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arXiv:quant-ph/0110030 · International Journal of Modern Physics B

LIMITATIONS ON QUANTUM CONTROL

A. I. Solomon, Sonia G. Schirmer

Published Jun 20, 2002• 12 citations
arXivHTMLPDF

Abstract

In this note we give an introduction to the topic of Quantum Control, explaining what its objectives are, and describing some of its limitations.
DOI
10.1142/s0217979202011858

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Some of the limitations of popular optimal control algorithms such as Lyapunov control, the Krotov method and other iterative algorithms based on functional gradients are considered. Lyapunov control, while elegant and generally effective for ideal systems, struggles for realistic systems. Iterative techniques such as the Krotov method tend to be more effective, but the inclusion of an energy penalty term implies that the resulting solutions are not critical points of the objective functional alone, and do not maximize it. These problems can be avoided by using functional gradients without penalty terms, but even these techniques fail for some problems.
Quantum information is an emerging field which has attracted a lot of attention in the last fifteen years or so. It is a broad subject which covers from the most applied questions (e.g., how to build quantum computers or secure cryptographic systems), to the most theoretical problems concerning the formalism of quantum mechanics, its complexity, and its potential to go beyond classical physics. I have written these notes as an introduction to the formalism of quantum information, trying to show how it is a natural progression from an undergraduate course on quantum mechanics (just as I felt it was for me). Hence, these are mainly notes about the language of quantum information and the meaning of its terms, not about its applications. They do not intend to be an exhaustive review about the field, but just a logical sequence of ideas, concepts and definitions, building up towards the continuous variables formalism, which actually takes over most of the notes. In this way, most of the results are stated without proof, and I haven't been careful with references; this is a work in progress, and I intend to add such things in the future. In any case, most of the topics that I introduce have been thoroughly covered in previous books and reviews, and with this notes I only intend to give my point of view, as well as to introduce the different topics and objects in exactly the way I work with and think about them, hoping it will be useful, especially for the people who are starting in the field. These notes originated in 2010 as a way of studying for my close collaboration with Raúl García-Patrón and Nicolas J. Cerf, to whom I am deeply grateful for guidance and incredibly clarifying discussions. They benefited greatly as well from the weeks I spent in Doha with Hyunchul Nha and his group in 2012, especially from discussions with him and Ho-Joon Kim.