  To gain deeper insight into the dynamics of complex quantum systems we need a
quantum leap in computer simulations. We can not translate quantum behaviour
arising with superposition states or entanglement efficiently into the
classical language of conventional computers. The final solution to this
problem is a universal quantum computer [1], suggested in 1982 and envisioned
to become functional within the next decade(s); a shortcut was proposed via
simulating the quantum behaviour of interest in a different quantum system,
where all parameters and interactions can be controlled and the outcome
detected sufficiently well. Here we study the feasibility of a quantum
simulator based on trapped ions [2]. We experimentally simulate the adiabatic
evolution of the smallest non-trivial spin system from the paramagnetic into
the (anti-)ferromagnetic order with a quantum magnetisation for two spins of
98%, controlling and manipulating all relevant parameters of the Hamiltonian
independently via electromagnetic fields. We prove that the observed transition
is not driven by thermal fluctuations, but of quantum mechanical origin, the
source of quantum fluctuations in quantum phase transitions [3]. We observe a
final superposition state of the two degenerate spin configurations for the
ferromagnetic and the antiferromagnetic order, respectively. These correspond
to deterministically entangled states achieved with a fidelity up to 88%. Our
work demonstrates a building block for simulating quantum spin-Hamiltonians
with trapped ions. The method has potential for scaling to a higher number of
coupled spins [2].
