  Exploiting quantum properties to outperform classical ways of
information-processing is an outstanding goal of modern physics. A promising
route is quantum simulation, which aims at implementing relevant and
computationally hard problems in controllable quantum systems. Here we
demonstrate that in a trapped ion setup, with present day technology, it is
possible to realize a spin model of the Mattis type that exhibits spin glass
phases. Remarkably, our method produces the glassy behavior without the need
for any disorder potential, just by controlling the detuning of the spin-phonon
coupling. Applying a transverse field, the system can be used to benchmark
quantum annealing strategies which aim at reaching the ground state of the spin
glass starting from the paramagnetic phase. In the vicinity of a phonon
resonance, the problem maps onto number partitioning, and instances which are
difficult to address classically can be implemented.
