The potential of analog quantum simulators: Tools for economical certification and assessment of their computational power

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Quantum simulators promise to gain an understanding of the physical world at the microlevel that seems unattainable using numerical simulations on classical computers. Specifically promising from a physical perspective are so-called analog simulators, which allow to probe complex quantum dynamics under precisely controlled conditions in the laboratory. Indeed, in recent years, analog simulation has become an extraordinarily active field of research. Still, many conceptual questions are wide open; if the ultimate promise is to be fulfilled, a satisfactory answer to these questions has to be found. This research proposal addresses these conceptual questions. Presumably, quantum simulators are more powerful than classical computers, but at the same time this constitutes a challenge when trying to find out whether a simulation has been correct. So how can one most economically and feasibly certify the correct functioning of quantum devices, employing classical and quantum techniques? Then, equally important, for what tasks is a quantum simulator more powerful than a classical computer? This proposal suggests a concerted programme to address these points. In particular, it introduces novel validation and certification tools, innovating tools of quantum certification using sampling techniques and interactive proof systems. In several ways, it approaches the question of the superior performance of quantum devices over classical machines from a variety of perspectives, ranging from insights into the dynamics of quantum many-body systems out of equilibrium, through novel ideas on quantum sampling problems, to new methods in computational complexity to assess analog simulators in the presence of realistic errors.