Simulating quantum mechanical systems is a classical hard problem
because the computational difficulties hinges on the exponential growth
of the size of Hilbert space with the number of particles in the system.
In the context of quantum information processing, this difficulty
becomes the main source of power: in some situations, information
processors based in quantum mechanics can process information
exponentially faster than classical systems. From the perspective of a
physicist, one of the most interesting applications of this type of
information processing is the simulation of quantum systems. We call a
quantum information processor that simulates other quantum systems a
quantum simulator.

Using a kind of nuclear magnetic resonance simulator, we implement the
simulations of the Heisenberg spin models by the use of average
Hamiltonian theory and observe the quantum phase transitions by using
different measurements, e.g., entanglement, fidelity decay and geometric
phase: the qualitative changes that the ground states of some quantum
mechanical systems exhibit when some parameters in their Hamiltonians
change through some critical points. In particular, we consider the
effect of the many-body interactions. Depending on the type and strength
of interactions, the ground states can be product states or they can be
maximally entangled states representing different types of entanglement.
When the many-body interaction (such as the three-body interaction)
takes part in the competition, new critical phenomena that cannot be
detected by the traditional two-spin correlation functions will occur.
By quantifying different types of entanglement, or by using suitable
entanglement witnesses, we successfully detect two types of quantum
transitions. Besides this, using such a NMR quantum simulator, we can
also simulate the static properties and dynamics of chemical systems,
such as the ground-state energy of Hydrogen molecule.