Possibility of further improvement in simulation complexity

Determine whether the asymptotic gate-count and depth scalings achieved for Trotter-based quantum simulation of molecular electronic-structure Hamiltonians with localized atomic orbitals can be further improved.

Background

The paper develops a Trotter-based quantum simulation algorithm for molecular electronic-structure Hamiltonians expressed in localized atomic orbitals. By combining orbital localization, a hierarchical decomposition inspired by the Fast Multipole Method, and shallow quantum arithmetic circuits, it achieves sublinear simulation depth and a total gate count of N^{(2d-1)/d+o(1)} for d-dimensional systems, including N^{5/3+o(1)} gates and N^{2/3+o(1)} depth in three dimensions.

After presenting these scalings, the discussion identifies as unresolved whether the resulting asymptotic performance can be improved further. The paper does not specify a particular target exponent or alternative method, so the open problem is to determine whether any improvement over the reported gate-count and depth scalings is possible.