  We present a quantum algorithm for simulating the dynamics of a
first-quantized Hamiltonian in real space based on the truncated Taylor series
algorithm. We avoid the possibility of singularities by applying various
cutoffs to the system and using a high-order finite difference approximation to
the kinetic energy operator. We find that our algorithm can simulate $\eta$
interacting particles using a number of calculations of the pairwise
interactions that scales, for a fixed spatial grid spacing, as
$\tilde{O}(\eta^2)$, versus the $\tilde{O}(\eta^5)$ time required by previous
methods (assuming the number of orbitals is proportional to $\eta$), and scales
super-polynomially better with the error tolerance than algorithms based on the
Lie-Trotter-Suzuki product formula. Finally, we analyze discretization errors
that arise from the spatial grid and show that under some circumstances these
errors can remove the exponential speedups typically afforded by quantum
simulation.
