Combining the synergistic control capabilities of modeling and experiments: Illustration of finding a minimum-time quantum objective

A common way to manipulate a quantum system, for example spins or artificial atoms, is to use properly tailored control pulses. In order to accomplish quantum information tasks before coherence is lost, it is crucial to implement the control in the shortest possible time. Here we report the near time-optimal preparation of a Bell state with fidelity higher than $99\%$ in an NMR experiment, which is feasible by combining the synergistic capabilities of modeling and experiments operating in tandem. The pulses preparing the Bell state are found by experiments that are recursively assisted with a gradient-based optimization algorithm working with a model. Thus, we exploit the interplay between model-based numerical optimal design and experimental-based learning control. Utilizing the balanced synergism between the dual approaches, as dictated by the case specific capabilities of each approach, should have broad applications for accelerating the search for optimal quantum controls.

Figure 1

Illustration of two general optimization approaches to find the controls achieving desired objective(s). In (I) the optimization loop is entirely based on numerical simulation-based design (red) followed by implementation of the design, whereas (II) makes exclusive use of experiments and data analysis (blue). The green dashed curve indicates a balanced synergistic combination of both approaches, which is used in our NMR experiment to (i) prepare a Bell state while (ii) also in minimum time. In this illustration the controls are updated using a gradient-based closed-loop learning algorithm, in which we utilize a model-based simulation to calculate the gradient and then experimental state tomography to determine the step size in each iteration cycle. In general, the best balanced way to combine the two approaches (I) and (II) depends on the experimental platform considered, the quality of the model describing the system, the objectives, and the algorithm(s) used to achieve the desired task. Thus, the balance between (I) and (II) is not static, as it rests on the future progress made in technology as well as theory and computational capabilities.

Figure 2

Experimental data for preparing a Bell state with high fidelity near the minimum time $T_{\text{min}}=2.30\mathrm{ms}$ (dashed red line) using shaped control pulses obtained by employing a synergistic combination of approaches (I) and (II) shown in Fig. 1. (a) shows the partial fidelity $J_{\mathrm{tomo}}$ obtained from 3 measurements on spin 1 and the time length $T$ of the control pulses as a function of the iteration step $n$ of the optimization algorithm. The curved black dashed line is the lower threshold $J_L$ and the straight horizontal black dashed line is the prescribed fidelity $J_H$. The vertical gray dashed lines indicate the two steps of the algorithm that was introduced in Sec. 4b, including a fidelity climb back to $J_{\text{H}}$ when a reduction to the threshold $J_{\text{L}}$ occurs. As explained below Eq. (6), when $J_{\mathrm{tomo}}$ drops beyond $J_L$ due to experimental or modeling errors, $T$ is kept fixed to bring the fidelity back to $J_H$ (i.e., indicated by the fidelity reclimbing in the early stages of the procedure), followed by further shrinking of $T$. (b) shows the optimized control pulses applied on spin 1 (blue open bars) and spin 2 (red solid bars) for a pulse length of $T=5.00\mathrm{ms}$ (left panel) and the pulses of length $T=2.31\mathrm{ms}$ near the minimum time (right panel).