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Complex System Modeling and Simulation

Optimization Landscape of Quantum Control Systems

Xiaozhen Ge, Re-Bing Wu, HERSCHEL A. RABITZ

Published Jun 1, 2021• 10 citations
Publisher

Abstract

Optimization is ubiquitous in the control of quantum dynamics in atomic, molecular, and optical systems. The ease or difficulty of finding control solutions, which is practically crucial for developing quantum technologies, is highly dependent on the geometry of the underlying optimization landscapes. In this review, we give an introduction to the basic concepts in the theory of quantum optimal control landscapes, and their trap-free critical topology under two fundamental assumptions. Furthermore, the effects of various factors on the search effort are discussed, including control constraints, singularities, saddles, noises, and non-topological features of the landscapes. Additionally, we review recent experimental advances in the control of molecular and spin systems. These results provide an overall understanding of the optimization complexity of quantum control dynamics, which may help to develop more efficient optimization algorithms for quantum control systems, and as a promising extension, the training processes in quantum machine learning.

DOI
10.23919/csms.2021.0014

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Optimization is ubiquitous in the control of quantum dynamics in atomic, molecular, and optical systems. The ease or difficulty of finding control solutions, which is practically crucial for developing quantum technologies, is highly dependent on the geometry of the underlying optimization landscapes. In this review, we give an introduction to the basic concepts in the theory of quantum optimal control landscapes, and their trap-free critical topology under two fundamental assumptions. Furthermore, the effects of various factors on the search effort are discussed, including control constraints, singularities, saddles, noises, and non-topological features of the landscapes. Additionally, we review recent experimental advances in the control of molecular and spin systems. These results provide an overall understanding of the optimization complexity of quantum control dynamics, which may help to develop more efficient optimization algorithms for quantum control systems, and as a promising extension, the training processes in quantum machine learning.
Quantum optimal control experiments and simulations have successfully manipulated the dynamics of systems ranging from atoms to biomolecules. Surprisingly, these collective works indicate that the effort (i.e., the number of algorithmic iterations) required to find an optimal control field appears to be essentially invariant to the complexity of the system. The present work explores this matter in a series of systematic optimizations of the state-to-state transition probability on model quantum systems with the number of states NNN ranging from 5 through 100. The optimizations occur over a landscape defined by the transition probability as a function of the control field. Previous theoretical studies on the topology of quantum control landscapes established that they should be free of suboptimal traps under reasonable physical conditions. The simulations in this work include nearly 5000 individual optimization test cases, all of which confirm this prediction by fully achieving optimal population transfer of at least 99.9% on careful attention to numerical procedures to ensure that the controls are free of constraints. Collectively, the simulation results additionally show invariance of required search effort to system dimension NNN. This behavior is rationalized in terms of the structural features of the underlying control landscape. The very attractive observed scaling with system complexity may be understood by considering the distance traveled on the control landscape during a search and the magnitude of the control landscape slope. Exceptions to this favorable scaling behavior can arise when the initial control field fluence is too large or when the target final state recedes from the initial state as NNN increases.
Quantum control landscape theory was formulated to assess the ease of finding optimal control fields in simulations and in the laboratory. The landscape is the observable as a function of the controls, and a primary goal of the theory is the analysis of landscape features. In what is referred to as the kinematic picture of the landscape, prior work showed that the landscapes are generally free of traps that could halt the search for an optimal control at a suboptimal observable value. The present paper considers the dynamical picture of the landscape, seeking the existence of singular controls, especially of a nonkinematic nature along with an assessment of whether they correspond to traps. We analyze the necessary and sufficient conditions for singular controls to be kinematic or nonkinematic critical solutions and the likelihood of their being encountered while maximizing an observable. An algorithm is introduced to seek singular controls on the landscape in simulations along with an associated Hessian landscape analysis. Simulations are performed for a large number of model finite-level quantum systems, showing that all the numerically identified kinematic and nonkinematic singular critical controls are not traps, in support of the prior empirical observations on the ease of finding high-quality optimal control fields.
Optimal Control Theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This review aims at providing an introduction to key concepts of optimal control theory which is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This review describes modern aspects of optimal control theory, with a particular focus on the Pontryagin Maximum Principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The review covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.
The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been shown to contain no critical point suboptimal traps, thereby enabling effective searches for fields that give the global maximum of the objective. This paper addresses the structure of the landscape as a complement to topological critical point features. Recent work showed that landscape structure is highly favorable for optimization of state-to-state transition probabilities, in that gradient-based control trajectories to the global maximum value are nearly straight paths. The landscape structure is codified in the metric R ≥ 1.0, defined as the ratio of the length of the control trajectory to the Euclidean distance between the initial and optimal controls. A value of R = 1 would indicate an exactly straight trajectory to the optimal observable value. This paper extends the state-to-state transition probability results to the quantum ensemble and unitary transformation control landscapes. Again, nearly straight trajectories predominate, and we demonstrate that R can take values approaching 1.0 with high precision. However, the interplay of optimization trajectories with critical saddle submanifolds is found to influence landscape structure. A fundamental relationship necessary for perfectly straight gradient-based control trajectories is derived, wherein the gradient on the quantum control landscape must be an eigenfunction of the Hessian. This relation is an indicator of landscape structure and may provide a means to identify physical conditions when control trajectories can achieve perfect linearity. The collective favorable landscape topology and structure provide a foundation to understand why optimal quantum control can be readily achieved.
The reliable and precise generation of quantum unitary transformations is essential for the realization of a number of fundamental objectives, such as quantum control and quantum information processing. Prior work has explored the optimal control problem of generating such unitary transformations as a surface-optimization problem over the quantum control landscape, defined as a metric for realizing a desired unitary transformation as a function of the control variables. It was found that under the assumption of nondissipative and controllable dynamics, the landscape topology is trap free, which implies that any reasonable optimization heuristic should be able to identify globally optimal solutions. The present work is a control landscape analysis, which incorporates specific constraints in the Hamiltonian that correspond to certain dynamical symmetries in the underlying physical system. It is found that the presence of such symmetries does not destroy the trap-free topology. These findings expand the class of quantum dynamical systems on which control problems are intrinsically amenable to a solution by optimal control.
A common goal of quantum control is to maximize a physical observable through the application of a tailored field. The observable value as a function of the field constitutes a quantum-control landscape. Previous papers have shown, under specified conditions, that the quantum-control landscape should be free of suboptimal critical points. This favorable landscape topology is one factor contributing to the efficiency of climbing the landscape. An additional complementary factor is the landscape structure, which constitutes all nontopological features. If the landscape's structure is too complex, then climbs may be forced to take inefficient convoluted routes to find optimal controls. This paper provides a foundation for understanding control-landscape structure by examining the linearity of gradient-based optimization trajectories through the space of control fields. For this assessment, a metric $R\ensuremath{\ge}1$ is defined as the ratio of the path length of the optimization trajectory to the Euclidean distance between the initial control field and the resultant optimal control field that takes an observable from the bottom to the top of the landscape. Computational analyses for simple model quantum systems are performed to ascertain the relative abundance of nearly straight control trajectories encountered when optimizing a state-to-state transition probability. The distribution of RRR values is found to be centered near remarkably low values upon sampling large numbers of randomly chosen initial control fields. Additionally, a stochastic algorithm is used to locate many distinct initial control fields, each of which corresponds to the start of an almost straight control trajectory with $R\ensuremath{\simeq}1.0$. The collected results indicate that quantum-control landscapes have very simple structural features. The favorable topology and the complementary simple structure of the control landscape provide a basis for understanding the generally observed ease of optimizing a state-to-state transition probability.