  Quantum control aims to manipulate quantum systems toward specific quantum
states or desired operations. Designing highly accurate and effective control
steps is vitally important to various quantum applications, including energy
minimization and circuit compilation. In this paper we focus on discrete binary
quantum control problems and apply different optimization algorithms and
techniques to improve computational efficiency and solution quality.
Specifically, we develop a generic model and extend it in several ways. We
introduce a squared $L_2$-penalty function to handle additional side
constraints, to model requirements such as allowing at most one control to be
active. We introduce a total variation (TV) regularizer to reduce the number of
switches in the control. We modify the popular gradient ascent pulse
engineering (GRAPE) algorithm, develop a new alternating direction method of
multipliers (ADMM) algorithm to solve the continuous relaxation of the
penalized model, and then apply rounding techniques to obtain binary control
solutions. We propose a modified trust-region method to further improve the
solutions. Our algorithms can obtain high-quality control results, as
demonstrated by numerical studies on diverse quantum control examples.
