  Quantum simulation is a prominent application of quantum computers. While
there is extensive previous work on simulating finite-dimensional systems, less
is known about quantum algorithms for real-space dynamics. We conduct a
systematic study of such algorithms. In particular, we show that the dynamics
of a $d$-dimensional Schr\"{o}dinger equation with $\eta$ particles can be
simulated with gate complexity $\tilde{O}\bigl(\eta d F
\text{poly}(\log(g'/\epsilon))\bigr)$, where $\epsilon$ is the discretization
error, $g'$ controls the higher-order derivatives of the wave function, and $F$
measures the time-integrated strength of the potential. Compared to the best
previous results, this exponentially improves the dependence on $\epsilon$ and
$g'$ from $\text{poly}(g'/\epsilon)$ to $\text{poly}(\log(g'/\epsilon))$ and
polynomially improves the dependence on $T$ and $d$, while maintaining best
known performance with respect to $\eta$. For the case of Coulomb interactions,
we give an algorithm using $\eta^{3}(d+\eta)T\text{poly}(\log(\eta
dTg'/(\Delta\epsilon)))/\Delta$ one- and two-qubit gates, and another using
$\eta^{3}(4d)^{d/2}T\text{poly}(\log(\eta dTg'/(\Delta\epsilon)))/\Delta$ one-
and two-qubit gates and QRAM operations, where $T$ is the evolution time and
the parameter $\Delta$ regulates the unbounded Coulomb interaction. We give
applications to several computational problems, including faster real-space
simulation of quantum chemistry, rigorous analysis of discretization error for
simulation of a uniform electron gas, and a quadratic improvement to a quantum
algorithm for escaping saddle points in nonconvex optimization.
