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arXiv:2201.05480

Quantum Control at the Boundary

Balmaseda, Aitor

Published Jan 14, 2022• 1 citation
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Abstract

This dissertation presents and prove the viability of a non-standard method for controlling the state of a quantum system by modifying its boundary conditions instead of relying on the action of external fields. The standard approach to quantum control bases on the use of an external field to manipulate the system. Some technological difficulties appear when controlling a quantum system in this way, due to the complications of manipulating a system made of few particles while maintaining the quantum correlations. As a consequence the systems need to be kept at very low temperatures and the interactions have to be performed very fast. The Quantum Control at the Boundary approach is radically different to the standard one. Instead of seeking the control of the quantum system by directly interacting with it through an external field, the control is achieved by manipulating the boundary conditions of the system. The spectrum of a quantum system, for instance an electron moving in a box, depends on the boundary conditions imposed on it. Hence, a modification of such boundary conditions modifies the state of the system allowing for its manipulation and, eventually, its control. This kind of interaction is weaker, which makes one to expect that it may help maintaining the quantum correlations. For showing the viability of the Quantum Control at the Boundary method, a family of boundary control systems on Quantum Circuits (a generalization of quantum grahs) is introduced. Before being able to address the problem of controllability, the problem of existence of solutions for the Schrödinger equation with time-dependent boundary conditions is addressed. The approximate controllability of the systems under study is proven using a controllability result by T. Chambrion et al. (2009) and a stability result which constitutes another original contribution of this dissertation.

DOI
10.48550/arxiv.2201.05480

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This dissertation presents and prove the viability of a non-standard method for controlling the state of a quantum system by modifying its boundary conditions instead of relying on the action of external fields. The standard approach to quantum control bases on the use of an external field to manipulate the system. Some technological difficulties appear when controlling a quantum system in this way, due to the complications of manipulating a system made of few particles while maintaining the quantum correlations. As a consequence the systems need to be kept at very low temperatures and the interactions have to be performed very fast. The Quantum Control at the Boundary approach is radically different to the standard one. Instead of seeking the control of the quantum system by directly interacting with it through an external field, the control is achieved by manipulating the boundary conditions of the system. The spectrum of a quantum system, for instance an electron moving in a box, depends on the boundary conditions imposed on it. Hence, a modification of such boundary conditions modifies the state of the system allowing for its manipulation and, eventually, its control. This kind of interaction is weaker, which makes one to expect that it may help maintaining the quantum correlations. For showing the viability of the Quantum Control at the Boundary method, a family of boundary control systems on Quantum Circuits (a generalization of quantum grahs) is introduced. Before being able to address the problem of controllability, the problem of existence of solutions for the Schrödinger equation with time-dependent boundary conditions is addressed. The approximate controllability of the systems under study is proven using a controllability result by T. Chambrion et al. (2009) and a stability result which constitutes another original contribution of this dissertation.
Abstract — The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finitedimensional submanifold of the unit sphere in Hilbert space – and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time-independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples. I.
Control in quantum systems is proving to be one of the top challenges in the computing world as scalability of qubits in quantum computers increase. This thesis has focused on understanding quantum control methods and contribute further understanding to progress this very important step in advancing computers. By applying principles from adaptive and Lyapunov-based control to the effect on particle states within a closed quantum system as defined by the Schrödinger equation, it will be possible to see the precise boundaries of creating various control laws as states change over time. The benefits for these methods will be used in trajectory tracking and state targeting as applied to Schrödinger’s time dependent equation and simulated with pre-defined evolving Hamiltonian eigenstates to create reference and error control between target and system states. The analysis of this closed quantum system will be useful in applying similar methods in more tedious, externally influenced, open quantum systems.
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 question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space—and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonian have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples.
The problem of quantum state preparation is one of the main challenges in achieving the quantum advantage. Furthermore, classically, for multi-level problems, our ability to solve the corresponding quantum optimal control problems is rather limited. The ability of the latter to feed into the former may result in significant progress in quantum computing. To address this challenge, we propose a formulation of quantum optimal control that makes use of artificial boundary conditions for the Schrödinger equation in combination with spectral methods. The resulting formulations are well suited for investigating periodic potentials and lend themselves to direct numerical treatment using conventional methods for bounded domains.
Abstract. We provide sufficient conditions for the approximate controllability of infinite-dimensional quantum control systems corresponding to form perturbations of the drift Hamiltonian modulated by a control function. We rely on previous results on controllability of quantum bilinear control systems and obtain a priori [Formula: see text]-bounds of the controls for generic initial and target states. We apply a stability result for the nonautonomous Schrödinger equation to extend the results to systems defined by form perturbations, including singular perturbations. As an application of our results, we prove approximate controllability of a quantum particle in a one-dimensional box with a point-interaction with tuneable strength at the center of the box.