta }}_{10},{\mathrm{\Delta }}_{20},A\}=\{6,0.7,0.7,0.33\}\times E_{\mathrm{ST}}$, and $r_1=r_2=1$, and the ac drive is applied to the tunnel coupling. The corresponding analytical solutions of the dressed-state theory with terms included up to $O\left[{(A/hf)}^3\right]$ are plotted as dashed white lines. (a) The probabilities $P_0\left(t\right)={\left|c_0\left(t\right)\right|}^2$ and $P_1\left(t\right)={\left|c_1\left(t\right)\right|}^2$ of the logical qubit states are plotted for the initial state given by $|0\rangle$. The sinusoidal envelopes correspond to the conventional Rabi oscillations that would occur under weak driving (i.e., the RWA), while the fast modulations are caused by strong driving. The analytical results shown here include corrections to the RWA up to $O\left[{(A/hf)}^3\right]$ and reproduce all observable features in the fast oscillations. (b) The corresponding evolution of the leakage probability $P_L\left(t\right)={\left|c_L\left(t\right)\right|}^2$. Again, the analytical solutions reproduce all observable features of the numerical simulations, and the square of the difference between the analytical and numerical results is less than ${10}^{-7}$ over the entire range. (c) Numerical and analytical calculations of the resonance frequency (analytical results are plotted with solid lines) and Rabi frequency (dashed lines) as a function of driving amplitude, for microwave driving of the detuning, with simulation parameters $E_{\mathrm{ST}}/h=12$ GHz and $\{\varepsilon ,{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2\}=\{6,0.7,0.7\}\times E_{\mathrm{ST}}$. The simulation results ($+$ markers for the resonance frequency and $\circ$ markers for the Rabi frequency) are obtained by performing a sweep of the driving frequency $f$; $f_{\text{res}}$ is identified as the driving frequency that minimizes the Rabi frequency, as in Eq. (13), while $f_{\text{Rabi}}$ is identified as the dominant peak in the Fourier spectrum. Analytical results are shown for the RWA (blue), corresponding to $f_0=(E_1-E_0)/h$, and corrections to the RWA up to $O\left[{(A/hf)}^4\right]$ (red). The difference between the actual resonance frequency and the RWA is called the Bloch-Siegert shift. Any remaining deviation of the numerical results from the analytical calculations comes from higher-order terms in the perturbation expansion.

Figure 4

Pulse envelopes and their Fourier spectra. (a) The three pulse shapes considered in this work are rectangular (black), truncated Gaussian (blue), and smoothed rectangular (red), as defined in Eqs. (16, 17, 18). The parameters used to generate the pulses shown here are $\{t_g,\sigma ,t_r\}=\{1,1,0.1\}$ ns, where $t_g$ is the gate time, $\sigma$ sets the width of the Gaussian pulse, and $t_r$ is the rise time of the smoothed rectangular pulse. (b) The Fourier spectra of the same three envelopes, using the same color scheme. Here, the frequencies $f\simeq (E_L-E_0)/h\pm (E_1-E_0)/h$, and $f\simeq (E_L-E_1)/h\pm (E_1-E_0)/h$ are associate with leakage; in our simulation, these are given by 48, 60, 72, and 84 GHz. At such high frequencies, the smoothed rectangular pulse has the lowest spectral density, and should therefore be the most effective at suppressing leakage.

