Grid-based quantum circuit simulation of the hydrogen atom: Coulomb singularity regularization and parameter-selection guidelines

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.