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\"{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

Analyzing the Error Bounds of Multipole-Based Treecodes

The problem of evaluating the potential due to a set of particles is an important and time-consuming one. The development of fast treecodes such as the Barnes-Hut and Fast Multipole Methods for n-body systems has enabled large scale simulations in astrophysics [9, 10, 13] and molecular dynamics [1]. Coupled with efficient parallel processing, these treecodes are capable of yielding several orders of magnitude improvement in performance [6, 14, 15]. In addition, treecodes have applications in the solution of dense linear systems arising from boundary element methods [3, 4, 5, 11, 12]. Using a p-term multipole expansion, the FMM reduces the complexity of a single timestep from O(n 2 ) to O(p 2 n) and Barnes-Hut method reduces it to O(p 2 n log n) for a uniform distribution. In this paper, we analyze the approximations introduced by these methods. We describe an algorithm that reduces the error significantly by selecting the multipole degree appropriately for different ..