Programmable Quantum Simulators with Atoms and Ions Quantum optical systems with cold atoms and ions provides one of the best ways to build controllable quantum many-body systems as quantum computers and quantum simulators. Here we report on recent developments in building, and in particular programming quantum simulators based on trapped ions as intermediate scale quantum devices. Our discussion will focus on hybrid classical-quantum scenarios: here the quantum part is the generation of highly entangled states on the quantum device in quench dynamics, which is combined with a classical post processing of measurement data, possibly run in a feedback loop with the quantum device. Examples highlighting these developments include the implementation of self-verifying variational quantum simulations, illustrated here by computing the ground state and quantum phase transition of a Schwinger Model as 1D QED. In addition, we develop and demonstrate a ‘randomized measurement toolbox’, allowing to access in experiments quantities like Renyi entanglement entropies and – as ongoing research – measure the entanglement spectrum, i.e. ‘seeing the Schmidt decomposition live’ in quench dynamics from an initial product state towards thermodynamic equilibrium.

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