Figure 1

(a) Typical run of the optimization routine to implement the entangling gate ${\mathcal{U}}_{(0.5,0.4,0.3)}$ for $g_x=5.40$ MHz, $g_y=9.95$ MHz, ${\omega }_1=0.13$ MHz, and ${\omega }_2=0.26$ MHz in minimal time. The time $t_f$ (solid line) is minimized, such that a gate fidelity (dashed line) of more than 99.99% is reached. (b) Result of the above optimization: logarithmic infidelity as a function of time with (solid line) and without (dashed line) control. Without control the gate can be realized with only modest fidelity.Reuse & Permissions

Figure 2

Logarithmic gate infidelity ${\log }_{10}(1-{\mathcal{F}}_0)$ as a function of the pulse duration $t_f$ (“$\times$”) for $n_{\max }=6$ frequency components. The symbol “$+$” depicts the time-optimal pulses for a required gate fidelity of ${\mathcal{F}}_0=1-{10}^{-4}$ and $1-{10}^{-6}$, respectively. They both lie on the graph ${\mathcal{F}}_0\left(t_f\right)$.Reuse & Permissions

Figure 3

Time evolution of the tangle $C^2$ for two interacting spins with $g_x=2.7\mathrm{MHz}$, $g_y=6.2\mathrm{MHz}$, ${\omega }_1=0.3\mathrm{MHz}$, and ${\omega }_2=0.2\mathrm{MHz}$. Without control (solid line) maximal entanglement can never be reached. The gray curves are obtained by maximizing the tangle at $t_f=0.2\mu \mathrm{s}$ (dotted curve) and $t_f=0.4\mu \mathrm{s}$ (dashed curve), respectively (these moments in time are emphasized by fat vertical lines), by using a control pulse with $n_{\max }=6$ frequency components and fundamental frequency $\Omega =\pi /t_f$. The curves obtained with an additional minimization of the curvature are depicted in black. The time interval for which the spins are highly entangled is considerably longer in these cases.Reuse & Permissions

Figure 4

(a) Time evolution of the entanglement of formation $E$ of the end spins in two random configurations of a chain with $N=3$ (solid line, $g_x^{\left(1\right)}=0.78$ MHz, $g_x^{\left(2\right)}=1.48$ MHz, $g_y^{\left(1\right)}=1.27$ MHz, $g_y^{\left(2\right)}=2.65$ MHz, ${\omega }_1=0.91$ MHz, ${\omega }_2=0.97$ MHz, and ${\omega }_3=0.40$ MHz) and $N=4$ spins (dotted line, $g_x^{\left(1\right)}=4.36$ MHz, $g_x^{\left(2\right)}=1.61$ MHz, $g_x^{\left(3\right)}=5.33$ MHz, $g_y^{\left(1\right)}=1.02$ MHz, $g_y^{\left(2\right)}=8.82$ MHz, $g_y^{\left(3\right)}=1.29$ MHz, ${\omega }_1=0.57$ MHz, ${\omega }_2=0.55$ MHz, ${\omega }_3=0.81$ MHz, and ${\omega }_4=0.42$ MHz). In both cases the intrinsic dynamics of the chain (gray) are not able to entangle the end spins, whereas a control pulse with $n_{\max }=6$ frequency components (black) can generate maximal entanglement at $t_f=1\mu$s (see the fat vertical line). (b) Entanglement of formation of the controlled system of $N=3$ spins (circles) if the pulse optimized for the above coupling configuration is applied to a system where the coupling constants are only known up to a certain error. The maximization of the averaged entanglement for a test ensemble of ten coupling configurations (crosses) significantly increases the amount of entanglement (for 5%, 94.2%$\to$ 99.3%; for 10%, 78.1%$\to$ 96.7%).Reuse & Permissions