Using nuclear magnetic resonance techniques with a solution of chlorof
orm molecules we implement Grover's search algorithm for a system with
four states. By performing a tomographic reconstruction of the densit
y matrix during the computation good agreement is seen between theory
and experiment. This provides the first complete experimental demonstr
ation of loading an initial state into a quantum computer, performing
a computation requiring fewer steps than on a classical computer, and
then reading out the final state.