How to learn a quantum Hamiltonian?

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Quantum dynamics of closed quantum systems is governed by Hamiltonians. These Hamiltonians are obviously usually assumed to be known, derived from a basic understanding of the systems at hand. However, basic as this assumption may sound, in various plausible physical situations, the underlying Hamiltonian and the concomitant equations of motion are not precisely known. This project sets out to comprehensively explore key questions that have received surprisingly little attention in the past: How can a Hamiltonian be certified, learned and recovered from data? Issues of data-driven recovery have for good reasons been moving into the focus of attention very recently, not the least due to the advent of quantum simulators that allow for the precise monitoring of Hamiltonian time evolution. They are of core interest at the same time for the foundations of quantum mechanics. After all, in experimental situations, data is ultimately all there is. In this sense, quantum mechanics is formulated the "wrong way round" when assuming perfect knowledge of a Hamiltonian in the first place. This project is organized along novel ideas of Hamiltonian certification, Hamiltonian learning and Hamiltonian tomography, guided by a hierarchy of knowledge to be acquired. In order for the ambitious aims to be accomplished, an interdisciplinary method development program is laid out, building upon ideas of superresolution, convex optimization and relaxation, tensor networks and quantum many-body dynamics. At the end of the project stands a reliable machinery of robust and scalable learning of quantum Hamiltonians from data.