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Algorithms for Quantum Control Based on of Lie Groups

Domenico D’Alessandro

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

Algorithms for the control of quantum mechanical systems are presented which are based on decompositions of Lie groups. The derived control laws drive the state of two and four level systems to any desired final configuration. The algorithms for the two-level case allow the use of an arbitrarily bounded control.

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This chapter discusses in detail the case of two-level systems and the associated Lie group. It gives an example of a general decomposition of SU(n), namely the decomposition into planar rotation, which is quite simple and can be obtained by straightforward computations. This decomposition gives a direct way to parametrize the special unitary group. The chapter presents the basics of Cartan decompositions of semisimple Lie groups. It gives some examples of applications of Lie group decompositions and introduces some notions of Lie group and Lie algebra theory. The chapter discusses Cartan decompositions of Lie groups. Cartan decompositions appear to be ubiquitous in quantum control and dynamics and in quantum information theory.
Constructive algorithms are presented for controlling quantum systems evolving on the SU (1, 1) Lie group. These procedures are performed via structured decomposition of SU (1, 1), which achieve precise controls without any approximations or iterative computations, under the sufficient condition that examines the existence of such decomposition. The technique is applied to controlling transitions between SU (1, 1) coherent states. These results open up new perspectives on the control design of infinite-dimensional quantum systems involving discrete or continuous spectra.
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.