Quantum simulation of real-space dynamics

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