2026/07/22 by Nisarga Paul, Haimeng Zhao, David D. Dai
#quant-ph #cond-mat.dis-nn #cond-mat.str-el
Learning compact, interpretable descriptions of quantum many-body states is an important task in quantum science. We study the task of learning the Slater determinant with maximum fidelity to an arbitrary fermionic many-body state, with motivation from both Hartree-Fock methods and agnostic tomography. Given an n-fermion wavefunction built from m fermionic modes, we provide classical and quantum algorithms returning a Slater determinant with fidelity within ε of maximal in time mpoly(n,1/ε). We prove matching hardness lower bounds, assuming standard complexity conjectures, along some parameter axes. Given access to quantum copies, we prove this can be accomplished with poly(m,n,1/ε) copies of ρ. We also show that above a fidelity of 2/3 any stationary point is the unique global maximum while below 2/3 the optimization landscape can have spurious stationary points, and hence 2/3 marks a transition point in the optimization landscape for this problem. We apply the algorithm to the Fermi-Hubbard model, extracting the closest Slater determinant from neural quantum state solutions. Together, our results provide algorithmic tools with provable guarantees in understanding fermionic many-body systems with classical or quantum simulation.