Optimal Quantum Control with Poor Statistics

Control of quantum systems is a central element of high-precision experiments and the development of quantum technological applications. Control pulses that are typically temporally or spatially modulated are often designed based on theoretical simulations. As we gain control over larger and more complex quantum systems, however, we reach the limitations of our capabilities of theoretical modeling and simulations, and learning how to control a quantum system based exclusively on experimental data can help us to exceed those limitations. Because of the intrinsic probabilistic nature of quantum mechanics, it is fundamentally necessary to repeat measurements on individual quantum systems many times in order to estimate the expectation value of an observable with good accuracy. Control algorithms requiring accurate data can thus imply an experimental effort that negates the benefits of avoiding theoretical modeling. We present a control algorithm that finds optimal control solutions in the presence of large measurement shot noise and even in the limit of single-shot measurements. The algorithm builds up on Bayesian optimization that is well suited to handle noisy data; but since the commonly used assumption of Gaussian noise is not appropriate for projective measurements with a low number of repetitions, we develop Bayesian inference that respects the binomial nature of shot noise. With several numerical and experimental examples we demonstrate that this method is capable of finding excellent control solutions with minimal experimental effort.

Popular Summary

Optimal quantum control, e.g., in terms of temporally shaped laser pulses, is essential for high-precision experiments and the development of quantum technologies. Quantum systems that can be experimentally controlled are of increasing complexity and numerically simulating their dynamics is becoming more and more demanding. Designing control pulses based exclusively on experimental observations allows one to avoid resorting to such prohibitively lengthy computations. However, since access to quantum mechanical hardware is a costly resource, it is essential that such pulses can be found using a minimal number of interactions with the experiment. For that purpose, we develop an algorithm that does not require accurate experimental data. Rather, it can tolerate data with any level of measurement noise, thereby avoiding the otherwise necessary repetitions of the same measurement.

Bayesian inference allows one to estimate the performance of any control pulse based on existing observations made with other control pulses. It can thus be used to identify the pulse that is the most promising. Once an experiment has been performed with this new pulse, the result is added to the total body of observations. An iteration of these steps constitutes the basic principle of Bayesian optimization. In our paper, we develop the tools that allow us to apply Bayesian optimization to situations in which all experimental observations are subject to the statistical nature of the quantum mechanical measurement. We demonstrate that taking into account the proper statistical fluctuations of quantum mechanical measurements helps our algorithm to substantially outperform more generic approaches.

Figure 1

Schematic example of the estimation of a control landscape based on a finite number of data points. In this example, the six given data points are consistent with a parabola shown in black, but they are also consistent with many other landscapes, such as that shown in red. Assuming the black landscape to be the correct landscape, one would predict the control parameter ${\theta }^{\mathrm{\prime }}$ to yield the maximum of the landscape, but one would predict the control parameter $\tilde{\theta }$ when assuming the red landscape to be the correct landscape. Even though the red landscape might not seem likely to be the correct one given the six data points, it can be worth considering since it promises a larger value of $F$ than the black landscape.

Figure 2

Three stages of a typical run of Bayesian optimization with extremely bad measurement statistics after (a) $30$, (b) $31$, and (c) $100$ steps. The top panel depicts the actual control landscape (dashed red line), the expected control landscape (solid blue line), and a $95\mathrm{\%}$ confidence interval (solid gray line) around the expected landscape. The actual probability density is represented by shades of blue. Simulated, random outcomes of a projective measurement are depicted by red circles and green squares, where green corresponds to the most recent data point. Assuming a sufficiently smooth landscape, it is possible to predict control landscapes ranging continuously between the values $0$ and $1$ despite the digital data. The lower panel depicts the acquisition function [Eq. (17) with $\alpha =4$ for (a) and (b) and $\alpha =0$ for (c)], the maximum of which determines the value of $\theta$ for the next upcoming step in the iterative algorithm. During the course of the optimization (from left to right), the surrogate model becomes a better approximation of the actual landscape in the vicinity of the maxima, but the algorithm avoids the effort that would be required to approximate the landscape well in other domains.

Figure 3

Gate sequence for the preparation of a three-qubit GHZ state in terms of two controlled-not gates and six single-qubit gates with adjustable parameter ${\theta }_j$. The addition of noisy unitaries $N_{\epsilon }$ and readout error helps to demonstrate optimal control in the presence of experimental noise in addition to the measurement shot noise.

Figure 4

Convergence of the control algorithms towards low infidelities $\mathcal{I}$ as a function of the total number of runs $N_r$ of the circuit required. The median values of $\mathcal{I}$ are indicated by filled symbols for the binomial modeling of measurement noise and by open symbols for Gaussian modeling. The different colors correspond to different numbers of repetitions of each measurement, ranging from $N=1$ in black to $N=1000$ in orange. Interquartile intervals are depicted by shaded regions. Optimizations with the proper binomial modeling converge substantially faster than optimizations with Gaussian modeling of measurement noise, except in the case of a large number of repetitions $N=1000$. The inset depicts results for fewer repetitions ranging from $N=1$ in black to $N=10$ in orange, and binomial modeling only. Interquartile intervals are not displayed for visual clarity.

Figure 5

Comparison of the adaptive strategy (black) explained in Sec. 3b3 with optimizations performed using SPSA with a fixed number of repetitions $N$ of each measurement given in legend. The adaptive method converges systematically faster than any of the other cases.

Figure 6

Convergence to low infidelities in the case of noisy circuits. The different colors correspond to different noise levels as specified in the legend where ${\sigma }_N$ denotes the amplitude of the noisy unitaries and $p_{\mathrm{ro}}$ the probability of a readout error. The noiseless case (black curve) is provided as a reference.

Figure 7

Convergence of SPSA in the presence of noise. All the results are obtained with single-shot measurements ($N=1$), since this gives the best results in the noiseless case.

Figure 8

Convergence of the filling error $\mathcal{E}$ of a bosonic chain as a function of the total number of runs $N_r$. Results obtained with binomial and Gaussian modeling are depicted with filled and open symbols, respectively. All optimizations are performed with a fixed number of repetitions $N$ of each measurement (specified in the legend) and a total number of iteration steps $M$ ranging from $250$ to 2000. The dashed red line shows the result of the best solution found with ${\mathcal{E}}_{\min }\approx 1.6\mathrm{\%}$.

Figure 9

Optimization results for the adaptive strategy (Sec. 3b3) and SPSA with different numbers of repetitions $N$. The dashed red line indicates the lowest obtained filling error.

Figure 10

Convergence of optimizations in the presence and absence of fluctuations in the particle number $N_b$ (purple and black). In both cases the final filling errors are very close to the best filling errors obtained with simulations assuming perfectly accurate measurements (dashed lines).