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The Journal of Chemical Physics

Comment concerning the optimum control of transformations in an unbounded quantum system

Meishan Zhao, Stuart A. Rice

Published Aug 15, 1991• 20 citations
Publisher

Abstract

We discuss the achievability of optimal control of the evolution of an unbounded quantum system by the action of external fields. Previous work has established that the evolution of a quantum system with a nondegenerate discrete and bounded spectrum of states can, in principle, be fully controlled, i.e., that the system wave function can be guided by external fields to approach arbitrarily closely a selected target state wave function. The optimal control of the evolution of a quantum system with a discrete and bounded spectrum of states has also been studied, using a method of analysis that depends on the localized character of bound state wave functions and the fact that the spectrum of states is bounded. In this paper, we examine whether it is possible to control, partially or fully, the evolution of an unbounded quantum system. We show that optimal control of the evolution of an unbounded quantum system is possible, in the sense that it is possible to minimize the difference between the product function formed at time t0 from a localizing function and a continuum wave function and a similarly defined target function at time tf. We have not been able to establish that such optimal control is equivalent to full control, i.e., that the difference between the initial and target functions can be made arbitrarily small, which would require showing that the set of control functions is complete with respect to the function space of the system. Our analysis establishes the existence of external fields that can optimally guide the unbounded system evolution in the absence of other constraints than the Schrödinger equation of motion, but does not provide an algorithm for the construction of such fields. The relationship between optimal control of the evolution of a quantum system and the existence of chaotic dynamics in the continuum domain is briefly discussed. We argue that the optimally controlled system cannot have chaotic dynamics even if the bare system does. As illustrations of the results obtained, we discuss briefly the optimal control of evolution in the subset of discrete states from a spectrum with both discrete and continuum states, the optimal control of evolution in resonant states, and the inversion of the optimally controlled localized product continuum target function to yield the corresponding system wave function.

DOI
10.1063/1.460950

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We discuss the achievability of optimal control of the evolution of an unbounded quantum system by the action of external fields. Previous work has established that the evolution of a quantum system with a nondegenerate discrete and bounded spectrum of states can, in principle, be fully controlled, i.e., that the system wave function can be guided by external fields to approach arbitrarily closely a selected target state wave function. The optimal control of the evolution of a quantum system with a discrete and bounded spectrum of states has also been studied, using a method of analysis that depends on the localized character of bound state wave functions and the fact that the spectrum of states is bounded. In this paper, we examine whether it is possible to control, partially or fully, the evolution of an unbounded quantum system. We show that optimal control of the evolution of an unbounded quantum system is possible, in the sense that it is possible to minimize the difference between the product function formed at time t0 from a localizing function and a continuum wave function and a similarly defined target function at time tf. We have not been able to establish that such optimal control is equivalent to full control, i.e., that the difference between the initial and target functions can be made arbitrarily small, which would require showing that the set of control functions is complete with respect to the function space of the system. Our analysis establishes the existence of external fields that can optimally guide the unbounded system evolution in the absence of other constraints than the Schrödinger equation of motion, but does not provide an algorithm for the construction of such fields. The relationship between optimal control of the evolution of a quantum system and the existence of chaotic dynamics in the continuum domain is briefly discussed. We argue that the optimally controlled system cannot have chaotic dynamics even if the bare system does. As illustrations of the results obtained, we discuss briefly the optimal control of evolution in the subset of discrete states from a spectrum with both discrete and continuum states, the optimal control of evolution in resonant states, and the inversion of the optimally controlled localized product continuum target function to yield the corresponding system wave function.
In this paper we investigate the limits of control for mixed-state quantum systems. The constraint of unitary evolution for nondissipative quantum systems imposes kinematical bounds on the optimization of arbitrary observables. We summarize our previous results on kinematical bounds and show that these bounds are dynamically realizable for completely controllable systems. Moreover, we establish improved bounds for certain partially controllable systems. Finally, the question of dynamical realizability of the bounds for arbitrary partially controllable systems is shown to depend on the accessible sets of the associated control system on the unitary group U(N)U(N)U(N) and the results of a few control computations are discussed briefly.
The control of matter and energy at a fundamental level will be a cornerstone of new technologies for years to come. This idea is exemplified in a distilled form by controlling the dynamics of quantum mechanical systems via a time-dependent potential. The contributions detailed within this work focus on the computational aspects of formulating and solving quantum control problems efficiently. The accurate numerical computation of optimal controls of infinite dimensional quantum control problems is a very difficult task that requires to take into account the features of the original infinite dimensional problem. An important issue is the choice of the functional space where the minimization process is defined. A systematic comparison of L2 versus H1 based minimization shows that the choice of the appropriate functional space matters and has many consequences in the implementation of some optimization techniques. A matrix-free cascadic BFGS algorithm is introduced in the L 2 and H1 settings and it is demonstrated that the choice of H1 over L2 results in a substantial performance and robustness increase. A comparison between optimal control resulting from function space minimization and the control obtained by minimization over Chebyshev and POD basis function coefficients is presented. A theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nano electronic devices and quantum computing. This method is based on a reduced Hessian Krylov-Newton scheme applied to a norm-preserving discrete model of a dipole quantum control problem. The use of second-order numerical methods for solving the control problem is justified proving existence of optimal solutions and analyzing first- and second-order optimality conditions. Criteria for the discretization of the non-convex optimization problem and for the formulation of the Hessian are given to ensure accurate gradients and a symmetric Hessian. Robustness of the Newton approach is obtained using a globalization strategy with a robust linesearch procedure. Results of numerical experiments demonstrate that the Newton approach presented in this dissertation is able to provide fast and accurate controls for high-energy state transitions. Control of bound-to-bound and bound-to-continuum transitions in open quantum systems and vector field control of two-dimensional systems is presented. An efficient space-time spectral discretization of the time-dependent Schrodinger equation and preconditioning strategy for a fast approximate solution with Krylov methods is outlined.
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.
The optimal creation of a targeted unitary transformation WWW is considered under the influence of an external control field. The controlled dynamics produces the unitary transformation UUU and the goal is to seek a control field that minimizes the cost $J=\ensuremath{\parallel}W\ensuremath{-}U\ensuremath{\parallel}$. The optimal control landscape is the cost JJJ as a functional of the control field. For a controllable quantum system with NNN states and without restrictions placed on the controls, the optimal control landscape is shown to have extrema with N+1N+1N+1 possible distinct values, where the desired transformation at U=WU=WU=W is a minimum and the maximum value is at $U=\ensuremath{-}W$. The other distinct $N\ensuremath{-}1$ extrema values of JJJ are saddle points. The results of this analysis have significance for the practical construction of unitary transformations.
This paper analyzes classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open- or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that a closed-loop feedback problem can describe it. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be considered constraints in a non-commutative space, then the momentum must depend on the position of any generic wave function. That implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. By way of completeness, we briefly review control theory and the classical mechanics of constrained systems and analyze some examples at the classical and quantum levels.
Abstract. Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time. Submitted to: New J. Phys. † Author to whom any correspondence should be addressed. ar