Error-resistant single-qubit gates with trapped ions

Coherent operations with individual trapped ${\text{Yb}}^{+}$ ions are demonstrated that are robust against variations in experimental parameters and intrinsically indeterministic system parameters. In particular, pulses developed using optimal control theory are demonstrated with trapped ions. Their performance as a function of error parameters is systematically investigated and compared to composite pulses. Such pulses are basic building blocks for single and multiqubit quantum gates.

Figure 1

(a) Ramsey interference fringes from sequence ${\text{A}}^{\prime }$ and (b) from sequence ${\text{B}}^{\prime }$ ($f=-1$, $g=0$). A fit (indicated by a solid line) of the experimental data using Eq. (1) gives ${\theta }_m=3.92\pm 0.05$, ${\varphi }_m=-0.418\pm 0.11$ (see text).

Figure 2

(a) Relative amplitude and (b) phase of an optimal control-based $\pi /2$ pulse consisting of 645 individual time steps [10] (available as supplementary online material) [17]. (c) shows the simulated time evolution of the pulse on the Bloch sphere, for $g=0$ and $f=-0.5$. At $t=0$ the state vector starts on the south pole of the Bloch sphere and reaches the desired state $(\theta =\pi /2$, $\varphi =-\pi /2)$ at the end of the pulse.

Figure 3

(a) Experimental fidelity $F$ for the optimal control-based $(\theta =\frac{\pi }{2},\varphi =-\frac{\pi }{2})$ pulse shown in Fig. 2 as a function of scaled detuning $f$ and relative error in the nutation angle $g$. The average statistical error of the measured points is ${\sigma }_{\text{av}}=0.023$. The center graphic is based on the same data and a white (black) rectangle indicates $F/F_m>0.90$ $\left(F/F_m<0.90\right)$ with the maximal fidelity $F_m=0.900\pm 0.014$. Finally on the right graph the results of the simulation generated by the time evolution of the density matrix are shown. (b) Corresponding data for a rectangular $\pi /2$ pulse (${\sigma }_{\text{av}}=0.035$, $F_m=0.899\pm 0.041$). Identical gray scales are used for plotting the experimental data (left column) and numerically simulated data (right column).

Figure 4

(a) Experimental fidelity for an optimal control-based $\theta =\pi$ pulse (445 steps) [17], ${\sigma }_{\text{av}}=0.025$, $F_m=0.821\pm 0.027$, white rectangles indicate $F/F_m>0.9$ and (b) for a rectangular $\pi$ pulse (${\sigma }_{\text{av}}=0.018$, $F_m=0.760\pm 0.026$). The simulated results are shown in the right column for both the optimal control-based pulse and the rectangular pulse.

Figure 5

(a) Experimental fidelity for an optimal control-based $\theta =\pi$ pulse (835 steps) [17], ${\sigma }_{\text{av}}=0.034$, $F_m=0.790\pm 0.035$, white rectangles indicate $F/F_m>0.9$ and (b) for a rectangular $\pi$ pulse (${\sigma }_{\text{av}}=0.024$, $F_m=0.720\pm 0.035$). The right column shows the simulated results of the optimal control-based pulse and the rectangular pulse.

Figure 6

(a) Experimental fidelity for a CORPSE-type composite pulse ($\theta =\pi$, ${\sigma }_{\text{av}}=0.030$, $F_m=0.853\pm 0.038$ white rectangles indicate $F/F_m>0.9$). The right column shows the simulated results using the same gray scale as the experimentally obtained fidelity. (b) Shows the same for the corresponding rectangular pulse $\left(F_m=0.839\pm 0.038,{\sigma }_{\text{av}}=0.027\right)$.

Figure 7

Experimental fidelity for the BB1 $R\left(\frac{\theta }{2}\right)-W-R\left(\frac{\theta }{2}\right)$ sequence ($\theta =\frac{\pi }{2},\varphi =0)$, $F_m=0.924\pm 0.008$, ${\sigma }_{\text{av}}=0.023$, white rectangles indicate $F/F_m>0.90$, the simulated values are shown in the right column. (b) Shows the same for the corresponding rectangular pulse, where $F_m=0.936\pm 0.009$, ${\sigma }_{\text{av}}=0.024$.