A collection of trapped atomic ions represents one of the most attractive platforms for the quantum simulation of interacting spin networks and quantum magnetism. Spin-dependent optical dipole forces applied to an ion crystal create long-range eective spin-spin interactions and allow the simulation of spin Hamiltonians

that possess nontrivial phases and dynamics.

We trap linear chains of 171Yb+ ions in a Paul trap, and constrain the occupation of energy levels to the ground hyperne clock-states, creating a qubit or pseudo-spin 1/2 system. We proceed to implement spin-spin couplings between two ions using the far detuned Mlmer-Srenson scheme and perform adiabatic

quantum simulations of Ising Hamiltonians with long-range couplings. We then demonstrate our ability to control the sign and relative strength of the interaction between three ions. Using this control, we simulate a frustrated triangular lattice, and for the first time establish an experimental connection between frustration and

quantum entanglement. We then scale up our simulation to show phase transitions from paramagnetism to ferromagnetism for nine ions, and to anti-ferromagnetism for sixteen ions.

The experimental work culminates with our most complicated Hamiltonian - a long range anti-ferromagnetic Ising interaction between 10 ions with a biasing axial field.

Theoretical work presented in this thesis shows how the approach to quantum simulation utilized in this thesis can be further extended and improved. It is shown how appropriate design of laser elds can provide for arbitrary multidimensional spin-spin interaction graphs even for the case of a linear spatial array of ions.

This scheme uses currently existing trap technology and is scalable to levels where classical methods of simulation are intractable.