  The prospect of quantum simulating lattice gauge theories opens exciting
possibilities for understanding fundamental forms of matter. Here, we show that
trapped ions represent a promising platform in this context when simultaneously
exploiting internal pseudo-spins and external phonon vibrations. We illustrate
our ideas with two complementary proposals for simulating lattice-regularized
quantum electrodynamics (QED) in (1+1) space-time dimensions. The first scheme
replaces the gauge fields by local vibrations with a high occupation number. By
numerical finite-size scaling, we demonstrate that this model recovers Wilson's
lattice gauge theory in a controlled way. Its implementation can be scaled up
to tens of ions in an array of micro-traps. The second scheme represents the
gauge fields by spins 1/2, and thus simulates a quantum link model. As we show,
this allows the fermionic matter to be replaced by bosonic degrees of freedom,
permitting small-scale implementations in a linear Paul trap. Both schemes work
on energy scales significantly larger than typical decoherence rates in
experiments, thus enabling the investigation of phenomena such as string
breaking, Coleman's quantum phase transition, and false-vacuum decay. The
underlying ideas of the proposed analog simulation schemes may also be adapted
to other platforms, such as superconducting qubits.

Quantum Physics;Quantum Gases;High Energy Physics - Lattice