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

Abstract

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 acoupled two-spin system in a l3C-labeled chloroform ( I3CHC13) 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.

Pareto-front shape in multiobservable quantum control

“…When J 1 reaches its global optima at ±1, the other objective J 2 can only take values within the range [−|r|, |r|], and vice versa. Further improvement of J 2 can be achieved via coherence transfer [41], but the value of J 1 will inevitably deviate from its optimum at the same time; e.g., the maximization of J 1 and J 2 obeys a linear tradeoff relation that J 1 + J 2 ≤ 1 + |r|. Experimental verification of the analysis in this example is given in Sec.…”

“…Two radio frequency pulses with an equal duration of T = 5ms irradiate the sample simultaneously as the control resources, whose carrier frequencies are resonant with 1 H or 13 C. The amplitudes and phases of the control pulses are modulated in the time domain for optimally driving the dynamics of the two-spin system in the desired manner. The detailed experimental setup is similar to that used in our previous works [41,46,47].…”

“…Consider a coupled two-spin system I-S with I and S denoting two discernible spins-1/2.We use the operators I a and S a (a = x, y, z) to represent the angular momenta of each spin in the x, y, z directions, defined as I a := (σ a /2) ⊗ I 2 and S a := I 2 ⊗ (σ a /2), where σ a is a Pauli matrix and I 2 is the 2 × 2 identity matrix. In conventional NMR experiments, the thermal equilibrium state of the system can be well approximated by ρ 0 = I z + rS z (i.e., the traceless portion of the density matrix), with 0 < |r| < 1 corresponding to the different gyromagnetic ratio of spins I and S[41]. For the purpose of simultaneously optimizing the signal intensities of both spins in their corresponding NMR spectra in a single measurement, we can consider the dual-observable optimization of J m (U ) = Tr(U ρ 0 U † O m ), m = 1, 2, with O 1 = I x and O 2 = S x .…”

Searching for an optimal control in the presence of saddles on the quantum-mechanical observable landscape

“…In this work, therefore, we focus on observable control, for which the landscape may have a much larger number of saddles. Recent OCEs performed on a two-spin system located saddles on the observable control landscape at the predicted objective values and of the right character [113], providing empirical support for the theoretical analysis.…”

Optimization Landscape of Quantum Control Systems

“…In contrast with the single-spin system, the two-spin control landscape possesses nontrivial kinematic saddle points. The saddles are addressed by measuring the Hessian and assessing how their presence can influence the search effort utilizing a gradient algorithm to seek an optimal control [68] . The 13 CHCl 3 sample was also used to experimentally demonstrate the existence of singular traps in systems with multiple control fields [55] .…”