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IET Control Theory and Applications

Optimal control of single spin‐1/2 quantum systems

Shizhou Fu, Michael Z. Q. Chen

Published Nov 14, 2013• 11 citations
Publisher

Abstract

The purpose of this study is to explore the optimal control problems for a class of single spin‐1/2 quantum ensembles. The system in question evolves on a manifold in ℛ 3 and is modelled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space SO (3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is first analysed in detail. Then, the maximum principle is applied to the optimal control for system evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. As an illustrative example, the authors apply their result to perform a reversible logic quantum operation NOT on single spin‐1/2 system. Explicit expressions for the optimal control are given which are linked to the initial state of the system.

DOI
10.1049/iet-cta.2013.0368

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The purpose of this study is to explore the optimal control problems for a class of single spin‐1/2 quantum ensembles. The system in question evolves on a manifold in ℛ 3 and is modelled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space SO (3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is first analysed in detail. Then, the maximum principle is applied to the optimal control for system evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. As an illustrative example, the authors apply their result to perform a reversible logic quantum operation NOT on single spin‐1/2 system. Explicit expressions for the optimal control are given which are linked to the initial state of the system.
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.
We consider the problem of steering control via an input electro-magnetic field for a system of two interacting spin 1/2 particles. This model is of interest in applications because it is used to perform logic operations in quantum computing that involve two quantum bits. The describing model is a bilinear system whose state varies on the Lie group of special unitary matrices of dimension 4, SU(4). By using decompositions of the latter Lie group, the problem can be decomposed into a number of subproblems for a system whose state varies on the (smaller) Lie group of 4/spl times/4 proper orthogonal matrices, SO(4). We tackle the time optimal control problem for this system and show that the extremals can be computed explicitly and they are the superposition of a constant field and a sinusoidal one.
This paper is concerned with the tracking control problem for the 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. An associated matrix Lie group equation describing transfers from the initial state being an identity matrix is naturally introduced. The tracking problem for the system in question is transformed to a stabilization problem such that the control techniques are able to be used. In order to deal with the tracking problem, the Lyapunov-based feedback design method is employed to obtain the control such that the dynamics of system can converge to the reference trajectory.
Given a compact, connected Lie group G with Lie algebra � . We discuss time-optimal control of bilinear systems of the form (I) where H d , H j ∈ � , U ∈ G , and the v j act as control variables. The case G = SU (2 n ) has found interesting appli- cations to questions of time-optimal control of spin systems. In this context Eq. (I) describes the dynamics of an n -particle system with fixed drift Hamiltonian H d , which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v j . The question of interest here is to transfer the sys- tem from a given initial state U 0 to a prescribed final state U 1 in least possible time. Denote bythe Lie algebra spanned by H 1 , …, H m , and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space G / K we discuss in detail time-optimal control strategies for system (I) in the case where G / K carries the structure of a Riemannian sym- metric space.
We consider the problem of steering control for the systems of one spin 1/2 particle and two interacting spin 1/2 particles in an electromagnetic field. The describing models are bilinear systems whose state varies on the Lie group of special unitary matrices of dimensions two and four, respectively. For both systems, we describe their controllability properties and provide constructive control algorithms. The algorithms based on optimal control theory consider the time of transfer as cost functional. Explicit expressions are derived for the time optimal controls. The algorithms based on Lie group decompositions improve over previously presented algorithms in that they allow an explicit analytic calculation of the parameters involved which does not require any numerics.
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 an energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices of dimension two, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. This includes, in particular, a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed.