Role of control constraints in quantum optimal control

The problems of optimizing the value of an arbitrary observable of a two-level system at both a fixed time and the shortest possible time is theoretically explored. Complete identification and classification along with comprehensive analysis of globally optimal control policies and traps (i.e., policies which are locally but not globally optimal) are presented. The central question addressed is whether the control landscape remains trap-free if control constraints of the inequality type are imposed. The answer is astonishingly controversial: Although the traps are proven always to exist in this case, in practice they become trivially escapable once the control time is fixed and chosen long enough.

Figure 1

Possible types of extremals $\tilde{u}\left(t\right)$ associated with nonkinematic optimal solutions and traps along with the locally time-optimal kinematic optimal solutions.

Figure 2

(a) The case $r_y^{-}{r}_y^{+}>0$: The equatorial singular arc $r^{-}\to r^{+}$ (thick black line) is more time-effective than the bang-bang extremal $r^{-}\to r^{\prime }\to r^{+}$ (thick orange line). The extremal $r^{-}\to r^{\prime \prime }\to r^{+}$ (thin blue curve) represents a local extremum (trap). (b) The case $r_y^{-}{r}_y^{+}<0$: The equatorial singular arc $r^{-}\to r^{+}$ (thick orange line) is suboptimal relative to the bang-singular extremal $r^{-}\to r^{\prime }\to r^{+}$ (thin black line).

Figure 3

The globally time-optimal ${}^0\mathrm{II}$ type trajectory ${\tilde{u}}_{\mathrm{anz}}\left(\tau \right)$ (thick yellow curve) and the locally time-optimal trapping solution (black curve) of the ${\mathcal{F}}^{\left[3\right]}({\tilde{u}}^{\text{anz}}\left(\tau \right))$ family connecting the points $r_0\propto \{1,1,-1\}$ and $o\propto \{-1,1,1\}$.

Figure 4

Illustration of the statement of Proposition 6. The thick colored curve depicts the bang-bang extremal. Its red and blue segments correspond to $u=+u_{\max }$ and $u=-u_{\max }$. All interior corner points (red and blue spheres) lie on two circles (associated with switchings $u_{\max }\to -u_{\max }$ and $-u_{\max }\to u_{\max }$, respectively) whose planes ${\lambda }_{\pm 1}$ intersect along the $z$ axis.

Figure 5

Signs of the parameters $q\left(\zeta \right)$ as a function of $\zeta$. Black circles indicate the values $\zeta ={\zeta }_i$ associated with the $i\mathrm{th}$ corner point.

Figure 6

Geometrical calculation of the value of $\mathrm{\Delta }{\theta }_{r,x}^{max}$. The rotation ${\mathcal{S}}_{{\vec{n}}_{-u_{\max }}}$ about vector ${\vec{n}}_{-u_{\max }}$ transfers any point $r_i$ on the Bloch sphere into a new point on the $A{A}^{\prime }$ plane. The $x$ coordinate of this new point is bounded by the planes ${\lambda }^{\prime }$ and ${\lambda }^{\prime \prime }$. Thus, the associated change in ${\theta }_{r,x}$ is less than $\angle \mathrm{AOB}=2\alpha$.

Figure 7

Distribution of types of globally optimal solutions according to Proposition 14. Note that the admissible values of ${\varphi }_x$ and ${\varphi }_z$ are restricted by the inequality ${\varphi }_x+{\varphi }_z\le \pi$.

Figure 8

Globally optimal solution (blue line), deadlock traps (light-red and light-green lines), and loop trap (black line) for the time-optimal control problem (2), (3), and (4) with $u_{\max }=\frac{1}{4},r\left(0\right)=\{\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\}$ (large green sphere), and $o=\{\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0\}$ (large black-yellow sphere). Small spheres indicate the positions of corner points. The parameters of extremals are listed in the table.

Figure 9

The optimal solution (medium-thick trajectory; $r_0\to r_1\to r_2\to o$), topological trap (thin trajectory; $r_0\to r^{-}\to r^{+}\to o$), and deadlock trap (thick trajectory; $r_0\to r^{\prime }$) for the time-optimal control problem (2), (3), (4) with $u_{\max }=8,r\left(0\right)\propto \{\frac{1}{2},\frac{1}{2},u_{\max }\}$, and $o\propto \{1,0,u_{\max }\}$. The blue, black, and red segments correspond to $u\left(\tau \right)=-u_{\max }$, 0, and $+u_{\max }$, respectively, and are associated with rotations about the axes ${\vec{n}}_{-u_{\max }},{\vec{\epsilon }}_x$, and ${\vec{n}}_{-u_{\max }}$. The durations $\mathrm{\Delta }{\tau }_i$ of the consequent bang arcs are summarized in the table.

Figure 10

Projections of the characteristic pieces of the original, varied, and reduced trajectories $\tilde{r}\left(\tau \right),r^{\prime }\left(\tau \right)$, and $r^{\prime \prime }\left(\tau \right)$ on the $xz$ plane (it is assumed that $y$ components of all shown parts of trajectories are greater than 0). The color associations are indicated in the inset.

Figure 11

Projection of the extremal on the $xz$ plane in the vicinity of the corner point ${\tilde{r}}_{i^{\prime }}$ in the case ${\tilde{r}}_{i^{\prime },x}<0$. The dashed (orange) ellipse is the projection of the intersection of the Bloch sphere with the planes ${\lambda }_{\pm 1}$. Arrows indicates the admissible routes of passing the point ${\tilde{r}}_{i^{\prime }}$ according to Proposition 3.