Gradient ascent pulse engineering (GRAPE)–a primer on quality functions and pulse shape penalties

Quantum optimal control is a versatile, powerful method to tailor quantum experiments. In NMR, numerical quantum optimal control algorithms such as the gradient ascent pulse engineering (GRAPE) algorithm have sparked many developments in pulse shaping, pushing the limits of pulse selectivity and robustness. To provide an overview of its core principles and the many expansions it has undergone ever since its introduction, the most common quality functions are benchmarked in their ability to guide the spin state transfer for facilitating excitation, inversion, and maximum quantum coherence. Explicit derivations of gradient and Hessian formulas are given, both summarizing what is known from the literature and complementing it. The derivations also give due consideration to the fact that spin states can be represented as density operators or spin operators. Analogously, the most common penalties to guide optimization toward experimentally feasible controls are discussed along with their explicit gradient and Hessian formula derivations. The discussion is complemented by thorough benchmarking for the most commonly used optimization algorithms.