Simultaneous approximation of multiple degenerate states using a single neural network quantum state

Abstract Neural network quantum states (NQSs) excel at approximating ground states of quantum many-body systems, but approximating all states of a degenerate manifold is nevertheless computationally expensive. We propose a single-trunk multi-head (ST-MH) NQS ensemble that shares a feature-extracting trunk while attaching lightweight heads for each target state. Using a cost function that also has an orthogonality term, we derive exact analytic gradients and overlap derivatives needed to train ST-MH within standard variational Monte Carlo (VMC) workflows. We prove that ST-MH can represent every degenerate eigenstate exactly whenever the feature map of latent width h , augmented with a constant, has column space containing the linear span of the targets’ log-moduli and (chosen) phase branches together with the constant on the common support where all states are non-vanishing. Under this condition, ST-MH reduces the parameter count and can reduce the leading VMC cost by a factor equal to the degeneracy K relative to other algorithms when K is modest and in trunk dominated regimes. As a numerical proof-of-principle, we validate and benchmark the ST-MH approach on the frustrated spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> </mml:mrow> </mml:math> J 1 – J 2 Heisenberg model at the Majumdar–Ghosh (MG) point on periodic ring lattices of up to 32 sites. By obtaining the two ground states that furnish the MG ground space, we demonstrate that ST-MH attains high fidelity and energy accuracy across degenerate ground state manifolds while using significantly lower computing resources. Lastly we provide a qualitative computational cost analysis which incentivise the applicability of the ST-MH ensemble under certain criteria on the latent width.