Ruixing Long

Sampling-based learning control of inhomogeneous quantum ensembles

Compensation for parameter dispersion is a significant challenge for control of inhomogeneous quantum ensembles. In this paper, we present the systematic methodology of sampling-based learning control (SLC) for simultaneously steering the members of inhomogeneous quantum ensembles to the same desired state. The SLC method is employed for optimal control of the state-to-state transition probability for inhomogeneous quantum ensembles of spins as well as � -type atomic systems. The procedure involves the steps of (i) training and (ii) testing. In the training step, a generalized system is constructed by sampling members according to the distribution of inhomogeneous parameters drawn from the ensemble. A gradient flow based learning and optimization algorithm is adopted to find an optimal control for the generalized system. In the process of testing, a number of additional ensemble members are randomly selected to evaluate the control performance. Numerical results are presented, showing the effectiveness of the SLC method.

A Motion-Planning Algorithm for the Rolling-Body Problem

In this paper, we consider the control system defined by the rolling of a strictly convex surface S of IR3 on a plane without slipping or spinning. The purpose of this paper is to present the numerical implementation of a constructive planning algorithm for Σ, which is based on a continuation method. The performances of that algorithm, both in robustness and convergence speed, are illustrated through several examples.

Sampling-based learning control for quantum systems with hamiltonian uncertainties

Robust control design for quantum systems has been recognized as a key task in the development of practical quantum technology. In this paper, we present a systematic numerical methodology of sampling-based learning control (SLC) for control design of quantum systems with Hamiltonian uncertainties. The SLC method includes two steps of “training” and “testing and evaluation”. In the training step, an augmented system is constructed by sampling uncertainties according to possible distributions of uncertainty parameters. A gradient flow based learning and optimization algorithm is adopted to find the control for the augmented system. In the process of testing and evaluation, a number of samples obtained through sampling the uncertainties are tested to evaluate the control performance. Numerical results demonstrate the success of the SLC approach. The SLC method has potential applications for robust control design of quantum systems.

Minimal time trajectories for two-level quantum systems with two bounded controls

In this paper we consider the minimum time population transfer problem for a two level quantum system driven by two external fields with bounded amplitude. The controls are modeled as real functions and we do not use the Rotating Wave Approximation. After projection on the Bloch sphere, we treat the time-optimal control problem with techniques of optimal synthesis on 2D manifolds. Based on the Pontryagin Maximum Principle, we characterize a restricted set of candidate optimal trajectories. Properties on this set, crucial for complete optimal synthesis, are illustrated by numerical simulations. Furthermore, when the two controls have the same bound and this bound is small with respect to the difference of the two energy levels, we get a complete optimal synthesis up to a small neighborhood of the antipodal point of the initial condition.

Time minimal trajectories for two-level quantum systems with two bounded controls

In this paper we consider the minimum time population transfer problem for a two level quantum system driven by two external fields with bounded amplitude. After projection on the so-called Bloch sphere, we tackle the problem with well-developed techniques of optimal synthesis on 2-D manifolds. Based on the Pontryagin Maximum Principle, we characterize a restricted set of candidate optimal trajectories. Properties on this set, crucial for a complete optimal synthesis, are illustrated by numerical simulations.

Searching for quantum optimal controls under severe constraints

The success of quantum optimal control for both experimental and theoretical objectives is connected to the topology of the corresponding control landscapes, which are free from local traps if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is of full rank, and (3) there are no constraints on the control field. This paper investigates how the violation of assumption (3) affects gradient searches for globally optimal control fields. The satisfaction of assumptions (1) and (2) ensures that the control landscape lacks fundamental traps, but certain control constraints can still introduce artificial traps. Proper management of these constraints is an issue of great practical importance for numerical simulations as well as optimization in the laboratory. Using optimal control simulations, we show that constraints on quantities such as the number of control variables, the control duration, and the field strength are potentially severe enough to prevent successful optimization of the objective. For each such constraint, we show that exceeding quantifiable limits can prevent gradient searches from reaching a globally optimal solution. These results demonstrate that careful choice of relevant control parameters helps to eliminate artificial traps and facilitate successful optimization.

A motion planning algorithm for the rolling-body problem

In this paper, we consider the control system ¿ defined by the rolling of a strictly convex surface S on a plane without slipping or spinning. It is well known that ¿ is completely controllable. The purpose of this paper is to present the numerical implementation of a constructive planning algorithm for ¿, which is based on a continuation method. The performances of that algorithm, both in robustness and convergence speed, are illustrated through two examples: rolling of a flattened ball and an egg on the plane.

A global steering method for general dynamical nonholonomic systems

In this paper, we extend the globally convergent steering algorithm for regular nonholonomic systems presented in to a much larger class of systems which contain singularities. This extension is based on the construction of a continuous first order approximation of the control system. We also propose an exact motion planning method for nilpotent systems. The method makes use of sinusoidal control laws and generalizes the algorithm presented in for steering chained-form systems. It gives rise to C1 trajectories, then makes easy dynamical extension.

An explicit desingularization procedure for general nonholonomic systems

Consider a control-affine nonholonomic system (Σ).We present in this paper a new desingularization procedure which constructs explicitly from (Σ) a regular control-affine nonholonomic system. The main idea is to add new variables, thus augmenting the dimension of the state space, such that its projection coincides with the original control system (Σ). This construction will allow us to devise a global and fully constructive motion planning method for general driftless and control-affine nonholonomic systems under the sole assumption of the Lie Algebraic Rank Condition.