Control of non-controllable quantum systems: A quantum control algorithm based on Grover iteration

A new notion of controllability, eigenstate controllability, is defined for finite-dimensional bilinear quantum mechanical systems which are neither strongly completely controllably nor completely controllable. And a quantum control algorithm based on Grover iteration is designed to perform a quantum control task of steering a system, which is eigenstate controllable but may not be (strongly) completely controllable, from an arbitrary state to a target state.Comment: 7 pages, no figures, submitte

Structural controllability of driftless bilinear control systems

This thesis addresses the structural controllability of driftless bilinear systems with sparse matrices. We begin with a rigorous introduction to the controllability of nonholonomic nonlinear systems. We present the notion of structural controllability and the fact that the controllability of linear systems is a generic property. We give a detailed presentation of the structural controllability of linear systems, based on Lin (1974). Afterwards, we proceed to the analysis of the structural controllability of driftless bilinear systems. We examine two cases; in the first case the matrices of the driftless bilinear system belong to a single vector space of matrices (single pattern case); in the second case the matrices belong to more than one vector spaces (multiple pattern case). After a rigorous presentation of the preliminaries of the theory of Lie algebras, we provide a theorem which claims that in the single pattern case, the driftless bilinear systems with more than two matrices can have a realization consisting of two matrices. This important result extends the theorem of Boothby (1975) about the realization of driftless bilinear systems. We prove that the controllability of driftless bilinear systems in both single and multiple pattern cases is a generic property. We define the notion of the graph which corresponds to a vector space of matrices and we establish necessary and sufficient conditions that relate the connectivity of this graph with the structural controllability of the driftless bilinear system in both cases. For the two patterns case, we provide a theorem which states that driftless bilinear systems with more than four matrices can have a realization with four matrices and we prove that similar propositions can be stated for more than two patterns

Finite Controllability of Infinite-Dimensional Quantum Systems

Quantum phenomena of interest in connection with applications to computation and communication almost always involve generating specific transfers between eigenstates, and their linear superpositions. For some quantum systems, such as spin systems, the quantum evolution equation (the Schr\"{o}dinger equation) is finite-dimensional and old results on controllability of systems defined on on Lie groups and quotient spaces provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, in an infinite-dimensional setting, controlling the evolution of quantum systems often presents difficulties, both conceptual and technical. In this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues. In particular, we analyze controllability for infinite-dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre-defined subspaces. This result, which we call the Finite Controllability Theorem, provides a set of sufficient conditions for controllability in an infinite-dimensional setting. We consider specific physical systems that are of interest for quantum computing, and provide insights into the types of quantum operations (gates) that may be developed.Comment: This is a much improved version of the paper first submitted to the arxiv in 2006 that has been under review since 2005. A shortened version of this paper has been conditionally accepted for publication in IEEE Transactions in Automatic Control (2009

Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability

The aim of this work is to study the controllability of the bilinear Schr\"odinger equation on compact graphs. In particular, we consider the equation (BSE) $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$, with $\mathscr{G}$ being a compact graph. The Laplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We provide a new technique leading to the global exact controllability of the (BSE) in $D(|\Delta|^{s/2})$ with $s\geq 3$. Afterwards, we introduce the "energetic controllability", a weaker notion of controllability useful when the global exact controllability fails. In conclusion, we develop some applications of the main results involving for instance star graphs

The Controllability of Planar Bilinear Systems

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