In previous chapters we have seen that quantum mechanics permits strong nonlocal correlations which are classically forbidden. It turns out that this makes it very difficult to engineer a classical digital computer to mimic the behaviour of quantum systems-it seems very likely that the general problem is classically intractable. However, we have good reason to believe that a quantum computer should be able to efficiently simulate most quantum systems of interest.In this chapter we provide a proof-of-principle demonstration of a new algorithm for quantum computers that would give precise calculations of chemical energies and configurations in regimes where classical techniques either fail to give good answers or require exponential computing power. We first examine existing methods for the simulation of quantum systems on classical and quantum computers, with particular focus on quantum chemistry. We then describe our algorithm, and discuss its distinguishing features with respect to existing techniques. Finally, we use this algorithm in a two-photon experiment, simulating the Helium Hydride molecule on the CNOT-MZ chip previously described.