Figure 5

Improving the fidelity of $X_{\pi }$ rotations using pulse shaping in the strong-driving regime. Simulations are performed by driving the tunnel coupling with the control parameters $E_{\text{ST}}/h=12$ GHz, $\{\varepsilon ,{\mathrm{\Delta }}_{10},{\mathrm{\Delta }}_{20}\}=\{6,0.7,0.7\}{E}_{\text{ST}}$, and $r_1=r_2=1$. Here, $E_{\text{ST}}$ is the singlet-triplet energy splitting of the doubly occupied dot, $\varepsilon$ is the detuning between the dots, ${\mathrm{\Delta }}_{10}$ (${\mathrm{\Delta }}_{20}$) is the time-independent part of the tunnel couplings between the states $|S·\rangle$ and $|·S\rangle$ ($|·T\rangle$), and $r_1$ ($r_2$) is the coefficient for the responses of the tunnel couplings to the ac signal. Panels (a) and (b) show typical dynamical evolutions for an $X_{\pi }$ gate with gate time $t_g=1$ ns, for the initial state with state amplitudes [see Eq. (12)] of $c_0=1, c_1=c_L=0$. Two results are shown, corresponding to a sharp rectangular pulse (black) or a smoothed rectangular pulse (red). The upper inset of (a) shows a blowup of times near $t=t_g$, where the smoothed pulse suppresses the fast oscillations. The lower inset shows estimates of $1-F$, where $F$ is the fidelity, for simulations including quasistatic charge noise of uniform distribution with zero mean and standard deviation ${\sigma }_{\varepsilon }$. In (b), we see that the leakage state occupation is suppressed over the whole gate evolution for the smoothed pulse, particularly near $t=0,t_g$. This is because leakage probability depends quadratically on the pulse envelope, $|c_L{|}^2\sim A^2\left(t\right)$, and therefore vanishes when $A\left(t\right)\to 0$ near the end points of the pulse. Panel (c) shows the process infidelity, $1-F$ [66], computed for several different scenarios: rectangular (black), truncated Gaussian (blue), and smoothed rectangular (red) pulse shapes with no dynamical corrections (RWA, solid lines), and dynamical corrections for ${\tilde{f}}_{\text{res}}$ up to order $O\left[{(A/hf)}^2\right]$ (dashed lines). Using these smooth pulse shapes and strong-driving corrections, we can achieve gate fidelities $>99.9\%$ for several different scenarios, with gate times as short as 1 ns (black oval). For the smoothed rectangular pulse, the fidelity is $>99.99\%$. [Note that these results, except the lower inset of (a), do not include noise.]

Figure 6

The dynamical evolution of a transversely driven two-level system under strong resonant driving. Exact numerical results are shown as thick red lines, while the analytical results obtained in Eqs. (B31) and (B32) are shown as thin white lines; the square of the difference between the two solutions is less than ${10}^{-4}$ over the whole range of the plot. Here, we take $|0\rangle$ as the initial state and plot the probabilities $P_0={\left|c_0\right|}^2$ and $P_1={\left|c_1\right|}^2$ of being in qubit states $|0\rangle$ and $|1\rangle$ as a function of time. The system parameters are given by $\{E_0,E_1,A\}/h=\{-10,10,3\}$ GHz, where $E_0$ and $E_1$ are the energies of the qubit states and $A$ is the amplitude of the drive.

Figure 7

Simulation results for the infidelity, $1-F$, of an $X_{\pi }$ rotation, obtained by driving the detuning. The results should be compared to Fig. 5, where tunnel coupling driving was used. In both cases, we assume the same control parameters, $E_{\text{ST}}/h=12$ GHz and $\{\varepsilon ,{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2\}=\{6,0.7,0.7\}{E}_{\text{ST}}$. The colors and line styles also have the same meaning as in Fig. 5, with one exception: in Fig. 5, the dashed lines corresponded to including strong-driving corrections for ${\tilde{f}}_{\text{res}}$ and $f_{\text{Rabi}}$ up to $O\left[{(A/hf)}^2\right]$, while here, we include corrections up to $O\left[{(A/hf)}^4\right]$.

Figure 8

Simulation results for the infidelity, $1-F$, of an $X_{\pi /2}$ rotation, using (a) tunnel coupling driving, and (b) detuning driving. The simulation parameters used here are the same as those mentioned in Figs. 5 and 7, where $X_{\pi }$ gates were investigated. The colors and line styles here have the same meaning as those figures.