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 Schrodinger equation with eta particles can be simulated with gate complexity(1) (O) over tilde (eta dF poly(log(g'/epsilon))), 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 poly (g'/epsilon) to 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 poly(log(eta dTg'/(Delta epsilon)))/Delta one- and two-qubit gates, and another using eta(3)(4d)T-d/2 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.