Fast control of semiconductor qubits beyond the rotating-wave approximation

We present a theoretical study of single-qubit operations by oscillatory fields on various semiconductor platforms. We explicitly show how to perform faster gate operations by going beyond the universally used rotating-wave approximation (RWA) regime, while using only two sinusoidal pulses. We first show for specific published experiments how much error is currently incurred by implementing pulses designed using standard RWA. We then show that an even modest increase in gate speed would cause problems in using RWA for gate design in the singlet-triplet (ST) and resonant-exchange (RX) qubits. We discuss the extent to which analytically keeping higher orders in the perturbation theory would address the problem. More strikingly, we give a new prescription for gating with strong coupling far beyond the RWA regime. We perform numerical calculations for the phases and the durations of two consecutive pulses to realize the key Hadamard and $\frac{\pi }{8}$ gates with coupling strengths up to several times the qubit splitting. Working in this manifestly non-RWA regime, the gate operation speeds up by two to three orders of magnitude and nears the quantum speed limit without requiring complicated pulse shaping or optimal control sequences.

Figure 1

Averaged infidelity due to RWA, $1-{\overline{\mathcal{F}}}_{\mathrm{rwa}}$, as a function of time for the zero detuning case $\left(\omega ={\omega }_z\right)$. Results for various driving strengths ${\omega }_x/{\omega }_z$ are shown: (a) 1, (b) 0.3, (c) 0.1, and (d) 0.01.

Figure 2

Averaged infidelity due to RWA, $1-{\overline{\mathcal{F}}}_{\mathrm{rwa}}$, as a function of time, for different relative detunings ($\delta \omega /{\omega }_z$ where $\delta \omega =\omega -{\omega }_z$) and coupling strengths $\left({\omega }_x/{\omega }_z\right)$, which are, respectively, (a) 0.1 and 0.01, (b) 0.01 and 0.001, (c) 0.1 and 0.1, and (d) 0.01 and 0.01. The insets plot $1-{\overline{\mathcal{F}}}_{\mathrm{rwa}}$'s behavior in the long time limit where another time period emerges.

Figure 3

Averaged infidelity $1-{\overline{\mathcal{F}}}_{\mathrm{ext}}$ from Eq. (15). The parameters in panels (a)–(c) are the same as in Figs. 1 and 1 and Fig. 2, respectively. In (d) we plot the difference in the return probability between the exact solution and both the RWA and the extended analytical results for an initial $|0\rangle$ state and ${\omega }_x/{\omega }_z=0.3$ and $\delta \omega /{\omega }_z=0.1$.

Figure 4

Contour plot for the infidelity amplitude in the first Rabi cycle, ${[1-{\overline{\mathcal{F}}}_{\mathrm{rwa}}]}^{\mathrm{Max}}$ in Eq. (21), as a function of relative detuning $\left(\delta \omega /{\omega }_z\right)$ and driving strength $\left({\omega }_x/{\omega }_z\right)$. We mark the ${10}^{-2}, {10}^{-3}$, and ${10}^{-4}$ contour lines. Three red crosses mark experimental operation points in the RX experiment (Figs. 2–4 in [1]), the light blue diamond marks the resonant ST experiment [2], the red solid circle marks the hybrid qubit [5], and the light blue square marks the weakly driven single-spin experiments [3, 4] (see Table 1).

Figure 5

Two-pulse solutions for the Hadamard gate in the resonant RWA limit. ${\mathrm{\Phi }}_{1,2}$ and $t_{1,2}$ are the phases and durations of the two pulses. The solution is plotted parametrically, with (a) $\{{\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2\}$, and (b) $\{t_1,t_2\}$ versus the total gate time, $t_{\mathrm{Tot}}=t_1+t_2$. Times are scaled by the driving frequency ${\omega }_x$. Note that all four curves are closed or periodic. The usual RWA solution, $\{{\mathrm{\Phi }}_1=\frac{\pi }{2},{\mathrm{\Phi }}_2=\pi ,\frac{{\omega }_x}{2}{t}_1=\frac{\pi }{2},\frac{{\omega }_x}{2}{t}_2=\pi \}$ in our notation, is marked by solid dots. These dots also give a sense of how to associate points on the multivalued curves at a given total time.

Figure 6

Two-pulse solutions for the resonant Hadamard gate with ${\omega }_x/{\omega }_z\le 4$. Times and frequencies are given in units such that ${\omega }_z=1$. (a) Minimal total time (scaled by driving strength) vs driving strength. The inset shows the same time in terms of the fixed unit. (b) Pulse segment times (scaled by driving strength) vs driving strength. Red solid and blue dashed curves represent the first and second pulses, respectively, throughout. (c) Pulse segment phases vs driving strength. (d) Parametric plot of pulse segment times $t_1$ and $t_2$ vs total time for ${\omega }_x={\omega }_z$. (e) Parametric plot of pulse segment phases ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ vs total time for ${\omega }_x={\omega }_z$. (Note the $2\pi$ periodicity in ${\mathrm{\Phi }}_{1,2}$.) The arrows are a guide to the eyes, indicating which values on the multivalued curves belong to the same solution. The minimal $t_{\mathrm{Tot}}$ solution is enclosed by the dotted cyan box. (f) Parametric plot of pulse segment times $t_1$ and $t_2$ vs total time for ${\omega }_x=0.1{\omega }_z$. (g) Parametric plot of pulse segment phases ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ vs total time for ${\omega }_x=0.1{\omega }_z$. (h)–(j) The evolution operators corresponding to the minimal $t_{\mathrm{Tot}}$ solutions for ${\omega }_x/{\omega }_z=0.1$ and 1 vs time. The axial vector $\tilde{\mathbf{\omega }}$ describes the cumulative effect of the evolution up to time $t$ in terms of an effective axis of rotation $\{{\tilde{\omega }}_{\theta },{\tilde{\omega }}_{\varphi }\}$ and an angle $|\tilde{\mathbf{\omega }}|$.

Figure 7

Two-pulse solutions for the resonant $\pi /8$ phase gate with ${\omega }_x/{\omega }_z\le 12$. Times and frequencies are given in units such that ${\omega }_z=1$. (a)–(j) follow the same structure as in Fig. 6 for the Hadamard gate. Most features are within ${\omega }_x\le 4{\omega }_z$, but the additional insets in (b) and (c) show the extended range $0<{\omega }_x\le 12{\omega }_z$. (e) and (g) only show ${\mathrm{\Phi }}_{1,2}$ over an interval of $\pi$ as that is the periodicity in this case (see Appendix pp1 for details).

Figure 8

Two-pulse solutions for the resonant Hadamard gate with ${\omega }_x/{\omega }_z\le 4$ for the shifted oscillations of Eq. (A1) with ${\omega }_{\mathrm{ave}}=2{\omega }_x$. Times and frequencies are given in units such that ${\omega }_z=1$. The arrangement of (a)–(j) and notations follow those in Fig. 6. In (i), the abrupt change of ${\tilde{\omega }}_{\varphi }$ in the ${\omega }_x=1{\omega }_z$ case, i.e., the jump from $-\pi$ to $\pi$ at around $t=4$, is simply due to the choice $-\pi <{\tilde{\omega }}_{\varphi }\le \pi$ here.

Figure 9

Two-pulse solutions for the resonant $\pi /8$ phase gate with ${\omega }_x/{\omega }_z\le 4$ for the shifted oscillations of Eq. (A1) with ${\omega }_{\mathrm{ave}}=2{\omega }_x$. Times and frequencies are given in units such that ${\omega }_z=1$. The arrangement of (a)–(j) and notations follow those in Fig. 6.