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IEEE Conference on Decision and Control (CDC)

Contributions of control theory to fundamental quantum mechanics and its applications

Domenico D’Alessandro

Published Jan 23, 2004• 3 citations
Publisher

Abstract

In this paper we survey some results in Control Theory that have had a significant impact on quantum mechanics and its applications. We shall consider three specific examples: 1) The concept of controllability, in its various forms, has provided a practical test to verify the feasibility of a number of putative experiments in quantum physics. The verification of controllability involves the study of the Lie algebra structure underlying the system. This study itself has proved a valuable tool to design experiments. 2) Results on the uniform generation of compact Lie groups have been applied to show that almost every set of quantum gates is universal for quantum computation. This result can be further refined by showing that the number of gates necessary to implement quantum logic operations is uniformly bounded by a value which only depends on the number of quantum bits processed. 3) Techniques of Optimal Control Theory have been employed to show that the pulses used to drive particles with spin 1/2 in Nuclear Magnetic Resonance are the optimal pulses which minimize the time of state transfer (subject to an energy constraint) and the energy (subject to a time constraint).

DOI
10.1109/cdc.2002.1185096

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In this paper we survey some results in Control Theory that have had a significant impact on quantum mechanics and its applications. We shall consider three specific examples: 1) The concept of controllability, in its various forms, has provided a practical test to verify the feasibility of a number of putative experiments in quantum physics. The verification of controllability involves the study of the Lie algebra structure underlying the system. This study itself has proved a valuable tool to design experiments. 2) Results on the uniform generation of compact Lie groups have been applied to show that almost every set of quantum gates is universal for quantum computation. This result can be further refined by showing that the number of gates necessary to implement quantum logic operations is uniformly bounded by a value which only depends on the number of quantum bits processed. 3) Techniques of Optimal Control Theory have been employed to show that the pulses used to drive particles with spin 1/2 in Nuclear Magnetic Resonance are the optimal pulses which minimize the time of state transfer (subject to an energy constraint) and the energy (subject to a time constraint).
In this paper, we define four different notions of controllability of physical interest for multilevel quantum mechanical systems. These notions involve the possibility of driving the evolution operator as well as the state of the system. We establish the connections among these different notions as well as methods to verify controllability. The paper also contains results on the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove that, for the system of two interacting spin 1/2 particles, not every state transfer can be obtained in arbitrary small time.
We investigate the problem of control of quantum systems. Various notions of controllability of quantum systems are discussed and Lie algebraic techniques are employed to derive necessary and sufficient conditions for the resulting degrees of controllability. The results are employed to study the degree of controllability for various model systems. The problem of optimal control is formulated as a problem of maximisation of the expectation value of an observable of the system subject to dynamical constraints and costs. A class of iterative algorithms is discussed, and shown to converge to a solution of the Euler-Lagrange equations, which is a necessary condition for optimality of a solution. A numerical realisation of the algorithm is presented and potential implementation of the resulting control fields in the laboratory is discussed. The algorithm is applied to find optimal control fields for a variety of applications including selective and simultaneous excitation of individual quantum dots in a globally addressed ensemble, creation of superposition states, and selective excitation of certain vibrational modes of a simple molecule such as Hydrogen Fluoride. In each case we systematically explore the parameter space of the algorithm to select solutions with desirable physical features from the multitude of possible choices. As there are many possible optimal control fields which give similar yields the question of whether some are more robust than others is investigated. A number of optimal fields are analysed quantitatively in terms of robustness with regard to temporal and spectral noise affecting the fields, uncertainty in the model parameters and the effect of experimental limitations such as restrictions on the bandwidth of the fields. It is shown that whilst the solutions tend to be sensitive to uncertainty in the energy levels chosen for the model, they appear to be surprisingly tolerant to even severe bandwidth limitations.
We study the manipulation of two-level quantum systems. This research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two-level quantum system, a quantum bit. We consider the problem of driving the evolution operator to a desired state, while minimizing the energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. In particular, we prove the 'normality' of the given problem and the 'regularity' of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed.
Abstract One of the main theoretical challenges in quantum computing is the design of explicit schemes that enable one to effectively factorize a given final unitary operator into a product of basic unitary operators. As this is equivalent to a constructive controllability task on a Lie group of special unitary operators, one faces interesting classes of bilinear optimal control problems for which efficient numerical solution algorithms are sought for. In this paper we give a review on recent Lie‐theoretical developments in finite‐dimensional quantum control that play a key role for solving such factorization problems on a compact Lie group. After a brief introduction to basic terms and concepts from quantum mechanics, we address the fundamental control theoretic issues for bilinear control systems and survey standard techniques fromLie theory relevant for quantum control. Questions of controllability, accessibility and time optimal control of spin systems are in the center of our interest. Some remarks on computational aspects are included as well. The idea is to enable the potential reader to understand the problems in clear mathematical terms, to assess the current state of the art and get an overview on recent developments in quantum control‐an emerging interdisciplinary field between physics, control and computation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
In this paper, we define three different notions of controllability for quantum mechanical systems involving the possibility of driving the evolution operator as well as the state of the system. By using general results on transitivity of transformation groups on spheres we establish the connections among these different notions of controllability. Motivated by the physical model of multilevel quantum systems, we also study the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove for the system of two interacting spin 1/2 particles the negative result that not every state transfer can be obtained in arbitrary time.
In this paper, we define three different notions of controllability for quantum mechanical systems involving the possibility of driving the evolution operator as well as the state of the system. By using general results on transitivity of transformation groups on spheres we establish the connections among these different notions of controllability. Motivated by the physical model of multilevel quantum systems, we also study the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove for the system of two interacting spin 1/2 particles the negative result that not every state transfer can be obtained in arbitrary time.