**Efficient quantum algorithm for simulating Hamiltonian evolution**

Feynman's motivated the quantum computer as an efficient quantum simulator for generic quantum systems. We present an efficient quantum algorithm for simulating the evolution of an arbitrary sparse Hamiltonian that is optimal in time (simulates in slightly superlinear time and we prove that simulations cannot be sublinear in time), and the cost of the simulatin is effectively independent of the number of qubits in the system being simulated.