Improved Grid-Based Simulation of Coulombic Dynamics

TL;DR

This paper introduces two correction schemes for grid-based quantum simulations of Coulombic systems, significantly improving accuracy and efficiency, and demonstrating their applicability to classical and quantum computing platforms.

Contribution

The paper presents novel correction methods that enhance grid-based Coulombic dynamics simulations, reducing errors and enabling quantum computing implementations.

Findings

Improved energy accuracy in hydrogenic systems.

Enhanced long-term time fidelity of simulations.

Feasible quantum circuit implementation with reduced errors.

Abstract

Accurate time-dependent quantum dynamics of Coulombic systems on grid-based representations remains computationally demanding due to the singularity of the Coulomb potential, which necessitates extremely fine spatial grids to mitigate discretisation errors. We propose two complementary correction schemes that, under identical resource budgets, consistently outperform the uncorrected counterparts. The first scheme modifies the potential operator to incorporate grid-basis structure into its representation, while the second introduces a corrected initial wavefunction inspired by analytical solutions of softened Coulomb potentials. Applied to hydrogenic systems, these corrections deliver improved energy accuracy and time fidelity across long evolutions. Beyond classical simulations, the proposed framework aligns naturally with quantum computing architectures, where the corrected operators and states can be encoded through truncated Walsh and Fourier series expansions. A resource analysis for the representative 2D hydrogen system yields a circuit depth of $1.5\times10^{8}$ gates over 6,000 Trotter steps. This study thus establishes practical strategies toward high-accuracy Coulombic dynamics on both classical and emerging quantum platforms.