  We present a route towards the quantum simulation of exotic quantum magnetism
in ion traps by exploiting dual relations between different spin models. Our
strategy allows one to start from Hamiltonians that can be realized with
current technology, while properties of an exotic dual model are inferred from
measurements of non-local, string-order-like, operators. The latter can be
achieved from fluorescence, or from certain spectroscopic measurements, both of
which can be combined with finite-size scaling by controlling the number of
ions in the dual quantum simulator. We apply this concept to propose quantum
simulators of frustrated quantum magnets, and Ising models with multi-spin
interactions. We test the validity of the idea by showing numerically that the
predictions of an ideal dual quantum simulator are not qualitatively modified
by relevant perturbations that occur naturally in the trapped-ion scenario.

Quantum Gases;Atomic Physics;Quantum Physics