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Journal of Physics A Mathematical and Theoretical

A stable toolkit method in quantum control

Mohamed Belhadj, Julien Salomon, Gabriel Turinici

Published Aug 1, 2008• 8 citations
Publisher

Abstract

Recently the 'toolkit' discretization introduced to accelerate the numerical resolution of the time-dependent Schrödinger equation arising in quantum optimal control problems demonstrated good results on a large range of models. However, when coupling this class of methods with the so-called monotonically convergent algorithms, numerical instabilities affect the convergence of the discretized scheme. We present an adaptation of the 'toolkit' method which preserves the monotonicity of the procedure. The theoretical properties of the new algorithm are illustrated by numerical simulations.

DOI
10.1088/1751-8113/41/36/362001

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Recently the 'toolkit' discretization introduced to accelerate the numerical resolution of the time-dependent Schrödinger equation arising in quantum optimal control problems demonstrated good results on a large range of models. However, when coupling this class of methods with the so-called monotonically convergent algorithms, numerical instabilities affect the convergence of the discretized scheme. We present an adaptation of the 'toolkit' method which preserves the monotonicity of the procedure. The theoretical properties of the new algorithm are illustrated by numerical simulations.
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 numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced by Tannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or their unified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. In Maday et al. [Num. Math. (2006) 323–338], a time discretization which preserves the property of monotonicity has been presented. This paper introduces a proof of the convergence of these schemes and some results regarding their rate of convergence.
Numerical propagation of the Schrödinger equation is the bottleneck in many quantum optimal control computations. For a quantum system of N states with an electric-field-dipole interaction, the use of a propagation toolkit introduced in a prior work yields an O(N) reduction in floating-point operations per wave function propagation. A concatenation scheme for the toolkit method is introduced, and a scaling analysis shows a significant additional reduction in computational cost. The method exploits the fact that the same sequences of discretized control field values are often repeated many times in a control simulation. The concatenated toolkit is benchmarked against the standard toolkit in a numerical simulation.
The non-linear optimization method developed by A. Konnov and V. Krotov [Autom. Remote Cont. (Engl. Transl.) 60, 1427 (1999)] has been used previously to extend the capabilities of optimal control theory from the linear to the non-linear Schrödinger equation [S. E. Sklarz and D. J. Tannor, Phys. Rev. A 66, 053619 (2002)]. Here we show that based on the Konnov-Krotov method, monotonically convergent algorithms are obtained for a large class of quantum control problems. It includes, in addition to nonlinear equations of motion, control problems that are characterized by non-unitary time evolution, nonlinear dependencies of the Hamiltonian on the control, time-dependent targets, and optimization functionals that depend to higher than second order on the time-evolving states. We furthermore show that the nonlinear (second order) contribution can be estimated either analytically or numerically, yielding readily applicable optimization algorithms. We demonstrate monotonic convergence for an optimization functional that is an eighth-degree polynomial in the states. For the "standard" quantum control problem of a convex final-time functional, linear equations of motion and linear dependency of the Hamiltonian on the field, the second-order contribution is not required for monotonic convergence but can be used to speed up convergence. We demonstrate this by comparing the performance of first- and second-order algorithms for two examples.
Most of the numerical simulation in quantum (bilinear) control have used one of the monotonically convergent algorithms of Krotov (introduced by Tannor et al.) or of Zhu and Rabitz. However, until now no explicit relationship has been revealed between the two algorithms in order to understand their common properties. Within this framework, we propose in this paper a unified formulation that comprises both algorithms and that extends to a new class of monotonically convergent algorithms. Numerical results show that the newly derived algorithms behave as well as (and sometimes better than) the well-known algorithms cited above.
The control of quantum dynamics is often concerned with finding time-dependent optimal control fields that can take a system from an initial state to a final state to attain the desired value of an observable. This paper presents a general method for formulating monotonically convergent algorithms to iteratively improve control fields. The formulation is based on a two-point boundary-value quantum control paradigm (TBQCP) expressed as a nonlinear integral equation of the first kind arising from dynamical invariant tracking control. TBQCP is shown to be related to various existing techniques, including local control theory, the Krotov method, and optimal control theory. Several accelerated monotonic convergence schemes for iteratively computing control fields are derived based on TBQCP. Numerical simulations are compared with the Krotov method showing that the new TBQCP schemes are efficient and remain monotonically convergent over a wide range of the iteration step parameters and the control pulse lengths, which is attributable to the trap-free character of the transition probability quantum dynamics control landscape.