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.
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.
An efficient and accurate computational framework for solving control problems governed by quantum spin systems is presented. Spin systems are extremely important in modern quantum technologies such as nuclear magnetic resonance spectroscopy, quantum imaging and quantum computing. In these applications, two classes of quantum control problems arise: optimal control problems and exact-controllability problems, with a bilinear con- trol structure. These models correspond to the Schrödinger-Pauli equation, describing the time evolution of a spinor, and the Liouville-von Neumann master equation, describing the time evolution of a spinor and a density operator. This thesis focuses on quantum control problems governed by these models. An appropriate definition of the optimiza- tion objectives and of the admissible set of control functions allows to construct controls with specific properties. These properties are in general required by the physics and the technologies involved in quantum control applications. A main purpose of this work is to address non-differentiable quantum control problems. For this reason, a computational framework is developed to address optimal-control prob- lems, with possibly L1 -penalization term in the cost-functional, and exact-controllability problems. In both cases the set of admissible control functions is a subset of a Hilbert space. The bilinear control structure of the quantum model, the L1 -penalization term and the control constraints generate high non-linearities that make difficult to solve and analyse the corresponding control problems. The first part of this thesis focuses on the physical description of the spin of particles and of the magnetic resonance phenomenon. Afterwards, the controlled Schrödinger- Pauli equation and the Liouville-von Neumann master equation are discussed. These equations, like many other controlled quantum models, can be represented by dynamical systems with a bilinear control structure. In the second part of this thesis, theoretical investigations of optimal control problems, with a possible L1 -penalization term in the objective and control constraints, are consid- ered. In particular, existence of solutions, optimality conditions, and regularity properties of the optimal controls are discussed. In order to solve these optimal control problems, semi-smooth Newton methods are developed and proved to be superlinear convergent. The main difficulty in the implementation of a Newton method for optimal control prob- lems comes from the dimension of the Jacobian operator. In a discrete form, the Jacobian is a very large matrix, and this fact makes its construction infeasible from a practical point of view. For this reason, the focus of this work is on inexact Krylov-Newton methods, that combine the Newton method with Krylov iterative solvers for linear systems, and allows to avoid the construction of the discrete Jacobian. In the third part of this thesis, two methodologies for the exact-controllability of quan- tum spin systems are presented. The first method consists of a continuation technique, while the second method is based on a particular reformulation of the exact-control prob- lem. Both these methodologies address minimum L2 -norm exact-controllability problems. In the fourth part, the thesis focuses on the numerical analysis of quantum con- trol problems. In particular, the modified Crank-Nicolson scheme as an adequate time discretization of the Schrödinger equation is discussed, the first-discretize-then-optimize strategy is used to obtain a discrete reduced gradient formula for the differentiable part of the optimization objective, and implementation details and globalization strategies to guarantee an adequate numerical behaviour of semi-smooth Newton methods are treated. In the last part of this work, several numerical experiments are performed to vali- date the theoretical results and demonstrate the ability of the proposed computational framework to solve quantum spin control problems.
The control of matter and energy at a fundamental level will be a cornerstone of new technologies for years to come. This idea is exemplified in a distilled form by controlling the dynamics of quantum mechanical systems via a time-dependent potential. The contributions detailed within this work focus on the computational aspects of formulating and solving quantum control problems efficiently.
The accurate numerical computation of optimal controls of infinite dimensional quantum control problems is a very difficult task that requires to take into account the features of the original infinite dimensional problem. An important issue is the choice of the functional space where the minimization process is defined. A systematic comparison of L2 versus H1 based minimization shows that the choice of the appropriate functional space matters and has many consequences in the implementation of some optimization techniques.
A matrix-free cascadic BFGS algorithm is introduced in the L 2 and H1 settings and it is demonstrated that the choice of H1 over L2 results in a substantial performance and robustness increase. A comparison between optimal control resulting from function space minimization and the control obtained by minimization over Chebyshev and POD basis function coefficients is presented.
A theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nano electronic devices and quantum computing. This method is based on a reduced Hessian Krylov-Newton scheme applied to a norm-preserving discrete model of a dipole quantum control problem. The use of second-order numerical methods for solving the control problem is justified proving existence of optimal solutions and analyzing first- and second-order optimality conditions. Criteria for the discretization of the non-convex optimization problem and for the formulation of the Hessian are given to ensure accurate gradients and a symmetric Hessian. Robustness of the Newton approach is obtained using a globalization strategy with a robust linesearch procedure. Results of numerical experiments demonstrate that the Newton approach presented in this dissertation is able to provide fast and accurate controls for high-energy state transitions.
Control of bound-to-bound and bound-to-continuum transitions in open quantum systems and vector field control of two-dimensional systems is presented. An efficient space-time spectral discretization of the time-dependent Schrodinger equation and preconditioning strategy for a fast approximate solution with Krylov methods is outlined.
