Experimental observation of saddle points over the quantum control landscape of a two-spin system

The growing successes in performing quantum control experiments motivated the development of control landscape analysis as a basis to explain these findings. When a quantum system is controlled by an electromagnetic field, the observable as a functional of the control field forms a landscape. Theoretical analyses have predicted the existence of critical points over the landscapes, including saddle points with indefinite Hessians. This paper presents a systematic experimental study of quantum control landscape saddle points. Nuclear magnetic resonance control experiments are performed on a coupled two-spin system in a ${}^{13}\mathrm{C}$-labeled chloroform $\left({}^{13}{\mathrm{CHCl}}_3\right)$ sample. We address the saddles with a combined theoretical and experimental approach, measure the Hessian at each identified saddle point, and study how their presence can influence the search effort utilizing a gradient algorithm to seek an optimal control outcome. The results have significance beyond spin systems, as landscape saddles are expected to be present for the control of broad classes of quantum systems.

Figure 1

(a) The controls at landscape critical points within three different submanifolds, with top, saddle A, and saddle B, corresponding to the final iteration in trajectory 1 and the initial iterations in 2 and 3 in Fig. 2, respectively. The initial iterations at two saddle points were found by simulation and then refined in the laboratory, as explained in Sec. 4b. The components $u_x^I$ and $u_y^I$ address spin $I$ (${}^1\mathrm{H}$), while $u_x^S$ and $u_y^S$ address spin $S$ (${}^{13}\mathrm{C}$). For the critical points in top and saddle A shown here, the $u_x^S$ and $u_y^S$ components are approximately zero. (b) The ${}^1\mathrm{H}$-detected NMR spectra in the chemical shift range of 7.79–8.79 ppm, acquired with control pulses at the three critical points. The small kink in the middle of the two lines is ascribed to the proton signal in unlabeled ${}^{12}{\mathrm{CHCl}}_3$, which is eliminated when integrating the peaks as the objective $J$.

Figure 2

Three gradient ascent trajectories starting from different initial controls. Trajectory 1: Zero control fields. Trajectory 2: Saddle A at $J\simeq 0$. Trajectory 3: Saddle B at $J\simeq 0.25$. The iteration step size is constant for all the three trajectories. Trajectory 1 starting at a noncritical point rapidly ascends on the landscape and climbs to the top, while trajectories 2 and 3 linger near their respective saddles. Furthermore, trajectories 2 and 3 are gradient ascent (instead of saddle-seeking) trajectories, so the final iterations are not necessarily better saddle controls than the initial ones. In fact, the late iterations of trajectory 2 are clearly moving farther away from the saddle.

Figure 3

A gradient ascent trajectory passing nearby saddle B at $J\simeq 0.25$. The initial control at the zeroth iteration is chosen by perturbing the control at saddle B, which is shown in Fig. 1. The trajectory slows down when again coming close to the saddle, but then continues up the landscape. The norm of the gradient is also shown.

Figure 4

Hessian eigenspectra at the three critical points in Fig. 1. Each Hessian was estimated by least squares with 1000 random samples taken in the neighborhood of the fields at their respective critical points. Theoretical analysis summarized in Table 1 shows that top, saddle A, and saddle B should have at most 0/8, 4/4, and 2/6 positive/negative Hessian eigenvalues, respectively. The predictions are affirmed within experimental error.

Figure 5

Estimation of the Hessian eigenvalues of the control at saddle B and extrapolation using the expected scaling as $\sim 1/\sqrt{M}$, where $M$ is the number of random samples taken for determining the Hessian. The extrapolated spectrum shows several clear positive and negative eigenvalues along with a set of null eigenvalues consistent with theoretical predictions. The error bars at a given $M$ are estimated from the eigenspectra determined from ten data subsets of size $M$, which are taken from the total $\sim 2600$ samples.

Figure 6

Driving off the critical points at saddle A (a) and saddle B (b), shown in Fig. 1, along their respective Hessian eigenvectors associated with the dominant eigenvalues. The eigenvector $v_i$ corresponds to the $i\text{th}$ eigenvalue in the Hessian spectrum (see Fig. 4). The presence of parabolas opening both upward and downward, resulting from positive and negative Hessian eigenvalues, respectively, affirms the saddle topology of these two critical points. The roving distance along each eigenvector is in reference to the starting field at the respective saddle points.

Figure 7

Two critical points over the landscape of the observable $O=S_x$ belonging to top and saddle B in Table 1. The objective $J$ at saddle B, determined from the peak integral in the ${}^{13}\mathrm{C}$ NMR spectrum, is $\sim 0.24$ when the $J$ value of top is normalized to 1.0. (a) The control fields at the two critical points. Note that for saddle B the field addressing spin $I$ is zero. (b) The Hessian eigenspectra at the two critical points. The solid and dashed lines in the inset give the eigenvectors of the 15th and 16th (almost degenerate) eigenvalues at saddle B, respectively, which turn on the field components for spin $I$ in order to enable climbing away from the saddle.