Quantum Control Landscapes

Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrodinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry, Vol. 26, Iss. 4, pp. 671-735 (2007

A linearized stabilizer formalism for systems of finite dimension

The stabilizer formalism is a scheme, generalizing well-known techniques developed by Gottesman [quant-ph/9705052] in the case of qubits, to efficiently simulate a class of transformations ("stabilizer circuits", which include the quantum Fourier transform and highly entangling operations) on standard basis states of d-dimensional qudits. To determine the state of a simulated system, existing treatments involve the computation of cumulative phase factors which involve quadratic dependencies. We present a simple formalism in which Pauli operators are represented using displacement operators in discrete phase space, expressing the evolution of the state via linear transformations modulo D <= 2d. We thus obtain a simple proof that simulating stabilizer circuits on n qudits, involving any constant number of measurement rounds, is complete for the complexity class coMod_{d}L and may be simulated by O(log(n)^2)-depth boolean circuits for any constant d >= 2.Comment: 25 pages, 3 figures. Reorganized to collect complexity results; some corrections and elaborations of technical results. Differs slightly from the version to be published (fixed typos, changes of wording to accommodate page breaks for a different article format). To appear as QIC vol 13 (2013), pp.73--11

Geometric Analysis of Nonlinear Partial Differential Equations

This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects

Computational Geometric Mechanics and Control of Rigid Bodies.

This dissertation studies the dynamics and optimal control of rigid bodies from two complementary perspectives, by providing theoretical analyses that respect the fundamental geometric characteristics of rigid body dynamics and by developing computational algorithms that preserve those geometric features. This dissertation is focused on developing analytical theory and computational algorithms that are intrinsic and applicable to a wide class of multibody systems. A geometric numerical integrator, referred to as a Lie group variational integrator, is developed for rigid body dynamics. Discrete-time Lagrangian and Hamiltonian mechanics and Lie group methods are unified to obtain a systematic method for constructing numerical integrators that preserve the geometric properties of the dynamics as well as the structure of a Lie group. It is shown that Lie group variational integrators have substantial computational advantages over integrators that preserve either one of none of these properties. This approach is also extended to mechanical systems evolving on the product of two-spheres. A computational geometric approach is developed for optimal control of rigid bodies on a Lie group. An optimal control problem is discretized at the problem formulation stage by using a Lie group variational integrator, and discrete-time necessary conditions for optimality are derived using the calculus of variations. The discrete-time necessary conditions inherit the desirable computational properties of the Lie group variational integrator, as they are derived from a symplectic discrete flow. They do not exhibit the numerical dissipation introduced by conventional numerical integration schemes, and consequently, we can efficiently obtain optimal controls that respect the geometric features of the optimality conditions. The approach that combines computational geometric mechanics and optimal control is illustrated by various examples of rigid body dynamics, which include a rigid body pendulum on a cart, pure bending of an elastic rod, and two rigid bodies connected by a ball joint. Since all of the analytical and computational results developed in this dissertation are coordinate-free, they are independent of a specific choice of local coordinates, and they completely avoid any singularity, ambiguity, and complexity associated with local coordinates. This provides insight into the global dynamics of rigid bodies.Ph.D.Aerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60804/1/tylee_1.pd

Numerical optimal control with applications in aerospace

This thesis explores various computational aspects of solving nonlinear, continuous-time dynamic optimization problems (DOPs) numerically. Firstly, a direct transcription method for solving DOPs is proposed, named the integrated residual method (IRM). Instead of forcing the dynamic constraints to be satisfied only at a selected number of points as in direct collocation, this new approach alternates between minimizing and constraining the squared norm of the dynamic constraint residuals integrated along the whole solution trajectories. The method is capable of obtaining solutions of higher accuracy for the same mesh compared to direct collocation methods, enabling a flexible trade-off between solution accuracy and optimality, and providing reliable solutions for challenging problems, including those with singular arcs and high-index differential-algebraic equations. A number of techniques have also been proposed in this work for efficient numerical solution of large scale and challenging DOPs. A general approach for direct implementation of rate constraints on the discretization mesh is proposed. Unlike conventional approaches that may lead to singular control arcs, the solution of this on-mesh implementation has better numerical properties, while achieving computational speedups. Another development is related to the handling of inactive constraints, which do not contribute to the solution of DOPs, but increase the problem size and burden the numerical computations. A strategy to systematically remove the inactive and redundant constraints under a mesh refinement framework is proposed. The last part of this work focuses on the use of DOPs in aerospace applications, with a number of topics studied. Using example scenarios of intercontinental flights, the benefits of formulating DOPs directly according to problem specifications are demonstrated, with notable savings in fuel usage. The numerical challenges with direct collocation are also identified, with the IRM obtaining solutions of higher accuracy, and at the same time suppressing the singular arc fluctuations.Open Acces

A Series Solution Framework for Finite-time Optimal Feedback Control, H-infinity Control and Games

