Molecular Properties in Quantum-Classical Auxiliary-Field Quantum Monte Carlo: Correlated Sampling with Application to Accurate Nuclear Forces

We extend correlated sampling to quantum-classical auxiliary-field quantum Monte Carlo (QC-AFQMC), enabling accurate nuclear force evaluation in strongly correlated systems. Computing forces via finite differences typically incurs prohibitive statistical noise in stochastic methods. We suppress this noise by maximizing correlation between geometries through synchronized random streams, orbital alignment, deterministic integral decomposition, and consistent classical shadow measurements. Crucially, a single shadow ensemble defined at the reference geometry suffices for all displaced structures, eliminating additional quantum measurements. This approach substantially reduces force variance while preserving accuracy. We validate the method on hydrogen chains across varying correlation regimes and demonstrate accurate forces for N₂ dissociation and stretched H₄ in strongly correlated regions where restricted coupled cluster methods fail qualitatively. Application to the MEA-CO₂ carbon capture reaction, integrating quantum information metrics for active space selection and matchgate shadows for overlap estimation, demonstrates that QC-AFQMC delivers accurate forces for complex reaction pathways in strongly correlated systems where conventional methods are unreliable or prohibitively expensive.