The purpose of this chapter is to revisit the quantum control problems and the solution methodologies discussed in the previous chapters presenting results of numerical experiments. In the following sections, we consider different model problems, discussed in Chapters 2, 3, and 4, and solution strategies based on the discretization schemes discussed in Chapter 5 and on the optimization methods illustrated in Chapter 6. We also provide some comparison of the strategies to provide information about which method is suitable in which context. Discussion on how to choose parameters characterizing the methods provide valuable insights for the practical implementation of quantum optimal control solvers.
The purpose of this chapter is to revisit the quantum control problems and the solution methodologies discussed in the previous chapters presenting results of numerical experiments. In the following sections, we consider different model problems, discussed in Chapters 2, 3, and 4, and solution strategies based on the discretization schemes discussed in Chapter 5 and on the optimization methods illustrated in Chapter 6. We also provide some comparison of the strategies to provide information about which method is suitable in which context. Discussion on how to choose parameters characterizing the methods provide valuable insights for the practical implementation of quantum optimal control solvers.
This chapter is devoted to numerical optimization methods with a focus on solution schemes that are used in the framework of quantum control problems. We first discuss classical optimization schemes that require the computation of reduced gradients and usually provide first-order convergence to a local minimum. These methods are supported by a large set of theoretical results that provide guarantee of convergence to local minima. The second section is devoted to monotonic schemes that originate from the works of Vadim Fedorovitch Krotov. While these methods remain popular in the quantum control application communities, in other frameworks the methods of choice are those discussed in Section 6.1.
In this chapter, we shall discuss some practical applications of quantum control and dynamics. Our goal is not to give an exhaustive introduction to areas of application, but to point out the role of control and dynamics so as to further motivate the field as well as the techniques described in the previous chapters. The basics of mathematical modeling of quantum mechanical control systems were described in Chapter 2. Here we focus on more practical aspects in the areas of nuclear magnetic resonance and control of molecular and atomic systems. We also discuss a possible implementation of quantum information processing, namely the one with trapped ions. The dynamics here is used to implement quantum logic gates. We shall show how, changing the physical set-up, it is possible to create different Hamiltonians. The control is obtained by switching among these different Hamiltonians.
Optimal Control Theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This review aims at providing an introduction to key concepts of optimal control theory which is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This review describes modern aspects of optimal control theory, with a particular focus on the Pontryagin Maximum Principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The review covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.
Optimization is a crucial challenge across various domains, including finance, resource allocation, and mobility. Quantum computing has the potential to redefine the way we handle complex problems by reducing computational complexity and enhancing solution quality. Optimization, particularly of objective functions, stands to benefit significantly from quantum solvers, which leverage principles of quantum mechanics like superposition, entanglement, and tunneling. The Ising and Quadratic Unconstrained Binary Optimization (QUBO) models are the most suitable formulations for these solvers, involving binary variables and constraints treated as penalties within the overall objective function. To harness quantum approaches for optimization, two primary strategies are employed: exploiting quantum annealers—special-purpose optimization devices—and designing algorithms based on quantum circuits. This review provides a comprehensive overview of quantum optimization methods, examining their advantages, challenges, and limitations. It demonstrates their application to real-world scenarios and outlines the steps to convert generic optimization problems into quantum-compliant models. Lastly, it discusses available tools and frameworks that facilitate the exploration of quantum solutions for optimization tasks.
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.