Control Aspects of Quantum Computing Using Pure and Mixed States

Steering quantum dynamics such that the target states solve classically hard problems is tan- tamount to quantum simulation and computation. And beyond, quantum control is also essential to pave the way to quantum technologies. Here, important control techniques are reviewed and presented in a unified frame covering quantum computational gate synthesis and spectroscopic state transfer alike. We emphasise that it does not matter whether the quantum states of interest are pure or not. While pure states underly the design of quantum circuits, ensemble mixtures of quantum states can be exploited in a more recent class of algorithms: it is illustrated by characterising the Jones polynomial in order to distinguish between different (classes of) knots. Further applications include Josephson elements, cavitiy grids, ion traps, NV-centres in scenarios of closed as well as open quantum systems.