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Kongzhi yu juece

Establishment of state space model in quantum systems control

Shuang Cong

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

A model of quantum mechanical systems and its ensemble are established from control theory viewpoint. The state vector of wave function and density matrix of quantum mechanical systems are transformed into evolution matrix state of geometry space. The manipulation problem of an abstract physical concept is changed into a control design problem on a state of geometry space, which is easy to deal with mathematically and suitable for control. (Finally,) five basic problems about quantum sysetms control, which may be solved based on the model, are (presented.)

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•Jan 1, 2015•0 citations
A model of quantum mechanical systems and its ensemble are established from control theory viewpoint. The state vector of wave function and density matrix of quantum mechanical systems are transformed into evolution matrix state of geometry space. The manipulation problem of an abstract physical concept is changed into a control design problem on a state of geometry space, which is easy to deal with mathematically and suitable for control. (Finally,) five basic problems about quantum sysetms control, which may be solved based on the model, are (presented.)
Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of quantum control systems must accord with Schrodinger equations, so it is foremost to obtain Hamiltonian operators of systems. There are corresponding relations between operators of quantum systems and corresponding physical quantities of classical systems, such as momentum, energy and Hamiltonian, so Schrodinger equation models of corresponding quantum control systems via quantization could been obtained from classical control systems, and then establish formal state space models through the suitable transformation from Schrodinger equations for these quantum control systems. This method provides a new kind of path for modeling in quantum control.
Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of quantum control systems must accord with Schrodinger equations, so it is foremost to obtain Hamiltonian operators of systems. There are corresponding relations between operators of quantum systems and corresponding physical quantities of classical systems,such as momentum, energy and Hamiltonian, so Schrodinger equation models of corresponding quantum control systems via quantization could been obtained from classical control systems, and then establish formal state space models through the suitable transformation from Schrodinger equations for these quantum control systems. This method provides a new kind of path for modeling in quantum control.
For quantum state trajectory tracking of density matrix in Liouville equation of quantum systems, with the help of concept in quantum system control, one can apply unitary transformation both to controlled system and free-evolutionary target system such as to change the time-variant and non-stationary target system into a stationary state. Therefore, the quantum state trajectory tracking problem becomes a steering one. State steering control law of the system transformed is designed by means of the Lyapunov stability theorem. Finally, numerical simulation experiments are given for a five-level energy quantum system. The comparison analysis of original system’s trajectory tracking with other method illustrates the advantage in control time of the method proposed.
Purpose This purpose of this paper is to present a new approach of solving quantum control problem. Design/methodology/approach State‐transition‐matrix (STM) is used to solute quantum control problem. Findings The paper obtains the matrix representation for wave function, normalization, and some other quantum operators, and then gives the possible methods to establish the STM model and solve it. Originality/value Based on this, many tools of the widely used control theory can be adopted to denote or analyze the quantum system.
Abstract. The robust manipulation of ensembles of quantum systems exhibiting variations or uncertainties in the system parameters and dynamics has been a long-standing problem in the field of quantum control. Existing methods for addressing quantum ensemble control primarily rely on the use of state-space models that describe the time evolution of quantum states. In this paper, we introduce a distinctive quantum control paradigm by mapping the quantum ensemble control problem to a dual problem defined on a coordinate system established by a moment transform. This transformation introduces a new type of quantization in quantum systems, enabling tractable systems-theoretic analysis and effective optimal control design, which are typically opaque. Subsequently, we develop an iterative computational algorithm to solve the transformed optimal moment control problems. We conduct a convergence analysis for the proposed algorithm and demonstrate its applicability and effectiveness on various challenging pulse design problems in quantum control.
This thesis is concerned with the control and filtering problems for quantum systems. Two kinds of control problems are considered for a class of quantum systems: optimal control and trajectory tracking control. Filtering of a class of linear quantum systems is also investigated. \n \nFor the optimal control of quantum systems, the system in question evolves on a manifold inR^3 and is modeled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space S O(3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is analyzed in detail. The maximum principle is applied to the optimal control for systems evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. Controlling the evolutions of coherence vector is equivalent to controlling the states of an associated matrix Lie group system. Thus, the optimal control problem for single spin-1/2 systems can be transformed to steer state of the matrix Lie group system from the identity matrix to a final matrix corresponding to a target state of single spin-1/2 systems in an optimal fashion. The obtained result is employed to perform a reversible logic quantum operation NOT on single spin-1/2 systems. Explicit expressions for the optimal control are given which are linked to the initial state of the system. The main contribution of the result is to extend the existing result to a larger class of systems, generalizing existing wave-function-based results to single spin-1/2 systems whose states are represented by density matrices such that it allows the inclusion of mixed states. \n \nFor the tracking control problem of closed single spin-1/2 quantum ensembles, based on the Liouville-von Neumann equation, mathematical model of dynamics for single spin-1/2 quantum ensembles are formulated as a bilinear form with states evolving on Bloch sphere. For a typical setting in NMR, one in general would not cancel the applied electromagnetic field along a specified direction, thus leading to the dynamics in a plane perpendicular to that direction on the Bloch sphere. One natural question arises that how can one design a feedback controller such that given an initial condition for quantum ensembles, the system evolutions can be steered to the pre-determined trajectory. An associated matrix Lie group differential equation describing transfers from the initial state being an identity matrix is naturally introduced. The tracking problem for the system of interest is transformed to a stabilization problem such that the control techniques are able to be used. A distance function as well as a distance vector defined on the Lie group SO(3) are introduced, which can be used to construct a Lyapunov function to deal with the tracking problem. In order to deal with the tracking problem, the Lyapunov-based feedback design method is employed to obtain the feedback control such that the dynamics of system can converge to the reference trajectory. \n \nFor filtering of quantum systems, the filter problem of a class of linear quantum stochastic systems with quantum noise is investigated. The quantum system model is first analyzed, then the necessary conditions of nondemolition are obtained by applying the observation principle. The relations between the input and output noise are necessary for the nondemolition measurement of the quantum output channels. Given prescribed quantum system, the filter of the system is derived using those necessary conditions.