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

TL;DR

This paper introduces a correlated sampling method within quantum-classical AFQMC to accurately compute nuclear forces, significantly reducing statistical noise and enabling reliable geometry and reaction studies in strongly correlated systems.

Contribution

The authors extend correlated sampling to QC-AFQMC, incorporating strategies to minimize variance and demonstrate improved force calculations in complex, strongly correlated molecular systems.

Findings

Validated accuracy across hydrogen chains in various correlation regimes.

Achieved significant variance reduction in force evaluations for N$_2$ and H$_4$.

Demonstrated robustness in challenging cases like stretched CO$_2$ and complex reaction pathways.

Abstract

We extend correlated sampling from classical auxiliary-field quantum Monte Carlo to the quantum-classical (QC-AFQMC) framework, enabling accurate nuclear force computations crucial for geometry optimization and reaction dynamics. Stochastic electronic structure methods typically encounter prohibitive statistical noise when computing gradients via finite differences. To address this, our approach maximizes correlation between nearby geometries by synchronizing random number streams, aligning orbitals, using deterministic integral decompositions, and employing a consistent set of classical shadow measurements defined at a single reference geometry. Crucially, reusing this single, reference-defined shadow ensemble eliminates the need for additional quantum measurements at displaced geometries. Together, these methodological choices substantially reduce statistical variance in computed forces. We validate the method across hydrogen chains, confirming accuracy throughout varying correlation regimes, and demonstrate significant improvements over single-reference methods in force evaluations for N$_2$ and stretched linear H$_4$, particularly in strongly correlated regions where conventional coupled cluster approaches qualitatively fail. Orbital-optimized trial wave functions further boost accuracy for demanding cases such as stretched CO$_2$, without increasing quantum resource requirements. Finally, we apply our methodology to the MEA-CO$_2$ carbon capture reaction, employing quantum information metrics for active space selection and matchgate shadows for efficient overlap evaluations, establishing QC-AFQMC as a robust framework for exploring complex reaction pathways.