The Bolza-form of the finite-time constrained optimal control problem leads to the Hamilton-Jacobi-Bellman (HJB) equation with terminal boundary conditions and tobe- determined parameters. In general, it is a formidable task to obtain analytical and/or numerical solutions to the HJB equation. This dissertation presents two novel polynomial expansion methodologies for solving optimal feedback control problems for a class of polynomial nonlinear dynamical systems with terminal constraints. The first approach uses the concept of higher-order series expansion methods. Specifically, the Series Solution Method (SSM) utilizes a polynomial series expansion of the cost-to-go function with time-dependent coefficient gains that operate on the state variables and constraint Lagrange multipliers. A significant accomplishment of the dissertation is that the new approach allows for a systematic procedure to generate optimal feedback control laws that exactly satisfy various types of nonlinear terminal constraints. The second approach, based on modified Galerkin techniques for the solution of terminally constrained optimal control problems, is also developed in this dissertation. Depending on the time-interval, nonlinearity of the system, and the terminal constraints, the accuracy and the domain of convergence of the algorithm can be related to the order of truncation of the functional form of the optimal cost function. In order to limit the order of the expansion and still retain improved midcourse performance, a waypoint scheme is developed. The waypoint scheme has the dual advantages of reducing computational efforts and gain-storage requirements. This is especially true for autonomous systems. To illustrate the theoretical developments, several aerospace application-oriented examples are presented, including a minimum-fuel orbit transfer problem. Finally, the series solution method is applied to the solution of a class of partial differential equations that arise in robust control and differential games. Generally, these problems lead to the Hamilton-Jacobi-Isaacs (HJI) equation. A method is presented that allows this partial differential equation to be solved using the structured series solution approach. A detailed investigation, with several numerical examples, is presented on the Nash and Pareto-optimal nonlinear feedback solutions with a general terminal payoff. Other significant applications are also discussed for one-dimensional problems with control inequality constraints and parametric optimization

Hybrid Modeling of the N-Body Problem with Applications to Astrophysics

Over the last years, the field of hybrid modeling, the concept of combining data-driven machine learning models and numerical solution methods to simulate a physical system, has seen an immense increase in research. This new paradigm within modeling uses its predictor capabilities from neural networks to uncover the unknown physics of the underlying system, and bridges these hidden physics with the strong mathematical foundation of numerical integrators and the governing equations of the physical system. Even though hybrid modeling is being introduced into many different fields of research, one field which so far has lacked a more detailed investigation is the field of n-body problems. This field also represents the class of non-linear systems of O.D.Es. with symplectic structure. The n-body problem has through the ages been a source of countless scientific discoveries and is still of great interest to this day. As an application, this thesis will look at the problem of n-body dynamics of planetary motion, more specifically, the simulation of the main celestial bodies of the Solar System. As the first to create a hybrid model for the n-body problem of more than 3 bodies, this thesis will show, through a series of important observations and modeling approaches, that hybrid modeling of the n-body problem can be achieved. The results will also show that the subsequent model can improve the results of a standard physics-based model, the standardized modeling approach for the n-body problem. To the best of the author’s knowledge, this thesis will also be the first to present a pure data-driven model for predicting the orbital motion of planets in our Solar System.Masteroppgave i matematikkMAB399KMAMN-MA

Essays on Numerical Integration in Hamiltonian Monte Carlo

This thesis considers a variety of topics broadly unified under the theme of geometric integration for Riemannian manifold Hamiltonian Monte Carlo. In chapter 2, we review fundamental topics in numerical computing (section 2.1), classical mechanics (section 2.2), integration on manifolds (section 2.3), Riemannian geometry (section 2.5), stochastic differential equations (section 2.4), information geometry (section 2.6), and Markov chain Monte Carlo (section 2.7). The purpose of these sections is to present the topics discussed in the thesis within a broader context. The subsequent chapters are self-contained to an extent, but contain references back to this foundational material where appropriate. Chapter 3 gives a formal means of conceptualizing the Markov chains corresponding to Riemannian manifold Hamiltonian Monte Carlo and related methods; this formalism is useful for understanding the significance of reversibility and volume-preservation for maintaining detailed balance in Markov chain Monte Carlo. Throughout the remainder of the thesis, we investigate alternative methods of geometric numerical integration for use in Riemannian manifold Hamiltonian Monte Carlo, discuss numerical issues involving violations of reversibility and detailed balance, and propose new algorithms with superior theoretical foundations. In chapter 4, we evaluate the implicit midpoint integrator for Riemannian manifold Hamiltonian Monte Carlo, presenting the first time that this integrator has been deployed and assessed within this context. We discuss attributes of the implicit midpoint integrator that make it preferable, and inferior, to alternative methods of geometric integration such as the generalized leapfrog procedure. In chapter 5, we treat an empirical question as to what extent convergence thresholds play a role in geometric numerical integration in Riemannian manifold Hamiltonian Monte Carlo. If the convergence threshold is too large, then the Markov chain transition kernel will fail to maintain detailed balance, whereas a convergence threshold that is very small will incur computational penalties. We investigate these phenomena and suggest two mechanisms, based on stochastic approximation and higher-order solvers for non-linear equations, which can aid in identifying convergence thresholds or suppress its significance. In chapter 6, we consider a numerical integrator for Markov chain Monte Carlo based on the Lagrangian, rather than Hamiltonian, formalism in classical mechanics. Our contributions include clarifying the order of accuracy of this numerical integrator, which has been misunderstood in the literature, and evaluating a simple change that can accelerate the implementation of the method, but which comes at the cost of producing more serially auto-correlated samples. We also discuss robustness properties of the Lagrangian numerical method that do not materialize in the Hamiltonian setting. Chapter 7 examines theories of geometric ergodicity for Riemannian manifold Hamiltonian Monte Carlo and Lagrangian Monte Carlo, and proposes a simple modification to these Markov chain methods that enables geometric ergodicity to be inherited from the manifold Metropolis-adjusted Langevin algorithm. In chapter 8, we show how to revise an explicit integration using a theory of Lagrange multipliers so that the resulting numerical method satisfies the properties of reversibility and volume-preservation. Supplementary content in chapter E investigates topics in the theory of shadow Hamiltonians of the implicit midpoint method in the case of non-canonical Hamiltonian mechanics and chapter F, which treats the continual adaptation of a parameterized proposal distribution in the independent Metropolis-Hastings sampler