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SIAM Journal on Scientific Computing

Quantum Control With Piecewise Constant Control Functions

Ilan Degani, Antonella Zanna, L. Sælen, Raymond Nepstad

Published Jan 1, 2009• 18 citations
Publisher

Abstract

One of the long-standing problems in the field of computational quantum control is to devise algorithms that converge to high yield controls with a desired structure; particularly, controls that can be reproduced by laboratory laser pulse shapers. In this paper we propose an approach that is a considerable step forward toward the solution of this problem. Our treatment employs piecewise constant controls. For these, the concept of “desired structure” is addressed by including appropriate quadratic forms in the performance functionals. Thereafter, correct extremal control equations are derived, whereas the equations appearing in the literature are correct only to first or second order. We illustrate numerically that our approach can give extremely high yield controls with a desired structure. Our results are valid not only in the context of quantum control, but also for similiar bilinear systems arising in different areas. In particular, they are suitable for controlling linear flows with drift on unitary matrix groups.

DOI
10.1137/080729839

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One of the long-standing problems in the field of computational quantum control is to devise algorithms that converge to high yield controls with a desired structure; particularly, controls that can be reproduced by laboratory laser pulse shapers. In this paper we propose an approach that is a considerable step forward toward the solution of this problem. Our treatment employs piecewise constant controls. For these, the concept of “desired structure” is addressed by including appropriate quadratic forms in the performance functionals. Thereafter, correct extremal control equations are derived, whereas the equations appearing in the literature are correct only to first or second order. We illustrate numerically that our approach can give extremely high yield controls with a desired structure. Our results are valid not only in the context of quantum control, but also for similiar bilinear systems arising in different areas. In particular, they are suitable for controlling linear flows with drift on unitary matrix groups.
This paper proposes a new numerical quantum optimal control method with Grover Iteration for two-level quantum systems. We show that our treatment of using the piecewise constant controls with desired control structure is implementable for laboratory laser experiments. And then, the robustness of the algorithm is also considered when parameter perturbations of quantum systems exist. The performance of the system is enhanced using the combination of the quantum optimal control and the Grover Iteration. In particular, the feasibility, effectiveness and priority of the new control scheme are presented by a comparison between our algorithm and the traditional quantum optimal control.
The purpose of this paper is four fold: 1) a brief description of how finite dimensional models arise in quantum control is given; 2) many notions in quantum engineering such as universality of logic gates, hard pulse approximation in NMR spectroscopy, etc., are related to controllability of systems on the unitary groups; 3) recent results of the authors, on constructive controllability, in the class of piecewise constant controls with constraints placed on them, for systems with drift on the unitary groups are sketched; and 4) a new decomposition of SU(2) matrices is provided and used to illustrate how problems caused by periodicity arguments, to address the presence of drift, may be circumvented.
In this article, the investigation of a class of quantum optimal control problems with L1 sparsity cost functionals is presented. The focus is on quantum systems modeled by Schrödinger-type equations with a bilinear control structure as it appears in many applications in nuclear magnetic resonance spectroscopy, quantum imaging, quantum computing, and in chemical and photochemical processes. In these problems, the choice of L1 control spaces promotes sparse optimal control functions that are conveniently produced by laboratory pulse shapers. The characterization of L1 quantum optimal controls and an efficient numerical semi-smooth Newton solution procedure are discussed.
The problem of finding the optimal control in numerical computer simulations of quantum control phenomena is usually addressed through the introduction of monotonically convergent algorithms that are guaranteed to improve the cost functional at each step. A recent extension of these algorithms implements a search for a control with given bounds. Within this context, this paper will present a generalisation of this algorithm to a more general class of local in time constraints. Numerical results that illustrate some of the properties of the algorithms introduced are given. KEY WORDS quantum control, bilinear control, monotonic algorithms, numerical methods, non-linear control 1
Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. We present such robust pulse designs as an optimal control problem of a continuum of bilinear systems with a common control function. We map this control problem of infinite dimension to a problem of polynomial approximation employing tools from geometric control theory. We then adopt this new notion and develop a unified computational method for optimal pulse design using ideas from pseudospectral approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem with spectral accuracy. Furthermore, this is a highly flexible and efficient numerical method that requires low order of discretization and yields inherently smooth solutions. We demonstrate this method by designing effective broadband π/2 and π pulses with reduced rf energy and pulse duration, which show significant sensitivity enhancement at the edge of the spectrum over conventional pulses in 1D and 2D NMR spectroscopy experiments.
This paper considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem of generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [48] for generic NNN-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage duration can be significantly less than the unmodified first stage duration, but at the cost of performing optimization in the class of piecewise constant controls.