Grassmann Variational Monte Carlo with neural wave functions

TL;DR

This paper introduces a Grassmann geometric framework for neural quantum states, enabling stable and accurate variational Monte Carlo optimization of multiple excited states in many-body quantum systems.

Contribution

It generalizes the stochastic reconfiguration method for multiple wave functions and introduces multidimensional operator variances and overlaps.

Findings

Accurately computed energies for many excited states of the Heisenberg model.

Validated the approach's stability and accuracy in excited state calculations.

Achieved results surpassing previous methods in excited state quantum simulations.

Abstract

Excited states play a central role in determining the physical properties of quantum matter, yet their accurate computation in many-body systems remains a formidable challenge for numerical methods. While neural quantum states have delivered outstanding results for ground-state problems, extending their applicability to excited states has faced limitations, including instability in dense spectra and reliance on symmetry constraints or penalty-based formulations. In this work, we rigorously formalize the framework introduced by Pfau et al.~\cite{pfau2024accurate} in terms of Grassmann geometry of the Hilbert space. This allows us to generalize the Stochastic Reconfiguration method for the simultaneous optimization of multiple variational wave functions, and to introduce the multidimensional versions of operator variances and overlaps. We validate our approach on the Heisenberg quantum spin model on the square lattice, achieving highly accurate energies and physical observables for a large number of excited states.