Differentiated Determinant Graphs Enable Large-Scale Multideterminant Auxiliary-Field Quantum Monte Carlo

Abstract

Multideterminant trial wave functions can reduce phaseless bias in auxiliary-field quantum Monte Carlo (AFQMC), but the repeated evaluation of their overlap, force bias, and local energy restricts systematic trial improvement. In this work, we organize particle--hole excitation determinants as a shared arithmetic directed acyclic graph of canonical minors. A coefficient-weighted reverse pass and batched directional passes evaluate the complete estimator without determinant-resolved cofactor tensors while preserving the sparse $α$--$β$ coefficient coupling exactly. Controlled complex-valued tests verify the complete kernel to machine precision and show orders-of-magnitude acceleration over direct determinant summation. In molecular calculations, the observed cost remains sublinear through million-configuration selected-CI trials and the same estimator remains usable at one hundred million configurations, where the eventual exact coefficient-contraction boundary becomes visible. For Cr$_2$, an orbital-optimized CAS(12e,42o) trial in a double-$ζ$ basis yields a potential-energy curve (PEC) and spectroscopic constants in close agreement with experiment. Differentiated determinant graphs therefore convert very large correlated trials from a formal accuracy hierarchy into a practical route for systematic AFQMC studies of strongly correlated molecules.