Variational Monte Carlo for Quantum Excited States via Nested Low-Rank Approximation

The electronic Schrödinger equation encodes a molecule's electronic structure and quantum properties, but solving it scales exponentially with the number of electrons. Neural-network variational Monte Carlo (VMC) has reached chemical accuracy on the ground state of small molecules; extending this success to excited states is constrained by two existing paradigms. Penalty-based methods are computationally scalable ($O(K)$ Laplacian count per step) but require tuning the overlap-penalty schedule against the energy gap they aim to compute. Natural-excited-states VMC (NES-VMC) is principled and penalty-free but couples all $K$ states through a joint $\det\Psi$ ansatz that forces $O(K^2)$ Laplacians per step. We resolve this trade-off via a low-rank approximation (LoRA) reparameterization of the rank-$K$ variational objective. The reparameterized objective admits an unbiased gradient estimator via MCMC sampling from the per-state Born density, and a sequential nesting scheme recovers the bottom-$K$ eigenfunctions in eigenvalue order without overlap penalties or post-training diagonalization. The resulting algorithm, NestedLoRA-VMC, is penalty-free, achieves $O(K)$ Laplacian count per step, and inherits a global optimality guarantee from the LoRA principle. On the first-row atoms (Li through Ne) at $K{=}10$, it stays within the chemical-accuracy threshold of NES-VMC on every atom and outperforms the penalty-based baseline of Szabó et al. on seven of the eight.