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Improved Grid-Based Simulation of Coulombic Dynamics

Xiaoning Feng, Hans Hon Sang Chan, David P. Tew

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.

Improved Grid-Based Simulation of Coulombic Dynamics

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 gates over 6,000 Trotter steps. This study thus establishes practical strategies toward high-accuracy Coulombic dynamics on both classical and emerging quantum platforms.
Paper Structure (16 sections, 18 equations, 7 figures, 1 table)

This paper contains 16 sections, 18 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Foundational Framework
  3. Grid-Based State Representation
  4. Model Systems
  5. Time Propagation
  6. Extraction of Observables
  7. Correction Schemes
  8. Performance of Correction Schemes
  9. Potential Operator Corrections
  10. Initial Wavefunction Corrections
  11. Quantum Implementation of Corrected Coulombic System Dynamics
  12. Preparation of Corrected Wavefunction
  13. Implementation of Time Propagation
  14. Conclusions
  15. Author Contributions
  16. ...and 1 more sections

Figures (7)

  • Figure 1: Absolute value and real part of autocorrelation functions recorded from simulating the 2-electron quantum ring with different $N_\text{simulation}$. Analytical profiles are plotted with dashed linestyle.
  • Figure 2: Absolute value and real part of autocorrelation functions recorded from simulations of (a) the 2-electron quantum ring and (b) the 2D hydrogen using $V_\text{corrected}$ computed from different $N_\text{correction}$. For both panels, $N_\text{simulation}$ is fixed at 128 per dimension. Analytical profiles and results from uncorrected runs are plotted with dashed and dotted linestyles, respectively.
  • Figure 3: Absolute value of autocorrelation functions recorded from 2D hydrogen simulations initialized with uncorrected and corrected wavefunctions across different $N_\text{simulation}$.
  • Figure 4: Schematic circuit for measuring the real part of autocorrelation functions at a specific time step.
  • Figure 5: Schematic overview of essential components for simulating the 2D hydrogen system.
  • ...and 2 more figures