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SIAM Journal on Scientific Computing

Optimal Quantum Control by an Adapted Coordinate Ascent Algorithm

Ilan Degani, Antonella Zanna

Published Jan 1, 2012• 2 citations
Publisher

Abstract

This paper explores an adaption of the coordinate ascent approach to quantum control problems. It was motivated by the observation that several of the existing monotone-converging schemes for quantum control may be viewed as approximations of the well-known coordinate ascent method. Our implementation employs line searches in coordinate directions in control space using only evaluations of the performance functional, without invoking its derivatives. It is based on recasting the performance functional as a local tracking function which gives the „future” quality of a control at each moment in time. Back propagation of a basis of the target space enables these performance functional (i.e., tracking function) evaluations to be done efficiently. The performance functional may include, or not include, regularization terms as needed. Convergence of the resulting algorithm and its relation to previous schemes are discussed, and numerical examples are provided. In our tests, the coordinate ascent algorithm exhibited rapid convergence to high yield controls. Moreover, it is readily adapted to different restrictions on the controls imposed by the physics of the system under study, e.g., controls with spectrum restrictions in laser control problems, and discrete controls on a specified time grid in electronically controlled nanodevices, or in NMR.

DOI
10.1137/11082467x

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This paper explores an adaption of the coordinate ascent approach to quantum control problems. It was motivated by the observation that several of the existing monotone-converging schemes for quantum control may be viewed as approximations of the well-known coordinate ascent method. Our implementation employs line searches in coordinate directions in control space using only evaluations of the performance functional, without invoking its derivatives. It is based on recasting the performance functional as a local tracking function which gives the „future” quality of a control at each moment in time. Back propagation of a basis of the target space enables these performance functional (i.e., tracking function) evaluations to be done efficiently. The performance functional may include, or not include, regularization terms as needed. Convergence of the resulting algorithm and its relation to previous schemes are discussed, and numerical examples are provided. In our tests, the coordinate ascent algorithm exhibited rapid convergence to high yield controls. Moreover, it is readily adapted to different restrictions on the controls imposed by the physics of the system under study, e.g., controls with spectrum restrictions in laser control problems, and discrete controls on a specified time grid in electronically controlled nanodevices, or in NMR.
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.
Numerical simulations of (bilinear) quantum control often rely on either monotonically convergent algorithms or tracking schemes. However, despite their mathematical simplicity, very limited intuitive understanding exists at this time to explain the former type of algorithms. Departing from the usual mathematical formalization, we present in this paper an interpretation of the monotonic algorithms as finite horizon, local in time, tracking schemes. Our purpose is not to present a new class of procedures but rather to introduce the necessary rigorous framework that supports this interpretation. As a by-product we show that at each instant, estimates of the future quality of the current control field are available and used in the optimization. When the target is expressed as reaching a prescribed final state, we also present an intuitive geometrical interpretation as the minimization of the distance between two correlated trajectories: one starting from the given initial state and the other backward in time from the target state. As an illustration, a stochastic monotonic algorithm is introduced. Numerical discretizations of the two procedures are also presented.
A gradient ascent method for optimal quantum control synthesis is presented that employs a gradient derived with respect to the coefficients of a functional basis expansion of the control. Restricting the space of allowable controls to weighted sums of the Slepian sequences efficiently parametrizes the control in terms of bandwidth, resolution, and pulse duration. A bound showing minimum time evolutions scaling with the inverse of the control bandwidth [S. Lloyd and S. Montangero, Phys. Rev. Lett. 113, 010502 (2014)] is recovered and the method is shown to achieve the scaling predicted by the bound on entangling two-qubit quantum gates.
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
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.
Optimal control theory is a powerful tool for solving control problems in quantum mechanics, ranging from the control of chemical reactions to the implementation of gates in a quantum computer. Gradient-based optimization methods are able to find high fidelity controls, but require considerable numerical effort and often yield highly complex solutions. We propose here to employ a two-stage optimization scheme to significantly speed up convergence and achieve simpler controls. The control is initially parametrized using only a few free parameters, such that optimization in this pruned search space can be performed with a simplex method. The result, considered now simply as an arbitrary function on a time grid, is the starting point for further optimization with a gradient-based method that can quickly converge to high fidelities. We illustrate the success of this hybrid technique by optimizing a geometric phase gate for two superconducting transmon qubits coupled with a shared transmission line resonator, showing that a combination of Nelder-Mead simplex and Krotov’s method yields considerably better results than either one of the two methods alone.