Pareto-front shape in multiobservable quantum control

Many scenarios in the sciences and engineering require simultaneous optimization of multiple objective functions, which are usually conflicting or competing. In such problems the Pareto front, where none of the individual objectives can be further improved without degrading some others, shows the tradeoff relations between the competing objectives. This paper analyzes the Pareto-front shape for the problem of quantum multiobservable control, i.e., optimizing the expectation values of multiple observables in the same quantum system. Analytic and numerical results demonstrate that with two commuting observables the Pareto front is a convex polygon consisting of flat segments only, while with noncommuting observables the Pareto front includes convexly curved segments. We also assess the capability of a weighted-sum method to continuously capture the points along the Pareto front. Illustrative examples with realistic physical conditions are presented, including NMR control experiments on a ${}^1\mathrm{H}-{}^{13}\mathrm{C}$ two-spin system with two commuting or noncommuting observables.

Figure 1

Pareto front of Example 1, a dual-observable problem defined in a two-spin system $I-S$, which has the shape of an octagon. The initial density matrix is chosen as ${\rho }_0=I_z+0.25S_z$, and the two commuting observables are $O_1=I_x$ and $O_2=S_x$. The points marked by $\circ$ and $\times$ are determined by permuting the diagonal entries of ${\rho }_0$ and calculating the corresponding $J_1$ and $J_2$ values; the former points are vertices of the octagon, while the latter ones are not.

Figure 2

Pareto front of Example 2 on manipulating the orbital angular momentum of the electron in a hydrogen atom. A pure initial state ${\rho }_0=|1,0,0\rangle \langle 1,0,0|$ is chosen, while the two commuting observables are $O_1=L^2$ and $O_2=L_z$. Both quantum and classical results of the Pareto front are provided to show the distinction caused by quantization.

Figure 3

Feasible region of Example 3, defined by ${\rho }_0=|1\rangle \langle 1|$ and three mutually commuting observables $O_m=|m\rangle \langle m|, m=1,2,3$. If the system only has the three levels $|1\rangle , |2\rangle$, and $|3\rangle$, the feasible region will be the colored triangle $ABC$. Once a fourth level (or more) is added in, the feasible region will be extended to the tetrahedron $OABC$.

Figure 4

Searching for the Pareto front by optimizing a weighted-sum objective $\mathcal{J}=w_1{J}_1+w_2{J}_2$. The dashed lines depict level sets with constant $\mathcal{J}$ values at different choices of $\mathbf{w}=(w_1,w_2)$, and in optimizing $\mathcal{J}$ along its own gradient we move across the level sets until stopping at a level set tangent (or nearly so, in practice) to the feasible region boundary. In (b) where the Pareto front (solid line) is smooth and convex, the search will stop where the normal vector (an arrow) is parallel to $\mathbf{w}$, while in (a) where the Pareto front consists of flat segments, the search will always stop at the vertex if $\mathbf{w}$ is aligned with any direction between the two dashed arrows.

Figure 5

Pareto front of the two-observable problem in (16). The observables are noncommuting except when $sin\theta =0$. As the parameter $\theta$ varies, the Pareto front may take on the extreme shapes of a circle, an ellipse, or a line segment.

Figure 6

Pareto front of the three-level example in Eq. (19) is a mixture of both flat and convexly curved segments. The nondifferentiable vertex $B$ corresponds to critical points of both individual objectives $J_1$ and $J_2$, and the Pareto front does not have a unique normal direction at point $B$.

Figure 7

Experimental NMR illustrations of the theoretical Pareto principles. Dual-observable quantum control experiment in the two-spin system of ${}^{13}\mathrm{CHCl}{}_3$ ($I{=}^1\mathrm{H}$; $S{=}^{13}\mathrm{C}$): the evolution of $(J_1,J_2)$ during the optimization of a weighted-sum objective function $\mathcal{J}\left(w_1\right)=w_1{J}_1+(1-w_1)J_2$. (a) $O_1=I_x$ and $O_2=S_x$ (commuting); (b) $O_1=I_x$ and $O_2=I_y$ (noncommuting). The dashed lines depict a portion of the theoretical Pareto fronts in the two cases. Red dotted and blue solid lines show optimization processes of an individual objective ($J_1$ or $J_2$) and combinations of both, respectively, with each point representing an experimental iteration.