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

Abstract

Neural network quantum states (NQS) 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 share a feature extracting trunk while attaching lightweight heads for each target state. Using a cost function which 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-$\tfrac{1}{2}$ $J_1-J_2$ Heisenberg model at the Majumdar-Ghosh point on periodic ring lattices of up to 8 sites. By obtaining the momentum eigenstates, 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.