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arXiv:2607.20623

Learning the closest Slater determinant

Nisarga Paul, Haimeng Zhao, David D. Dai

quant-phcond-mat.dis-nncond-mat.str-elSubmitted Jul 22, 2026• Announced Jul 24, 2026• 0 citations
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Abstract

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 $\varepsilon$ of maximal in time $m^{\text{poly}(n,1/\varepsilon)}$. 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 $\text{poly}(m,n,1/\varepsilon)$ 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.

Comments
12 + 14 pages, 9 figures

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We study the problem of finding a (pure) product state with optimal fidelity to an unknown nnn-qubit quantum state ρρρ, given copies of ρρρ. This is a basic instance of a fundamental question in quantum learning: is it possible to efficiently learn a simple approximation to an arbitrary state? We give an algorithm which finds a product state with fidelity ε\varepsilonε-close to optimal, using N=npoly(1/ε)N = n^{\text{poly}(1/\varepsilon)}N=npoly(1/ε) copies of ρρρ and poly(N)\text{poly}(N)poly(N) classical overhead. We further show that estimating the optimal fidelity is NP-hard for error ε=1/poly(n)\varepsilon = 1/\text{poly}(n)ε=1/poly(n), showing that the error dependence cannot be significantly improved. For our algorithm, we build a carefully-defined cover over candidate product states, qubit by qubit, and then demonstrate that extending the cover can be reduced to approximate constrained polynomial optimization. For our proof of hardness, we give a formal reduction from polynomial optimization to finding the closest product state. Together, these results demonstrate a fundamental connection between these two seemingly unrelated questions. Building on our general approach, we also develop more efficient algorithms in three simpler settings: when the optimal fidelity exceeds 5/65/65/6; when we restrict ourselves to a discrete class of product states; and when we are allowed to output a matrix product state.
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