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Journal of Physics B Atomic Molecular and Optical Physics

Topological and statistical properties of quantum control transition landscapes

Michael Hsieh, Re-Bing Wu, Carey M. Rosenthal, HERSCHEL A. RABITZ

Published Mar 26, 2008• 17 citations
Publisher

Abstract

A puzzle arising in the control of quantum dynamics is to explain the relative ease with which high-quality control solutions can be found in the laboratory and in simulations. The emerging explanation appears to lie in the nature of the quantum control landscape, which is an observable as a function of the control variables. This work considers the common case of the observable being the transition probability between an initial and a target state. For any controllable quantum system, this landscape contains only global maxima and minima, and no local extrema traps. The probability distribution function for the landscape value is used to calculate the relative volume of the region of the landscape corresponding to good control solutions. The topology of the global optima of the landscape is analysed and the optima are shown to have inherent robustness to variations in the controls. Although the relative landscape volume of good control solutions is found to shrink rapidly as the system Hilbert space dimension increases, the highly favourable landscape topology at and away from the global optima provides a rationale for understanding the relative ease of finding high-quality, stable quantum optimal control solutions.

DOI
10.1088/0953-4075/41/7/074020

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A puzzle arising in the control of quantum dynamics is to explain the relative ease with which high-quality control solutions can be found in the laboratory and in simulations. The emerging explanation appears to lie in the nature of the quantum control landscape, which is an observable as a function of the control variables. This work considers the common case of the observable being the transition probability between an initial and a target state. For any controllable quantum system, this landscape contains only global maxima and minima, and no local extrema traps. The probability distribution function for the landscape value is used to calculate the relative volume of the region of the landscape corresponding to good control solutions. The topology of the global optima of the landscape is analysed and the optima are shown to have inherent robustness to variations in the controls. Although the relative landscape volume of good control solutions is found to shrink rapidly as the system Hilbert space dimension increases, the highly favourable landscape topology at and away from the global optima provides a rationale for understanding the relative ease of finding high-quality, stable quantum optimal control solutions.
An optimally controlled quantum system possesses a search landscape defined by the physical objective as a functional of the control field. This paper particularly explores the topological structure of quantum mechanical transition probability landscapes. The quantum system is assumed to be controllable and the analysis is based on the Euler-Lagrange variational equations derived from a cost function only requiring extremizing the transition probability. It is shown that the latter variational equations are automatically satisfied as a mathematical identity for control fields that either produce transition probabilities of zero or unit value. Similarly, the variational equations are shown to be inconsistent (i.e., they have no solution) for any control field that produces a transition probability different from either of these two extreme values. An upper bound is shown to exist on the norm of the functional derivative of the transition probability with respect to the control field anywhere over the landscape. The trace of the Hessian, evaluated for a control field producing a transition probability of a unit value, is shown to be bounded from below. Furthermore, the Hessian at a transition probability of unit value is shown to have an extensive null space and only a finite number of negative eigenvalues. Collectively, these findings show that (a) the transition probability landscape extrema consists of values corresponding to no control or full control, (b) approaching full control involves climbing a gentle slope with no false traps in the control space and (c) an inherent degree of robustness exists around any full control solution. Although full controllability may not exist in some applications, the analysis provides a basis to understand the evident ease of finding controls that produce excellent yields in simulations and in the laboratory.
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.
The generation of specific unitary transformations is central to a variety of quantum control problems. Given a target unitary transformation, the optimal control landscape is defined as the Hilbert-Schmidt distance between the target and controlled unitary transformation as a function of the control variables. The critical topology of the landscape is analyzed for controllable quantum systems evolving under unitary dynamics over a finite dimensional Hilbert space. It is found that the critical regions of the landscape corresponding to global optima are isolated points, and the local optima are Grassmannian submanifolds. The volumes of the critical submanifolds corresponding to suboptimal critical values asymptotically vanish in the limit of large Hilbert space dimension. Furthermore, these critical submanifolds have saddle-point topology, which cannot act as traps when searching for optimal controls. These favorable properties of the local optima suggest that the landscape topology is generally amenable to optimization. The analysis is independent of the particular structure of the system Hamiltonian, except for the assumption of full controllability, and the results are universal to the control of unitary transformations of any quantum system.
A broad class of quantum control problems entails optimizing the expectation value of an observable operator through tailored unitary propagation of the system density matrix. Such optimization processes can be viewed as a directed search over a quantum control landscape. The attainment of the global extrema of this landscape is the goal of quantum control. Local optima will generally exist, and their enumeration is shown to scale factorially with the system's effective Hilbert space dimension. A Hessian analysis reveals that these local optima have saddlepoint topology and cannot behave as suboptimal extrema traps. The implications of the landscape topology for practical quantum control efforts are discussed, including in the context of nonideal operating conditions.
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.
A large number of experimental studies and simulations show that it is surprisingly easy to find excellent quality control over broad classes of quantum systems. We now prove that for controllable quantum systems with no constraints placed on the controls, the only allowed extrema of the transition probability landscape correspond to perfect control or no control. Under these conditions, no suboptimal local extrema exist as traps that would impede the search for an optimal control. The identified landscape structure is universal for all controllable quantum systems of the same dimension when seeking to maximize the same transition probability, regardless of the detailed nature of the system Hamiltonian. The presence of weak control field noise or environmental decoherence is shown to preserve the general structure of the control landscape, but at lower resolution.