In this thesis, two aspects of control theory, namely controllability and optimal control, are applied to quantum systems. The presented results are based on group theoretical techniques and numerical studies. By Lie-algebraic analysis, the controllability properties of systems with an arbitrary topology are described and related to the symmetries existing in these systems. We find that symmetry precludes full controllability. Our work investigates well-known control systems and gives rules for the design of new systems. Furthermore, theoretical and numerical concepts are instrumental to studying quantum channels: Their capacities are optimised using gradient flows on the unitary group in order to find counterexamples to a long-established additivity conjecture. The last part of this thesis presents and benchmarks a modular optimal control algorithm known as GRAPE. Numerical tests show how the interplay of its modules can be optimised for higher performance, and how the algorithm performs in comparison to a Krotov-type optimal control algorithm. It is found that GRAPE performs particularly well when aiming for high qualities.
In this thesis, two aspects of control theory, namely controllability and optimal control, are applied to quantum systems. The presented results are based on group theoretical techniques and numerical studies. By Lie-algebraic analysis, the controllability properties of systems with an arbitrary topology are described and related to the symmetries existing in these systems. We find that symmetry precludes full controllability. Our work investigates well-known control systems and gives rules for the design of new systems. Furthermore, theoretical and numerical concepts are instrumental to studying quantum channels: Their capacities are optimised using gradient flows on the unitary group in order to find counterexamples to a long-established additivity conjecture. The last part of this thesis presents and benchmarks a modular optimal control algorithm known as GRAPE. Numerical tests show how the interplay of its modules can be optimised for higher performance, and how the algorithm performs in comparison to a Krotov-type optimal control algorithm. It is found that GRAPE performs particularly well when aiming for high qualities.
This thesis is concerned with the control of quantum systems.Given a Hamiltonian model of a quantum system, we are interested in finding controls-typically shaped electromagnetic pulses-that steer the evolution of the system toward a desired target operation.For this we employ a numerical optimisation method known as the GRAPE algorithm.For particular experimental systems, we design control schemes that respect constraints of robustness and addressability, and are within the reach of the experimental hardware.A general procedure is given for specifying a Hamiltonian model of a driven N -level system and converting it to an appropriate rotating frame.This is then applied together with the numerical algorithm to design improved schemes for two different systems, where laser fields manipulate orbital and hyperfine states of Pr 3+ and Rb.The generation of cluster states in Ising-coupled systems is also studied.We find that, in the ideal case, the solution of evolving only under the coupling Hamiltonian is not time-optimal.This surprising result is in contrast to the known cases for unitary gates.For a symmetrised three-qubit example, we provide a geometrical interpretation of this.Numerically optimised control schemes are then developed for a nonideal coupling topology, modelling an experimental configuration of trapped ions.Controls for the implementation of the two-qubit Deutsch and Grover algorithms are designed for a pair of 13 C nuclear spins at a nitrogen vacancy center in diamond.These implementations are robust to experimental errors, and found to be reproduced with high accuracy on a VFG-150 pulse generator.We also consider two-qubit gate synthesis in a system of superconducting qubits coupled by mii xii CONTENTS C.4 Examples of time-optimal gates . . . . . . . .
For paving the way to novel applications in quantum simulation, computation, and technology, increasingly large quantum systems have to be steered with high precision. It is a typical task amenable to numerical optimal control to turn the time course of pulses, i.e., piecewise constant control amplitudes, iteratively into an optimized shape. Here, we present a comparative study of optimal-control algorithms for a wide range of finite-dimensional applications. We focus on the most commonly used algorithms: GRAPE methods which update all controls concurrently, and Krotov-type methods which do so sequentially. Guidelines for their use are given and open research questions are pointed out. Moreover, we introduce a unifying algorithmic framework, DYNAMO (dynamic optimization platform), designed to provide the quantum-technology community with a convenient matlab-based tool set for optimal control. In addition, it gives researchers in optimal-control techniques a framework for benchmarking and comparing newly proposed algorithms with the state of the art. It allows a mix-and-match approach with various types of gradients, update and step-size methods as well as subspace choices. Open-source code including examples is made available at http://qlib.info.
