Quantum simulation of real-space dynamics

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"odinger equation with η particles can be simulated with gate complexity tildeObigl(η d F poly(log(g'/ε))bigr), where ε 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 ε and g' from poly(g'/ε) to poly(log(g'/ε)) and polynomially improves the dependence on T and d, while maintaining best known performance with respect to η. For the case of Coulomb interactions, we give an algorithm using η3(d+η)Tpoly(log(η dTg'/(Deltaε)))/Delta one- and two-qubit gates, and another using η3(4d)d/2Tpoly(log(η dTg'/(Deltaε)))/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.