Bounding the costs of quantum simulation of many-body physics in real space

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.