Entanglement and optimization within autoregressive neural quantum states

Abstract: Neural quantum states (NQS) are powerful variational ans\"atze capable of representing highly entangled quantum many-body wavefunctions. While the average entanglement properties of ensembles of restricted Boltzmann machines are well understood, the entanglement structure of autoregressive NQS such as recurrent neural networks and transformers remains largely unexplored. We perform large-scale simulations of ensembles of random autoregressive wavefunctions for chains of up to $256$ spins and uncover signatures of transitions in their average entanglement scaling, entanglement spectra, and correlation functions. We show that the standard softmax normalization of the wavefunction suppresses entanglement and fluctuations, and introduce a square modulus normalization function that restores them. Finally, we connect the insights gained from our entanglement and activation function analysis to initialization strategies for finding the ground states of strongly correlated Hamiltonians via variational Monte Carlo.