Error-corrected phase estimation averaged over variable grids on a trapped-ion quantum computer: hyperacuity spectra of a CO molecule adsorbed onto $χ$-Fe$_5$C$_2$

Abstract: Quantum phase estimation (QPE) is an underlying technology for extracting the excitation spectra of many-electron systems, yet its practical use on current hardware is hindered by low grid resolution and environmental noises. Here we propose QPE averaged over variable grids (QAVG), a vernier-type approach that combines low-resolution QPE with multiple origin shifts and physically motivated continuous parametrization to reconstruct the spectra accurately. We introduce this approach into an end-to-end workflow for the {\it ab initio}-based model system for a CO molecule adsorbed onto the $χ$-Fe$_5$C$_2$ surface. We perform experiments on Quantinuum H2-2 using both physical QPE circuits and logical QPE circuits encoded in the Steane code with offline bit-flip correction. We demonstrate that QAVG accurately reconstructs the spectra with deviations much smaller than the nominal QPE resolution, even when the noisy histograms are used. The cost landscapes averaged over the shifted grids substantially suppress the local minima arising from the spectral leakage, thereby stabilizing the optimization of trial parameters. These results indicate that QAVG provides a robust route to quantum simulations of correlated spectra toward the era of early-fault-tolerant quantum computers.