The process of obtaining solutions to optimal control problems via mesh based techniques suffers from the well known problem of the curse of dimensionality. This issue is especially severe in quantum systems whose dimensions grow exponentially with the number of interacting elements (qubits) that they contain. In this article we develop a modification of recently introduced curse-of-dimensionality-free max-plus techniques, to the control of quantum systems. This method yields a more manageable growth that is related to the cardinality of the control set. Its efficacy is demonstrated by obtaining an approximate solution to a previously intractable problem on a 2 qubit system.
This thesis is concerned with the control and filtering problems for quantum systems. Two kinds of control problems are considered for a class of quantum systems: optimal control and trajectory tracking control. Filtering of a class of linear quantum systems is also investigated. \n \nFor the optimal control of quantum systems, the system in question evolves on a manifold inR^3 and is modeled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space S O(3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is analyzed in detail. The maximum principle is applied to the optimal control for systems evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. Controlling the evolutions of coherence vector is equivalent to controlling the states of an associated matrix Lie group system. Thus, the optimal control problem for single spin-1/2 systems can be transformed to steer state of the matrix Lie group system from the identity matrix to a final matrix corresponding to a target state of single spin-1/2 systems in an optimal fashion. The obtained result is employed to perform a reversible logic quantum operation NOT on single spin-1/2 systems. Explicit expressions for the optimal control are given which are linked to the initial state of the system. The main contribution of the result is to extend the existing result to a larger class of systems, generalizing existing wave-function-based results to single spin-1/2 systems whose states are represented by density matrices such that it allows the inclusion of mixed states. \n \nFor the tracking control problem of closed single spin-1/2 quantum ensembles, based on the Liouville-von Neumann equation, mathematical model of dynamics for single spin-1/2 quantum ensembles are formulated as a bilinear form with states evolving on Bloch sphere. For a typical setting in NMR, one in general would not cancel the applied electromagnetic field along a specified direction, thus leading to the dynamics in a plane perpendicular to that direction on the Bloch sphere. One natural question arises that how can one design a feedback controller such that given an initial condition for quantum ensembles, the system evolutions can be steered to the pre-determined trajectory. An associated matrix Lie group differential equation describing transfers from the initial state being an identity matrix is naturally introduced. The tracking problem for the system of interest is transformed to a stabilization problem such that the control techniques are able to be used. A distance function as well as a distance vector defined on the Lie group SO(3) are introduced, which can be used to construct a Lyapunov function to deal with the tracking problem. In order to deal with the tracking problem, the Lyapunov-based feedback design method is employed to obtain the feedback control such that the dynamics of system can converge to the reference trajectory. \n \nFor filtering of quantum systems, the filter problem of a class of linear quantum stochastic systems with quantum noise is investigated. The quantum system model is first analyzed, then the necessary conditions of nondemolition are obtained by applying the observation principle. The relations between the input and output noise are necessary for the nondemolition measurement of the quantum output channels. Given prescribed quantum system, the filter of the system is derived using those necessary conditions.
Solutions to optimal control problems can be obtained via grid-based dynamic programming iterations. However, such an approach is prohibitively expensive for problems with higher state dimensions, due to the exponential increase in grid points required in approximating the state space. This issue is especially severe in quantum systems whose dimensions grow exponentially with the number of interacting subsystems therein. In this paper, we study state synchronisation of two coupled qubits as an optimal control problem by tuning the interaction Hamiltonian of the system, whose dynamic programming equation is solved via a min-plus curse-of-dimensionality-free method. An illustration in the context of numerical examples is also provided.
An efficient and robust computational framework for solving closed quantum spin optimal-control and exact-controllability problems with control constraints is presented. Closed spin systems are of fundamental importance in modern quantum technologies such as nuclear magnetic resonance (NMR) spectroscopy, quantum imaging, and quantum computing. These systems are modeled by the Liouville--von Neumann master (LvNM) equation describing the time evolution of the density operator representing the state of the system. A unifying setting is provided to discuss optimal-control and exact-controllability results. Different controllability results for the LvNM model are given, and necessary optimality conditions for the LvNM control problems are analyzed. Existence and regularity of optimal controls are proved. The computational framework is based on matrix-free reduced-Hessian semismooth Krylov--Newton schemes for solving optimal-control problems of the LvNM equation in a real vector space rotating-frame representation. A continuation technique is designed to solve closed spin exact-controllability problems that is based on the solution of an appropriately formulated optimal-control problem. These computational strategies are put into a rigorous theoretical framework, proving convergence to the solutions sought. Results of numerical experiments validate the theoretical results and demonstrate the computational ability of the proposed framework to solve closed quantum spin control problems.