This thesis is devoted to the theoretical and numerical study of bilinear control systems modelled by ordinary differential equations arising from applications to quantum computing and to parametric resonance. We focus on time optimal transfers between two states, either exact or approximate, subject to pointwise constraints on the controls. By working on specific models such as Schrödinger's equation, the GKSL equation or a classical oscillator, the aim is to investigate both qualitatively and quantitatively the structure of optimal controls. This thesis provides quantitative estimates on the minimal time needed to achieve the best approximation of a logical gate on noisy quantum computers, a numerical proof of concept demonstrating the fast and smooth control of quantum gates using high-dimensional isolated spin-qudits and evidence that parametric resonance is a time optimal way to destabilise a system.
This thesis is devoted to the theoretical and numerical study of bilinear control systems modelled by ordinary differential equations arising from applications to quantum computing and to parametric resonance. We focus on time optimal transfers between two states, either exact or approximate, subject to pointwise constraints on the controls. By working on specific models such as Schrödinger's equation, the GKSL equation or a classical oscillator, the aim is to investigate both qualitatively and quantitatively the structure of optimal controls. This thesis provides quantitative estimates on the minimal time needed to achieve the best approximation of a logical gate on noisy quantum computers, a numerical proof of concept demonstrating the fast and smooth control of quantum gates using high-dimensional isolated spin-qudits and evidence that parametric resonance is a time optimal way to destabilise a system.
Contrôle en temps minimal : applications à l'informatique quantique et à la résonance paramétrique Cette thèse est dédiée à l'étude théorique et numérique de systèmes de contrôle bilinéaires modélisés par des équations différentielles ordinaires issues d'applications à l'informatique quantique et à la résonance paramétrique. On s'intéresse à réaliser en temps minimal le transfert entre deux états, exact ou approché, en tenant compte de contraintes ponctuelles sur les contrôles. En travaillant sur des modèles particuliers tels que l'équation de Schrödinger, de GKSL ou d'un oscillateur classique, l'objectif est d'étudier qualitativement et quantitativement la structure des contrôles optimaux. Cette thèse apporte des estimations sur le temps minimal nécessaire pour réaliser la meilleure approximation d'une porte logique sur un ordinateur quantique bruité, une preuve de concept illustrant le contrôle rapide et lisse de portes quantiques en grande dimension sur des qudits de spin isolés et des preuves que la résonance paramétrique est un moyen temps optimal de déstabiliser un système.
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.
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.
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.
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 article provides a review of recent developments in the formulation and execution of optimal control strategies for the dynamics of quantum systems. A brief introduction to the concept of optimal control, the dynamics of of open quantum systems, and quantum information processing is followed by a presentation of recent developments regarding the two main tasks in this context: state-specific and state-independent optimal control. For the former, we present an extension of conventional theory (Pontryagin's principle) to quantum systems which undergo a non-Markovian time-evolution. Owing to its importance for the realization of quantum information processing, the main body of the review, however, is devoted to state-independent optimal control. Here, we address three different approaches: an approach which treats dissipative effects from the environment in lowest-order perturbation theory, a general method based on the time--evolution superoperator concept, as well as one based on the Kraus representation of the time-evolution superoperator. Applications which illustrate these new methods focus on single and double qubits (quantum gates) whereby the environment is modeled either within the Lindblad equation or a bath of bosons (spin-boson model). While these approaches are widely applicable, we shall focus our attention to solid-state based physical realizations, such as semiconductor- and superconductor-based systems. While an attempt is made to reference relevant and representative work throughout the community, the exposition will focus mainly on work which has emerged from our own group.