Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of the ... American Control Conference/Proceedings of the American Control Conference

Constructive controllability of one and two spin 1/2 particles

Domenico D’Alessandro

Published Jan 1, 2001• 18 citations
Publisher

Abstract

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.

DOI
10.1109/acc.2001.945978

Related papers

Control of one and two homonuclear spins

Domenico D’Alessandro

We consider the problem of steering control for the systems of one spin 1/2 particle and two interacting homonuclear spin 1/2 particles in an electro-magnetic field. The describing models are bilinear systems whose state varies on the Lie group of special unitary matrices of dimensions two and four, respectively. By performing decompositions of Lie groups, taking into account the describing equations at hand, we derive control laws to steer the state of the system to any desired final configuration. Explicit formulas are given for the parameters involved in the control algorithms. Moreover, the proposed algorithms allow for arbitrary bounds on the magnitude of the controls and for some flexibility in the specification of the final time which must be greater than a given value but otherwise arbitrary.

arXiv:quant-ph/0106127•Jun 22, 2001•4 citations

The optimal control problem on SO(4) and its applications to quantum control

Domenico D’Alessandro

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.

IEEE Transactions on Automatic Control•Jan 1, 2002•37 citations

Lie Theory for Quantum Control

Gunther Dirr, Uwe Helmke B

Abstract One of the main theoretical challenges in quantum computing is the design of explicit schemes that enable one to effectively factorize a given final unitary operator into a product of basic unitary operators. As this is equivalent to a constructive controllability task on a Lie group of special unitary operators, one faces interesting classes of bilinear optimal control problems for which efficient numerical solution algorithms are sought for. In this paper we give a review on recent Lie‐theoretical developments in finite‐dimensional quantum control that play a key role for solving such factorization problems on a compact Lie group. After a brief introduction to basic terms and concepts from quantum mechanics, we address the fundamental control theoretic issues for bilinear control systems and survey standard techniques fromLie theory relevant for quantum control. Questions of controllability, accessibility and time optimal control of spin systems are in the center of our interest. Some remarks on computational aspects are included as well. The idea is to enable the potential reader to understand the problems in clear mathematical terms, to assess the current state of the art and get an overview on recent developments in quantum control‐an emerging interdisciplinary field between physics, control and computation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

GAMM-Mitteilungen•Feb 29, 2008•62 citations

Optimal control of single spin‐1/2 quantum systems

Shizhou Fu, Michael Z. Q. Chen

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.

IET Control Theory and Applications•Nov 14, 2013•11 citations

SPACE-TIME EVOLUTION OF WAVE FUNCTIONS; CONTROL OF QUANTUM DYNAMICS WITH THE HELP OF COHERENT PROCESSES

Jessica Swoboda

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.

•Jan 1, 2007•0 citations

Control of Heisenberg spin systems; Lie algebraic decompositions and action-angle variables

Umesh Vaidya, Deanna M. D’Alessandro, Igor Mezić

In this paper, we consider the control of two coupled spin 1/2 particles that interact via Heisenberg interaction. We decompose the problem into a sequence of subproblems on a lower dimensional Lie group whose Lie algebra is the direct sum of su(2) and a two dimensional Abelian subalgebra. The associated lower dimensional control problems amount to controlling a two level system. To this aim, we derive a control algorithm which, from an energy function, calculates a nominal feedback law. The results guarantee convergence of the state from any initial condition to the desired final configuration.

IEEE Conference on Decision and Control (CDC)•Apr 23, 2004•12 citations
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 consider the problem of steering control for the systems of one spin 1/2 particle and two interacting homonuclear spin 1/2 particles in an electro-magnetic field. The describing models are bilinear systems whose state varies on the Lie group of special unitary matrices of dimensions two and four, respectively. By performing decompositions of Lie groups, taking into account the describing equations at hand, we derive control laws to steer the state of the system to any desired final configuration. Explicit formulas are given for the parameters involved in the control algorithms. Moreover, the proposed algorithms allow for arbitrary bounds on the magnitude of the controls and for some flexibility in the specification of the final time which must be greater than a given value but otherwise arbitrary.
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.
Abstract One of the main theoretical challenges in quantum computing is the design of explicit schemes that enable one to effectively factorize a given final unitary operator into a product of basic unitary operators. As this is equivalent to a constructive controllability task on a Lie group of special unitary operators, one faces interesting classes of bilinear optimal control problems for which efficient numerical solution algorithms are sought for. In this paper we give a review on recent Lie‐theoretical developments in finite‐dimensional quantum control that play a key role for solving such factorization problems on a compact Lie group. After a brief introduction to basic terms and concepts from quantum mechanics, we address the fundamental control theoretic issues for bilinear control systems and survey standard techniques fromLie theory relevant for quantum control. Questions of controllability, accessibility and time optimal control of spin systems are in the center of our interest. Some remarks on computational aspects are included as well. The idea is to enable the potential reader to understand the problems in clear mathematical terms, to assess the current state of the art and get an overview on recent developments in quantum control‐an emerging interdisciplinary field between physics, control and computation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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.
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.
In this paper, we consider the control of two coupled spin 1/2 particles that interact via Heisenberg interaction. We decompose the problem into a sequence of subproblems on a lower dimensional Lie group whose Lie algebra is the direct sum of su(2) and a two dimensional Abelian subalgebra. The associated lower dimensional control problems amount to controlling a two level system. To this aim, we derive a control algorithm which, from an energy function, calculates a nominal feedback law. The results guarantee convergence of the state from any initial condition to the desired final configuration.