Time-optimal control of a dissipative qubit

A formalism based on Pontryagin's maximum principle is applied to determine the time-optimal protocol that drives a general initial state to a target state by a Hamiltonian with limited control, i.e., there is a single control field with bounded amplitude. The coupling between the bath and the qubit is modeled by a Lindblad master equation. Dissipation typically drives the system to the maximally mixed state; consequently, there generally exists an optimal evolution time beyond which the decoherence prevents the system from getting closer to the target state. For some specific dissipation channel, however, the optimal control can keep the system from the maximum entropy state for infinitely long. The conditions under which this specific situation arises are discussed in detail. The numerical procedure to construct the time-optimal protocol is described. In particular, the formalism adopted here can efficiently evaluate the time-dependent singular control which turns out to be crucial in controlling either an isolated or a dissipative qubit.

Figure 1

The optimal control for the evolution time $t_f=0.42\pi$, for (a), (b) $\xi =0$; (c), (d) $\xi =0.2$; and (e), (f) $\xi =0.8$. (a), (c), and (e) show that all necessary conditions are satisfied. Dashed curves: scaled control; solid curves: c-Hamiltonian; dotted curves: switching function. (b), (d), (f) The corresponding singular arc (dashed curves) and the optimal trajectory (solid curves) on the Bloch sphere. Note that the optimal control goes from XSY to YSY upon increasing $\xi$.

Figure 2

$\xi =0.2$ and $\mathrm{\Gamma }=0.1$, the uniform dissipation described in Eq. (32). (a) The target-state overlap as a function of the evolution time $t_f$. The vertical dashed (black) line is the boundaries of different optimal control structures; the corresponding optimal controls are indicated in red. The optimal evolution time is around $0.42\pi$, indicated by the solid (red) vertical line. (b1)–(b4) The optimal controls at four representative computational times: (b1) $t_f=0.2\pi$, (b2) $t_f=0.35\pi$, (b3) $t_f=0.419\pi$ (optimal $t_f$), and (b4) $t_f=0.43\pi$. Dashed curves: scaled control; solid curves: c-Hamiltonian; dotted curves: switching function. As the evolution time increases, the optimal control changes from (b1) $XY$ to (b2)–(b4) XSY. For (b1)–(b3), all the necessary conditions are satisfied. (b3) At the optimal evolution time, the c-Hamiltonian is zero. (b4) Beyond the optimal evolution time, the c-Hamiltonian becomes positive.

Figure 3

(a) The target-state overlap $\langle {\mathit{\rho }}_f,\mathit{\rho }\left(t_f\right)\rangle$ as a function of evolution time $t_f$. Vertical dashed (black) lines are boundaries of different optimal control structures; the corresponding optimal controls are indicated in red. (b1)–(b4) The optimal controls at four representative computational times: (b1) $t_f=0.2\pi$, (b2) $t_f=0.35\pi$, (b3) $t_f=0.42\pi$, and (b4) $t_f=0.90\pi$. Dashed curves: scaled control; solid curves: c-Hamiltonian; dotted curves: switching function. As the evolution time increases, the optimal control changes from (b1) $XY$ to (b2) XSY to (b3) XSXY to (b4) XYSXY. Numerically, no finite optimal $t_f$ is found in this case.

Figure 4

$\mathit{\rho }\left(t\right)$ during the optimal control for different dissipation channels. (a) Uniform dissipation with $t_f=0.4\pi$; (b) ${\sigma }_z$ dissipation channel with $t_f=0.4\pi$. The trajectories are very similar for these cases. (c),(d) ${\sigma }_x$ dissipation channel for (c) $t_f=0.38\pi$ and (d) $t_f=2.00\pi$. At $t_f=0.38\pi$, the trajectory is still similar to those in (a) and (b). As $t_f$ increases, $\mathit{\rho }\left(t\right)$ approaches the point ${\mathit{\rho }}_c=({\rho }_x,0,0)$, where $u\to 0$ is an admissible control that leaves ${\rho }_c$ unchanged. During the singular control in the ${\sigma }_x$ dissipation channel, the amplitude of $\left|\mathit{\rho }\right(t\left)\right|$ decays slowly because of the small ${\rho }_y$ and ${\rho }_z$ components. Vertical lines indicate the switching times.

Figure 5

(a) The target-state overlap $\langle {\mathit{\rho }}_f,\mathit{\rho }\left(t_f\right)\rangle$ as a function of evolution time $t_f$ using $\xi =0.8$. The optimal control structure for $t_f\ge 0.5\pi$ is found to be YSXY. An optimal $t_f$ is around 0.73 $\pi$. As $t_f$ increases, $\langle {\mathit{\rho }}_f,\mathit{\rho }\left(t_f\right)\rangle$ does not decay to zero. (b) The $\mathit{\rho }\left(t\right)$ for $t_f=2.0\pi$. $\mathit{\rho }\left(t\right)$ spends almost 50% of the time close to ${\rho }_c=({\rho }_x,0,0)$, the point where the dissipation has no effect. Three vertical dotted lines indicate the switching times when the control protocol changes from $Y$ to $S$, from $S$ to $X$, and from $X$ to $Y$.

Figure 6

The optimal control obtained from the gradient-based numerical optimization (cross symbol, 500 points) and the numerically exact procedure (solid). A good agreement is seen, especially for the singular part. The parameters are the same as those used in Fig 3, and the exact optimal control is taken from Fig. 3. The c-Hamiltonian obtained from numerical optimization is also given. There is a small discontinuity across each switching time because the exact switching times are not captured.

Figure 7

The quantum state retention with the ${\sigma }_x$ dissipation channel. (a) The target-state overlap $\langle {\mathit{\rho }}_f,\mathit{\rho }\left(t_f\right)\rangle$ as a function of evolution time $t_f$. The solid-circle curve is obtained using optimal control; the dotted curve is obtained without any control [$u\left(t\right)=0$]. The target-state overlap obtained using optimal control is larger than that obtained with no control. Vertical dashed (black) lines are boundaries of different optimal control structures; the corresponding optimal controls are indicated in red. A local maximum occurs at $t_f\approx 0.58\pi$ for the optimal control, and at $t_f\approx \pi$ for zero control. As $t_f$ increases, $\langle {\mathit{\rho }}_f,\mathit{\rho }\left(t_f\right)\rangle$ does not decay to zero. (b1)–(b3) The optimal control for three different evolution times: (b1) $t_f=0.2\pi$, (b2) $t_f=0.5\pi$, and (b3) $t_f=0.7\pi$. Dashed curves: scaled control; solid curves: c-Hamiltonian; dotted curves: switching function. As the evolution time increases, the optimal control changes from (b1) $Y$ to (b2) XYX, to (b3) XYSYX. (c) $\mathit{\rho }\left(t\right)$ for $t_f=1.6\pi$. Around $t=0.8\pi , \mathit{\rho }$ is close to ${\mathit{\rho }}_c=({\mathit{\rho }}_x,0,0)$.