  We describe a simple, efficient method for simulating Hamiltonian dynamics on
a quantum computer by approximating the truncated Taylor series of the
evolution operator. Our method can simulate the time evolution of a wide
variety of physical systems. As in another recent algorithm, the cost of our
method depends only logarithmically on the inverse of the desired precision,
which is optimal. However, we simplify the algorithm and its analysis by using
a method for implementing linear combinations of unitary operations to directly
apply the truncated Taylor series.
