Systematic improvement of trial states in phaseless auxiliary-field quantum Monte Carlo

TL;DR

The paper tackles the bias in phaseless AFQMC arising from the trial state by introducing a hierarchy of trial states built from coupled-cluster expansions up to CCSDTQ. It develops and implements trial constructions projected onto CI subspaces, enabling efficient evaluation of overlaps and local energies via automatic differentiation and Wick contractions, with scalable trial-building costs of $N^8$ and $N^{10}$ for triples and quadruples. Benchmarking on the HEAT dataset shows that AFQMC energies improve systematically as the CC trial is refined, with the most substantial gains for weaker correlation regimes; however, the relative improvement over the underlying CC energy diminishes as higher excitations are added, and strong correlation can reduce or reverse the benefit of higher-order terms. The study thus clarifies when high-order CC trials are advantageous for AFQMC, provides practical scaling guidance, and points to further exploration of the trial–bias relationship in diverse chemical problems.

Abstract

We extend the use of coupled cluster (CC) trial states in the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method beyond single and double excitations to include both triple and quadruple excitations. With this AFQMC/CC hierarchy, we are able to systematically benchmark the method's performance on molecular systems as the quality of the trial is improved. Our results show that the phaseless AFQMC energy improves systematically and is typically significantly more accurate than the energy of the underlying trial state. However, the relative improvement compared to the trial CC energy decreases as we ascend the CC hierarchy. As the CC wavefunction is usually further approximated when used as an AFQMC trial, we also explore the relationship between the components of the CC wavefunction and the resulting AFQMC/CC error. Our results suggest that improving the representation of the CC wave function in the AFQMC trial does not always lower the bias even when it increases the fidelity of the trial with the exact ground state.

Figures (3)

• Figure 1: Errors in the ground state energy for the HEAT dataset. Shaded areas indicate chemical accuracy $E_{ca} = 1.596$ m$E_h$$\approx 1$ kcal/mol (blue) and 0.1 kcal/mol (green). Errors are given relative to CCSDTQP (for F$_2$) and CCSDTQPH (for all other systems).
• Figure 2: Hydrogen cluster (H$_8$/STO-3G). The left and right columns show results for H--H bond distances of $R = 1.00$ Å and $R = 1.75$ Å. The upper panels show errors in energies for fully expanded and projected trials, where the truncation of the projected trials is given as the truncation of the cluster operator. The middle panels show absolute errors for varying truncations of the trial. The bottom panels show the infidelity of the truncated trials (see text).
• Figure 3: Errors along potential energy curve for HF/6-31G.