Exploring the top and bottom of the quantum control landscape

Vincent Beltrani

Vincent Beltrani, Jason Dominy, Tak‐San Ho +1 more

A controlled quantum system possesses a search landscape defined by the target physical objective as a function of the controls. This paper focuses on the landscape for the transition probability P(i → f) between the states of a finite level quantum system. Traditionally, the controls are applied fields; here, we extend the notion of control to also include the Hamiltonian structure, in the form of time independent matrix elements. Level sets of controls that produce the same transition probability value are shown to exist at the bottom P(i → f)=0.0 and top P(i → f)=1.0 of the landscape with the field and/or Hamiltonian structure as controls. We present an algorithm to continuously explore these level sets starting from an initial point residing at either extreme value of P(i → f). The technique can also identify control solutions that exhibit the desirable properties of (a) robustness at the top and (b) the ability to rapidly rise towards an optimal control from the bottom. Numerical simulations are presented to illustrate the varied control behavior at the top and bottom of the landscape for several simple model systems.

Vincent Beltrani, Jason Dominy, Tak-San Ho +1 more

The transition probability between the states of a controlled quantum system is a basic physical observable, and the associated control landscape is specified by the transition probability as a function of the applied field. An initial control likely will produce a transition probability near the bottom of the landscape, while the final goal is to find a field that results in a high transition probability value at the top. For controls producing either of the latter extreme landscape values, the Hessian of the transition probability with respect to the control field characterizes the curvature of the landscape and the ease of leaving either limit. Prior work showed that the Hessian spectrum possesses an upper bound on the number of non-zero eigenvalues as well as an infinite number of zero eigenvalues. The associated eigenfunctions accordingly specify the coordinated control field changes that either maximally or minimally influence the transition probability. We show in this paper that there exists a lower bound on the number of non-zero Hessian eigenvalues at either the top or bottom of the landscape. In particular, there is at least one non-zero eigenvalue at the top and generally one at the bottom. Under special circumstances, the Hessian may be identically zero at the bottom (i.e. it possesses no non-zero eigenvalues). These results dictate the curvature of the top and bottom of the landscape, which has important physical significance for seeking optimal control fields. At the top, a field that produces a single non-zero Hessian eigenvalue of small magnitude will generally exhibit a high degree of robustness to field noise. In contrast, at the bottom, working with a field producing the maximum number of non-zero eigenvalues will generally assure the most rapid climb towards a high transition probability.

Arun Nanduri, Ashley Donovan, Tak‐San Ho +1 more

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 $R$ 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.

Michael Hsieh