Since no physical system can ever be completely isolated from its environment, the study of open quantum systems is pivotal to reliably and accurately control complex quantum systems. In practice, reliability of the control field needs to be confirmed via certification of the target evolution while accuracy requires the derivation of high-fidelity control schemes in the presence of decoherence. In the first part of this thesis an algebraic framework is presented that allows to determine the minimal requirements on the unique characterisation of arbitrary unitary gates in open quantum systems, independent on the particular physical implementation of the employed quantum device. To this end, a set of theorems is devised that can be used to assess whether a given set of input states on a quantum channel is sufficient to judge whether a desired unitary gate is realised. This allows to determine the minimal input for such a task, which proves to be, quite remarkably, independent of system size. These results allow to elucidate the fundamental limits regarding certification and tomography of open quantum systems. The combination of these insights with state-of-the-art Monte Carlo process certification techniques permits a significant improvement of the scaling when certifying arbitrary unitary gates. This improvement is not only restricted to quantum information devices where the basic information carrier is the qubit but it also extends to systems where the fundamental informational entities can be of arbitary dimensionality, the so-called qudits. The second part of this thesis concerns the impact of these findings from the point of view of Optimal Control Theory (OCT). OCT for quantum systems utilises concepts from engineering such as feedback and optimisation to engineer constructive and destructive interferences in order to steer a physical process in a desired direction. It turns out that the aforementioned mathematical findings allow to deduce novel optimisation functionals that significantly reduce not only the required memory for numerical control algorithms but also the total CPU time required to obtain a certain fidelity for the optimised process. The thesis concludes by discussing two problems of fundamental interest in quantum information processing from the point of view of optimal control the preparation of pure states and the implementation of unitary gates in open quantum systems. For both cases specific physical examples are considered: for the former the vibrational cooling of molecules via optical pumping and for the latter a superconducting phase qudit implementation. In particular, it is illustrated how features of the environment can be exploited to reach the desired targets.
One main task, which is considerably important in many applications in quantum control, is to explore the possibilities of steering a quantum system from an initial state to a target state. This thesis focuses on fundamental control-theoretical issues of quantum dynamics described by the Lindblad-Kossakowski master equation which arises as a bilinear control system on some underlying real vector spaces, e.g controllability aspects and the structure of reachable sets. Based on Lie-algebraic methods from nonlinear control theory, the thesis presents a unified approach to control problems of finite dimensional closed and open quantum systems. In particular, a simplified treatment for controllability of closed quantum systems as well as new accessibility results for open quantum systems are obtained. The main tools to derive the results are the well-known classifications of all matrix Lie groups which act transitively on Grassmann manifolds, and respectively, on real vector spaces without the origin. It is also shown in this thesis that accessibiity of the Lindblad-Kossakowski master equation is a generic property. Moreover, based on the theoretical accessibility results, an algorithm is developed to decide when the Lindblad-Kossakowski master equation is accessible.
This thesis focuses on the optimal control of a class of closed 1 quantum systems. It encompasses and addresses the following themes 1. The theoretical underpinnings of the dynamic programming method for quantum spin systems. 2. The efficient numerical implementation of algorithms to generate optimal control laws. The use of the dynamic programming principle from optimal control theory for the control of quantum systems is introduced, while taking into account the properties of the manifolds on which they evolve. This enables the unified application of dynamic programming to disparate problems which were previously approached using distinct tools. Moreover the approach herein is applicable, in principle, to an arbitrarily large quantum system - thereby overcoming the limitations present in some other commonly used methods in the literature, on the size of the quantum system being analyzed. As the optimal cost function is non-differentiable in several cases of interest, this thesis provides a rigorous interpretation of this cost function in terms of viscosity solutions of the associated Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) over the manifolds of interest. This framework is applied to explore optimal control strategies for drift and drift-less quantum systems that arise in various applications - thus demonstrating the efficacy of a unified approach to address problems on qualitatively different systems. This thesis also investigates, via two distinct avenues, the efficient numerical solution to optimal control problems of interest. Firstly, it brings together the Markov chain ap- proximation methods of Kushner-Dupuis and the triangulation of manifolds introduced by Munkres, in order to numerically solve the optimal control problem on quantum systems via a value iteration approach. Secondly, the exponential growth in computational resources required - termed the curse of dimensionality (COD) -inherent in any spatial discretization approach to solving the HJB PDE is drastically lessened via a new reduced complexity method motivated by the COD free max-plus technique recently developed by McEneaney et al. The dramatic speedup obtained by this approach enables an approximate solution to a previously intractable optimal control problem for a two qubit2 spin system evolving on a 15 dimensional Lie group.