Optimal control design of NMR and dynamic nuclear polarization experiments using monotonically convergent algorithms

Abstract

Optimal control theory has recently been introduced to nuclear magnetic resonance (NMR) spectroscopy as a means to systematically design and optimize pulse sequences for liquid- and solid-state applications. This has so far primarily involved numerical optimization using gradient-based methods, which allow for the optimization of a large number of pulse sequence parameters in a concerted way to maximize the efficiency of transfer between given spin states or shape the nuclear spin Hamiltonian to a particular form, both within a given period of time. Using such tools, a variety of new pulse sequences with improved performance have been developed, and the NMR spin engineers have been challenged to consider alternative routes for analytical experiment design to meet similar performance. In addition, it has lead to increasing demands to the numerical procedures used in the optimization process in terms of computational speed and fast convergence. With the latter aspect in mind, here we introduce an alternative approach to numerical experiment design based on the Krotov formulation of optimal control theory. For practical reasons, the overall radio frequency power delivered to the sample should be minimized to facilitate experimental implementation and avoid excessive sample heating. The presented algorithm makes explicit use of this requirement and iteratively solves the stationary conditions making sure that the maximum of the objective is reached. It is shown that this method is faster per iteration and takes different paths within a control space than gradient-based methods. In the present work, the Krotov approach is demonstrated by the optimization of NMR and dynamic nuclear polarization experiments for various spin systems and using different constraints with respect to radio frequency and microwave power consumption.

Optimal control for non-Markovian open quantum systems

“…The goal of quantum optimal control here is to reach a desired targetQ with maximum objective function J (or fidelity F ) in a certain time t f . The optimal algorithm following the Krotov method [15] is summarized as follows [2,3,5]. (i) Guess an initial control sequence ε 0 (t).…”

“…A somewhat different QOCT approach from the standard gradient optimization methods is the Krotov it- * Corresponding author: goan@phys.ntu.edu.tw erative method [2,5,7,15]. The Krotov method has several appealing advantages [2,5,7] over the gradient methods: (a) monotonic increase of the objective with iteration number, (b) no requirement for a line search, and (c) macrosteps at each iteration.…”

“…The Krotov method has several appealing advantages [2,5,7] over the gradient methods: (a) monotonic increase of the objective with iteration number, (b) no requirement for a line search, and (c) macrosteps at each iteration. A version of the Krotov optimization method has been used recently in Ref.…”

“…Quantum optimal control theory (QOCT) [1][2][3][4][5][6][7] is a powerful tool that provides a variational framework for calculating the optimal shaped pulse to maximize a desired physical objective (or minimize a physical cost function). It has been applied to various open quantum systems or models to obtain control sequences for quantum gate operations [8][9][10][11][12][13].…”

Fast-kick-off monotonically convergent algorithm for searching optimal control fields

“…The existing monotonically convergent approaches include the well-known Krotov method [10,11], Zhu-Rabitz method [12,13], MadayTurinici method [14], and a recently formulated two-point boundary-value quantum control paradigm (TBQCP) method based on the local control theory [15][16][17]. These monotonically convergent methods, especially the TBQCP-based schemes [16], allow for much larger search steps throughout iterations and converge superlinearly, in contrast to the usual gradient-based methods [18,19]. In practice, the resultant optimal control fields calculated using these algorithms often require further frequency constraint [20,21], which could in turn hinder the convergence rate or even the monotonicity.…”