Fast quantum gates based on Landau-Zener-Stückelberg-Majorana transitions

Fast quantum gates are of paramount importance for enabling efficient and error-resilient quantum computations. In the present work we analyze Landau-Zener-Stückelberg-Majorana (LZSM) strong driving protocols, tailored to implement fast gates with particular emphasis on small-gap qubits. We derive analytical equations to determine the specific set of driving parameters for the implementation of single-qubit and two-qubit gates employing single-period sinusoidal pulses. Our approach circumvents the need to scan experimentally a wide range of parameters and instead it allows to concentrate on fine-tuning the device near the analytically predicted values. We analyze the dependence of relaxation and decoherence on the amplitude and frequency of the pulses, obtaining the optimal regime of driving parameters to mitigate the effects of the environment. Our study focus on the single-qubit $X_{\frac{\pi }{2}}, Y_{\frac{\pi }{2}}$ and identity gates. Also, we propose the $\sqrt{\mathrm{bSWAP}}$ as the simplest two-qubit gate attainable through a robust LZSM driving protocol.

Figure 1

Intensity plot of $P_{01}=|\langle 1|U_{\mathrm{num}}{\left(T\right)|0\rangle |}^2$ as a function of the driving frequency $\omega$ and amplitude $A$, both normalized by the qubit gap $\mathrm{\Delta }$. $P_{01}$ is computed numerically using a fourth-order Trotter-Suzuki algorithm. See text for details. The color bar indicates the probability $P_{01}$.

Figure 2

(a) Transition probability $P_{01}$ obtained from Eq. (11) as a function of driving frequency $\omega /\mathrm{\Delta }$ and amplitude $A/\mathrm{\Delta }$. The color bar indicates the probability $P_{01}$. (b) Intensity plot of the error $\mathcal{E}$ of the CHRW approximation. The color bar indicates the logarithmic error $-{log}_{10}\left|\mathcal{E}\right|$. Regions in white correspond to $\mathcal{E}<{10}^{-8}$.

Figure 3

Location of the parameters for implementation of the $Y_{\frac{\pi }{2}}$ gate. Intensity plot of the error function $\mathcal{E}$ that compares the numerically exact evolution operator $U_{\mathrm{num}}$ with $Y_{\frac{\pi }{2}}$. The color bar corresponds to $-{log}_{10}\left|\mathcal{E}\right|$. The point where $\mathcal{E}<{10}^{-7}$ (practically zero within numerical precision) gives the operational parameters of the gate, frequency ${\omega }_Y$ and amplitude $A_Y$, which are in agreement with the analytical estimates of Eq. (14), ${\omega }_Y\approx 2.07\mathrm{\Delta }$ and $A_Y\approx 2.87\mathrm{\Delta }$.

Figure 4

(a) Plot of the parameter sets $\omega$ and $A$ for the implementation of $X_{\frac{\pi }{2}}$ and $Y_{\frac{\pi }{2}}$ gates. The plotted points satisfy the analytical expressions of Eqs. (17) and (22) with a ${10}^{-3}$ accuracy (except for the low-frequency points, which correspond to the evaluation of the condition $P_{01}=1/2$ with the same accuracy as the numerically exact evolution, since the CHRW approximation does not apply in this case). Inset: Schematic representation of the pulsing protocol with an idle time $t_i$ added before the sinusoidal drive and a second idle time $t_f$ afterwards. (b) Values of $\omega$ and $A$ to implement the identity operation determined from Eq. (23), obtained with the same accuracy. In the regions shaded in gray the CHRW approximation has an error of 0.01, as analyzed in Fig. 2.

Figure 5

(a) Plot, as a function of the parameters $A$ and $\omega$, of the error $\mathcal{E}$ that compares $\sqrt{\mathrm{bSWAP}}$ with the numerically exact $U_{\mathrm{num},2q}$, computed from $H_{2q}$ Eq. (24). The color bar scale corresponds to $-{log}_{10}\left|\mathcal{E}\right|$. The point where $\mathcal{E}<{10}^{-7}$ (practically zero within numerical precision) gives the operational parameters of the gate, frequency ${\omega }_{bS}$ and amplitude $A_{bS}$. It agrees with the analytical estimate of ${\omega }_{bS}\approx 4.14\mathrm{\Delta }, A_{bS}\approx 5.74\mathrm{\Delta }$. (b) Time evolution of the population transfers during a driving period for ${\omega }_{bS},A_{bS}$. Continuous lines correspond to the evolution of the populations of the $|00\rangle$ (black) and the $|11\rangle$ states (blue), after the initial state $|00\rangle$. Dashed lines correspond to the evolution of the populations of the $|01\rangle$ (green) and the $|10\rangle$ states (red), after the initial state $|01\rangle$. (c) Same as (a) but for two slightly different qubits with ${\mathrm{\Delta }}_2=1.05{\mathrm{\Delta }}_1$. See text for more details.

Figure 6

(a) Plot of the relaxation rate ${\mathrm{\Gamma }}_1$ as a function of $A/\mathrm{\Delta }$ for $\omega =1.92\mathrm{\Delta }$. The black continuous line corresponds to the numerically exact value and the dashed line the CHRW approximation, Eq. (37). (b) Plot of the dephasing rate ${\mathrm{\Gamma }}_{\varphi }$ as a function of $A/\mathrm{\Delta }$ for $\omega =1.92\mathrm{\Delta }$. The black continuous line corresponds to the numerically exact value and the dashed line to the CHRW approximation. Also, the decoherence rate ${\mathrm{\Gamma }}_2$ is plotted (blue continuous line, numerically exact values only). In both (a) and (b), the arrows indicate the values of $A$ where the $X_{\frac{\pi }{2}}, Y_{\frac{\pi }{2}}$ gates can be implemented, as obtained from Fig. 4; the rates ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_{\varphi },{\mathrm{\Gamma }}_2$ are normalized by the noise strength parameter $\gamma$ and correspond to a thermal bath at temperature $T_b=0.1\mathrm{\Delta }$.

Figure 7

Intensity plot of (a) the relaxation rate ${\mathrm{\Gamma }}_1$ and (b) the decoherence rate ${\mathrm{\Gamma }}_2$, as a function of $A/\mathrm{\Delta }$ and $\omega /\mathrm{\Delta }$. The black dots indicate the parameter sets $\omega$ and $A$ for the implementation of $X_{\frac{\pi }{2}}$ and $Y_{\frac{\pi }{2}}$ gates, as in Fig. 4. The white circle shows the optimal parameter region to minimize environmental effects. All rates correspond to a bath at temperature $T_b=0.1\mathrm{\Delta }$. The color bar scale corresponds to the rates ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2$ normalized by the noise strength parameter $\gamma$.

Figure 8

Plot of the transition probability $P_{01}$ as a function of $\omega /\mathrm{\Delta }$ for $A=1.16\mathrm{\Delta }$ comparing different approximation methods. Black squares: numerically exact values. Red circles: CHRW, as given by Eq. (11). Blue triangles: double rotating frame rotating wave approximation (DR), as given by Eq. (B8). Green stars: Magnus expansion approximation (ME), first order, as given by Eq. (B13).