In this talk, an introduction into the general functionality of a quantum computer will be given and the state-of-the-art of the Innsbruck trapped-ion quantum computer is briefly reviewed. We present an overview on the available quantum toolbox and discuss the scalability of the approach. With strings of trapped Ca+ ions, we present both the digital quantum simulation and a hybrid quantum-classical simulation of the Lattice Schwinger model, a gauge theory of 1D quantum electrodynamics. Employing universal quantum computations, we investigate the dynamics of the pair-creation [1] and using a hybrid-classical ansatz, we determine steady-state properties of the Hamiltonian. Hybrid classical-quantum algorithms aim at solving optimization problems variationally, using a feedback loop between a classical computer and a quantum co-processor, while benefitting from quantum resources [2].

[1] E. A. Martinez et al., Nature 534, 516 (2016)
[2] C. Kokail et al., Nature 569, 355–360 (2019)