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Trapped-ion quantum simulators, in analog and digital modes, are considered aprimary candidate to achieve quantum advantage in quantum simulation andquantum computation. The underlying controlled ion-laser interactions induceall-to-all two-spin interactions via the collective modes of motion throughCirac-Zoller or Molmer-Sorensen schemes, leading to effective two-spinHamiltonians, as well as two-qubit entangling gates. In this work, theMolmer-Sorensen scheme is extended to induce three-spin interactions viatailored first- and second-order spin-motion couplings. The scheme enablesengineering single-, two-, and three-spin interactions, and can be tuned via anenhanced protocol to simulate purely three-spin dynamics. Analytical resultsfor the effective evolution are presented, along with detailed numericalsimulations of the full dynamics to support the accuracy and feasibility of theproposed scheme for near-term applications. With a focus on quantum simulation,the advantage of a direct analog implementation of three-spin dynamics isdemonstrated via the example of matter-gauge interactions in the U(1) latticegauge theory within the quantum link model. The mapping of degrees of freedomand strategies for scaling the three-spin scheme to larger systems, aredetailed, along with a discussion of the expected outcome of the simulation ofthe quantum link model given realistic fidelities in the upcoming experiments.The applications of the three-spin scheme go beyond the lattice gauge theoryexample studied here and include studies of static and dynamical phase diagramsof strongly interacting condensed-matter systems modeled by two- and three-spinHamiltonians.

Engineering an Effective Three-spin Hamiltonian in Trapped-ion Systems for Applications in Quantum Simulation