Grid-based quantum circuit simulation of the hydrogen atom: Coulomb singularity regularization and parameter-selection guidelines
4 affiliations
- 11Managed and Platform Services Business Division, Hitachi, Ltd., 292, Yoshida, Totsuka-ku, Yokohama 244-0817, Japan2Institute of Industrial Science, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8505, Japan
- 22Institute of Industrial Science, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8505, Japan3Next Research, Research and Development Group, Hitachi, Ltd., Kokubunji, Tokyo 185-8601, Japan
- 32Institute of Industrial Science, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8505, Japan4Tokyo Metropolitan College of Industrial Technology, 8-17-1, Minami-Senju, Arakawa-ku, Tokyo 116-8523, Japan
- 42Institute of Industrial Science, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8505, Japan
Abstract
A quantum circuit for first-quantization-based Hamiltonian simulation, capable of running on a small-scale quantum computer emulator with fewer than 30 qubits, was constructed and demonstrated for one- and two-dimensional hydrogen-atom models, the simplest settings sufficient to check the proposed circuits without the confounding effects of electron-electron interaction and correlation. For these systems, Uniformly Controlled Rotation Z (UCRZ) circuits were found valuable for constructing a Coulomb potential circuit compact enough to fit on a small-scale emulator. Although this approach does not scale to larger models, we compare it against a conventional arithmetic-circuit implementation to identify the regime in which UCRZ is preferable. To deal with the Coulomb singularity, we propose regularizing it via a geometric average of the potential over a small circular region around the pole, avoiding divergence and minimizing the resulting energy error. To select parameters minimizing error for a given number of qubits, we determine the optimal spatial grid spacing from the trade-off between spatial discretization error and domain truncation error, and determine the time evolution step size not from the formal worst-case Trotter error bound, but from a sampling-theorem-based bound on the wave function’s maximum frequency component, avoiding unnecessary circuit depth without sacrificing accuracy. While the validity of a UCRZ-based circuit structure is restricted to the small-scale regime, the singularity treatment and parameter-selection procedure address the grid discretization itself and apply broadly to grid-based first-quantization simulations.
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BibTeX
@article{takahashi2026gridbased,
title = {{Grid-based quantum circuit simulation of the hydrogen atom: Coulomb singularity regularization and parameter-selection guidelines}},
author = {Hideo Takahashi and Tatsuya Tomaru and Toshiyuki Hirano and Saisei Tahara and Fumitoshi Sato},
journal = {AIP Advances},
year = {2026},
month = oct,
doi = {10.1063/5.0357248},
url = {https://doi.org/10.1063/5.0357248},
note = {CC BY},
}
RIS
TY - JOUR
TI - Grid-based quantum circuit simulation of the hydrogen atom: Coulomb singularity regularization and parameter-selection guidelines
AU - Hideo Takahashi
AU - Tatsuya Tomaru
AU - Toshiyuki Hirano
AU - Saisei Tahara
AU - Fumitoshi Sato
PY - 2026
DA - 2026/10/23
JO - AIP Advances
DO - 10.1063/5.0357248
UR - https://doi.org/10.1063/5.0357248
AB - A quantum circuit for first-quantization-based Hamiltonian simulation, capable of running on a small-scale quantum computer emulator with fewer than 30 qubits, was constructed and demonstrated for one- and two-dimensional hydrogen-atom models, the simplest settings sufficient to check the proposed circuits without the confounding effects of electron-electron interaction and correlation. For these systems, Uniformly Controlled Rotation Z (UCRZ) circuits were found valuable for constructing a Coulomb potential circuit compact enough to fit on a small-scale emulator. Although this approach does not scale to larger models, we compare it against a conventional arithmetic-circuit implementation to identify the regime in which UCRZ is preferable. To deal with the Coulomb singularity, we propose regularizing it via a geometric average of the potential over a small circular region around the pole, avoiding divergence and minimizing the resulting energy error. To select parameters minimizing error for a given number of qubits, we determine the optimal spatial grid spacing from the trade-off between spatial discretization error and domain truncation error, and determine the time evolution step size not from the formal worst-case Trotter error bound, but from a sampling-theorem-based bound on the wave function’s maximum frequency component, avoiding unnecessary circuit depth without sacrificing accuracy. While the validity of a UCRZ-based circuit structure is restricted to the small-scale regime, the singularity treatment and parameter-selection procedure address the grid discretization itself and apply broadly to grid-based first-quantization simulations.
SN - 2158-3226
ER -