The first quantum revolution has dramatically changed our daily lives with technological developments in semi-conductor physics (micro chips), lasers, and lots of other things. As the first quantum revolution has been such a great success, we want of course a second one. While the technological devices of the first revolution rely on special quantum effects, the goal of the second one is to use superposition and entanglement to perform tasks that cannot be done with present technologies.
A lot can be learned about quantum systems by simulating them on classical computer. Here, it is crucial to find computation friendly parametrizations of the relevant degrees of freedom. Tensor networks provide a practical way to do this. Most prominently, matrix product states (MPS) (a.k.a. tensor trains) are the workhorse of DMRG simulations. Much of my research is concerned with extending these methods to allow for local Markovian noise, which can be given in GKSL form.
A central and ambitious goal is the development of a quantum computer allowing to run quantum algorithms. Basic building blocks of a quantum computer are quantum gates, which are simple and fixed quantum processes. These gates need to be implemented with hight accuracy. In order to learn what leading error in a gate implementation are, quantum process tomography is an indispensable tool: Here one fully measures the implemented quantum process, or at least its leading "components".
A quantum simulator is a quantum system that helps to solve a specific problem that is difficult to solve otherwise. Hence, its classical simulation and also its tomographic characterization are difficult. But if one cannot know the outcome, how can one then be sure that it works as one hopes it works? Solving this problem is the goal of certification.