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Chapman & Hall/CRC eBooks

More Tools for Quantum Control

Domenico D'Alessandro

Published Aug 3, 2007• 0 citations
Publisher

Abstract

In this chapter, we discuss some more tools for the control of quantum systems. Some of these are commonly considered in the physics and chemistry literature. Some others, like the Lyapunov control method, have been very well studied in the control literature and have then been extended to quantum systems.
DOI
10.1201/9781584888833-11

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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.
Abstract The development of quantum information science has brought great changes to the fields of computer and communication, and deepened people’s understanding of basic physical problems. The core problems of overcoming decoherence, preparing and manipulating quantum states are quantum control problems in essence, so the research on quantum control will attract more and more attention and be important in the development of quantum information. This paper studies several applications of quantum Lyapunov control and quantum feedback control in quantum information processing by using Schrodinger equation, and some principal equations as tools for finite-dimensional systems. In this paper, a brief introduction of quantum control and its applications is presented to review the recent development and its prospect.
In the design of complex quantum systems like ion traps for quantum computing, it is usually desired to stabilize a particular system state or make the system state track a desired trajectory. Several control theoretical approaches based on feedback seem attractive to solve such problems. But the uncertain dynamics introduced by measurement on quantum systems makes the synthesis of feedback control laws very complicated. Although we have not explicitly modeled the change in system dynamics due to measurement (we have assumed weak measurements), this is a first step towards a more detailed analysis and closed-loop feedback design. Here, we present a Lyapunov-based control approach on the lines of that developed by Mirrahimi, Rouchon, Turnici (2005). The states are assumed to be obtained from weak measurements. The Lyapunov control technique has not been applied to realistic quantum systems so far. We have extended and applied the technique to two realistic physical systems - the quantum harmonic oscillator and the n-qubit system. We also propose to extend this concept to ion traps.
In the last two decades there has been an enormous progress in the experimental investigation of single quantum systems. This progress covers fields such as quantum optics, quantum computation, quantum cryptography, and quantum metrology, which are sometimes summarized as `quantum technologies'. A key issue there is entanglement, which can be considered as the characteristic feature of quantum theory. As disparate as these various fields maybe, they all have to deal with a quantum mechanical treatment of the measurement process and, in particular, the control process. Quantum control is, according to the authors, `control for which the design requires knowledge of quantum mechanics'. Quantum control situations in which measurements occur at important steps are called feedback (or feedforward) control of quantum systems and play a central role here. This book presents a comprehensive and accessible treatment of the theoretical tools that are needed to cope with these situations. It also provides the reader with the necessary background information about the experimental developments. The authors are both experts in this field to which they have made significant contributions. After an introduction to quantum measurement theory and a chapter on quantum parameter estimation, the central topic of open quantum systems is treated at some length. This chapter includes a derivation of master equations, the discussion of the Lindblad form, and decoherence – the irreversible emergence of classical properties through interaction with the environment. A separate chapter is devoted to the description of open systems by the method of quantum trajectories. Two chapters then deal with the central topic of quantum feedback control, while the last chapter gives a concise introduction to one of the central applications – quantum information. All sections contain a bunch of exercises which serve as a useful tool in learning the material. Especially helpful are also various separate boxes presenting important background material on topics such as the block representation or the feedback gain-bandwidth relation. The two appendices on quantum mechanics and phase-space and on stochastic differential equations serve the same purpose. As the authors emphasize, the book is aimed at physicists as well as control engineers who are already familiar with quantum mechanics. It takes an operational approach and presents all the material that is needed to follow research on quantum technologies. On the other hand, conceptual issues such as the relevance of the measurement process for the interpretation of quantum theory are neglected. Readers interested in them may wish to consult instead a textbook such as Decoherence and the Quantum-to-Classical Transition by Maximilian Schlosshauer. Although the present book does not contain applications to gravity, part of its content might become relevant for the physics of gravitational-wave detection and quantum gravity phenomenology. In this respect it should be of interest also for the readers of this journal.
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