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Hybrid quantum computing combines qubits and qumodes and allows for direct access and manipulation of fermionic and bosonic degrees of freedom, both of which are present in many field theories, including QCD. We collect a set of universal gates for hybrid quantum computing using trapped ions and elaborate on several of their experimental realizations. We demonstrate the utility of this approach by applying it towards the many-body Jaynes Cummings model where we use a variational quantum algorithm (VQA) to find ground states and to time evolve states in the combined Hilbert space.