Abstract
Efficient algorithms are of fundamental importance for the discovery of optimal control designs for coherent control of quantum processes. One important class of algorithms is sequential update algorithms, generally attributed to Krotov. Although widely and often successfully used, the associated theory is often involved and leaves many crucial questions unanswered, from the monotonicity and convergence of the algorithm to discretization effects, leading to the introduction of ad hoc penalty terms and suboptimal update schemes detrimental to the performance of the algorithm. We present a general framework for sequential update algorithms including specific prescriptions for efficient update rules with inexpensive dynamic search length control, taking into account discretization effects and eliminating the need for ad hoc penalty terms. The latter, despite being necessary for regularizing the problem in the limit of infinite time resolution, i.e. the continuum limit, are shown to be undesirable and unnecessary in the practically relevant case of finite time resolution. Numerical examples show that the ideas underlying many of these results extend even beyond what can be rigorously proved.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Quantum physics has developed from a fundamental scientific theory to the point where engineering quantum processes is becoming a realistic possibility, with many promising applications from quantum chemistry to artificial nanostructures. A fundamental requirement for harnessing this potential is the control of quantum dynamics and optimal control design. One of the main techniques for solving such control problems is a general class of methods attributed to Krotov. Unfortunately, there are many variants and their effectiveness and efficiency depend strongly on making good parameter choices; however, there are few guidelines for doing so.
Main results. We study the class of algorithms both theoretically and numerically. The analysis reveals that the most intuitive choices here are among the worst and provides prescriptions for good parameter choices and more effective and efficient variants of the algorithm.
Wider implications. The development of efficient algorithms for solving control optimization problems for quantum systems is crucial from a practical point of view—to enable the solution of a wider range of problems for realistic systems, which is currently a significant challenge due to the intrinsic complexity of quantum dynamics and the substantial computational overheads required to solve the resulting control optimization problems numerically. The ability to solve complex control optimization problems efficiently on a larger scale also facilitates the understanding of quantum dynamics and the discovery of new control mechanisms through control simulations, and thus the development of novel quantum technologies.