  Quantum simulation uses a well-known quantum system to predict the behavior
of another quantum system. Certain limitations in this technique arise,
however, when applied to specific problems, as we demonstrate with a
theoretical and experimental study of an algorithm to find the low-lying
spectrum of a Hamiltonian. While the number of elementary quantum gates does
scale polynomially with the size of the system, it increases inversely to the
desired error bound $\epsilon$. Making such simulations robust to decoherence
using fault-tolerance constructs requires an additional factor of $1/ \epsilon$
gates. These constraints are illustrated by using a three qubit nuclear
magnetic resonance system to simulate a pairing Hamiltonian, following the
algorithm proposed by Wu, Byrd, and Lidar.
