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IEEE International Workshop on Workload Characterization (WWC)

Geometric quantum control

J. W. Clark, TZYH JONG TARN, Dennis G. Lucarelli

Published Apr 23, 2004• 3 citations
Publisher

Abstract

The property of controllability of quantum systems is explored, and conditions for controllability based on Lie-algebraic properties are enunciated. Due consideration is given to the unbounded character of some of the operators that arise and to the generally noncompact nature of the operator algebras associated with the system. Results for both finite and infinite-dimensional systems are provided; in the latter case the results are not limited to cases having purely discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated through examples, namely the Poschl-Teller potential and the repulsive radial oscillator, which reveal some of the subtleties of the controllability issue. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for its controllability.

DOI
10.1109/phycon.2003.1237008

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The property of controllability of quantum systems is explored, and conditions for controllability based on Lie-algebraic properties are enunciated. Due consideration is given to the unbounded character of some of the operators that arise and to the generally noncompact nature of the operator algebras associated with the system. Results for both finite and infinite-dimensional systems are provided; in the latter case the results are not limited to cases having purely discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated through examples, namely the Poschl-Teller potential and the repulsive radial oscillator, which reveal some of the subtleties of the controllability issue. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for its controllability.
The property of controllability of quantum systems is revisited. In the case of a system having this property, couplings to external agents are available whose adjustment can guide the state to any chosen target on a suitably defined manifold (or arbitrarily close to any such target) at any chosen time. With due consideration to unbounded operators corresponding to physical observables possessing continuous spectra, sufficient conditions for controllability based on Lie-algebraic arguments are obtained. The results are not limited to finite-dimensional systems, nor to infinite-dimensional systems with discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated for the case of the one-dimensional Poschl-Teller potential. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for controllability.
Quantum control over scattering states has potentially important applications in the future quantum technologies. However, a rigorous theoretical study is difficult due to the nonlocal nature of scattering states. This paper introduces a mathematical framework to deal with general quantum control systems that allow noncompact and infinite dimensional controllability Lie algebras, within which the generated infinite dimensional Lie groups are used to represent flows of scattering states. Then we present the concept of smooth controllability on infinite dimensional manifolds, and a criterion for approximate smooth strong controllability. This is a nontrivial extension of the existing results of controllability based on the analysis over finite dimensional vector spaces to analysis over infinite dimensional manifolds. It also opens up many interesting problems for future studies.
The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space—and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonian have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples.
Abstract — The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finitedimensional submanifold of the unit sphere in Hilbert space – and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time-independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples. I.
In this work, we study operator controllability of finite-dimensional closed quantum systems driven by a single coherent control. Two general Lie-algebraic results on controllability of such systems are proved. The first gives sufficient conditions under which controllability of the system is preserved when controlled energy-level shifts are added. The second gives a sufficient basis-element condition under which the dynamical Lie algebra contains su(N)\mathfrak{su}(N)su(N) and the system is therefore operator controllable. These results, together with known controllability conditions, are applied to obtain a complete classification of operator controllability of all three-level quantum systems in terms of their free and interaction Hamiltonian matrix elements. This classification covers all combinations of energy-level and level-spacing degeneracies, two or three allowed transitions, as well as the presence or absence of controlled energy-level shifts, and gives necessary and sufficient conditions for operator controllability in each case.
A quantum system subject to external fields is said to be controllable if these fields can be adjusted to guide the state vector to a desired destination in the state space of the system. Fundamental results on controllability are reviewed against the background of recent ideas and advances in two seemingly disparate endeavours: (i) laser control of chemical reactions and (ii) quantum computation. Using Lie-algebraic methods, sufficient conditions have been derived for global controllability on a finite-dimensional manifold of an infinite-dimensional Hilbert space, in the case that the Hamiltonian and control operators, possibly unbounded, possess a common dense domain of analytic vectors. Some simple examples are presented. A synergism between quantum control and quantum computation is creating a host of exciting new opportunities for both activities. The impact of these developments on computational many-body theory could be profound.