Modeling and Control of Quantum Systems: An Introduction

Abstract

Abstract-The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedbackdesign problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies.

Controlling open quantum systems: tools, achievements, and limitations

“…Rigorous controllability analyses for reduced dynamics have been limited to date to the Markovian case [112][113][114][115][116][117]. In particular, the sets of reachable states were characterized [112], and the Lindblad operators were shown to form a Lie wedge [114].…”

Horizons of cybernetical physics

“…Surveys of control theoretic achievements can be found, for example, in [159][160][161]. Optimal control techniques have also been successfully applied to multi-dimensional nuclear magnetic resonance targeting at improving the sensitivity of these systems [162,163].…”

Flag-based control of quantum purity forn=2systems

“…The Hamiltonian appears only in the transverse equation (9), while the radial component (8) has no explicit Hamiltonian dependence. We are interested in the question of how to steer the transverse component in order to influence the radial.…”

“…One of the key challenges of quantum control is counter-acting the influence of the environment, which causes decoherence (loss of coherence between quantum states) and dissipation (loss of energy) (see [6], [7], [8] and [9] for surveys). If one models an open system as Markovian and time-independent, the dynamics are described by a quantum dynamical semigroup and the Lindblad master equation [10][11] [12].…”