Limitations of quantum simulation examined by simulating a pairing hamiltonian using nuclear magnetic resonance

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 proposed by Wu, Byrd, and Lidar [Phys. Rev. Lett. 89, 057904 (2002).] to find the low-lying spectrum of a pairing Hamiltonian. While the number of elementary quantum gates required scales 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 requires an additional factor of similar to 1/epsilon gates. These constraints, along with the effects of control errors, are illustrated using a three qubit NMR system.