  Trapped-ion quantum simulators, in analog and digital modes, are considered a
primary candidate to achieve quantum advantage in quantum simulation and
quantum computation. The underlying controlled ion-laser interactions induce
all-to-all two-spin interactions via the collective modes of motion through
Cirac-Zoller or Molmer-Sorensen schemes, leading to effective two-spin
Hamiltonians, as well as two-qubit entangling gates. In this work, the
Molmer-Sorensen scheme is extended to induce three-spin interactions via
tailored first- and second-order spin-motion couplings. The scheme enables
engineering single-, two-, and three-spin interactions, and can be tuned via an
enhanced protocol to simulate purely three-spin dynamics. Analytical results
for the effective evolution are presented, along with detailed numerical
simulations of the full dynamics to support the accuracy and feasibility of the
proposed scheme for near-term applications. With a focus on quantum simulation,
the advantage of a direct analog implementation of three-spin dynamics is
demonstrated via the example of matter-gauge interactions in the U(1) lattice
gauge theory within the quantum link model. The mapping of degrees of freedom
and strategies for scaling the three-spin scheme to larger systems, are
detailed, along with a discussion of the expected outcome of the simulation of
the quantum link model given realistic fidelities in the upcoming experiments.
The applications of the three-spin scheme go beyond the lattice gauge theory
example studied here and include studies of static and dynamical phase diagrams
of strongly interacting condensed-matter systems modeled by two- and three-spin
Hamiltonians.
