Optimization difficulty and approximation rates for universal-determinant factors

Quantify the difficulty of reconstructing the true nodal structure through the exact smooth bosonic factors in the universal-determinant representation, together with approximation rates as a function of the particle number.

Background

The paper proves that a fixed universal Slater determinant, multiplied by a smooth bosonic factor, is dense in the fermionic H1 space, while at most two such fixed determinants suffice in H2 for three-dimensional systems. This establishes variational completeness at the level of representation but does not address whether practical optimization procedures can discover the required bosonic factors.

The unresolved issue concerns both the optimization challenge of reconstructing the target wave function’s nodal structure and the quantitative convergence or approximation rates as the number of particles increases. These questions are relevant to assessing the practical scalability of generalized Slater–Jastrow neural quantum states beyond their existence-level universality guarantee.