  Quantum field theories are the cornerstones of modern physics, providing
relativistic and quantum mechanical descriptions of physical systems at the
most fundamental level. Simulating real-time dynamics within these theories
remains elusive in classical computing. This provides a unique opportunity for
quantum simulators, which hold the promise of revolutionizing our simulation
capabilities. Trapped-ion systems are successful quantum-simulator platforms
for quantum many-body physics and can operate in digital, or gate-based, and
analog modes. Inspired by the progress in proposing and realizing quantum
simulations of a number of relativistic quantum field theories using
trapped-ion systems, and by the hybrid analog-digital proposals for simulating
interacting boson-fermion models, we propose hybrid analog-digital quantum
simulations of selected quantum field theories, taking recent developments to
the next level. On one hand, the semi-digital nature of this proposal offers
more flexibility in engineering generic model interactions compared with a
fully-analog approach. On the other hand, encoding the bosonic fields onto the
phonon degrees of freedom of the trapped-ion system allows a more efficient
usage of simulator resources, and a more natural implementation of intrinsic
quantum operations in such platforms. This opens up new ways for simulating
complex dynamics of e.g., Abelian and non-Abelian gauge theories, by combining
the benefits of digital and analog schemes.
