Selecting optimal unrestricted Hartree-Fock trial wavefunctions for phaseless auxiliary-field quantum Monte Carlo: Accuracy and limitations in modeling three iron-sulfur clusters
Don Danilov, Brad Ganoe, Leon Otis, Zhi Gong, Zixiang Lu, James Shee
physics.chem-phcond-mat.str-elquant-phSubmitted May 5, 2026• Announced May 6, 2026• 0 citations
Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has emerged as a promising electronic structure method for correlated electronic systems. However, the quality of its predictions depends critically on the choice of trial wavefunction, and it is not obvious how to make an optimal choice especially for strongly correlated states of large systems. Mean-field wavefunctions are compelling trial wavefunction candidates as they map directly to chemical concepts and can be obtained with $O(N^4)$ cost. Yet in the strongly correlated regime one faces a symmetry dilemma and the existence of multiple nearly-degenerate solutions. In this work we investigate active space models of [2Fe-2S]$^{2+}$, mixed-valent [4Fe-4S]$^{2+}$, and [4Fe-4S]$^{4+}$ and explore the sensitivity of ph-AFQMC to the choice of unrestricted Hartree-Fock trial wavefunction. We find that chemical properties and physical symmetries, rather than the variational energy, ought to guide the choice of mean-field trial for ph-AFQMC (or reference state for coupled cluster models), and show that surprisingly accurate ground-state energies for these systems can be obtained. However, in all cases we find a rapidly vanishing overlap between the stochastic wavefunction and the UHF trial, indicating that the trials are suboptimal importance functions. By analogy to a similar situation in the stretched helium dimer cation, we show how this sampling bias pushes ph-AFQMC towards artificially negative energies, which evidently can be compensated for by the phaseless bias in certain cases.
Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has emerged as a promising electronic structure method for correlated electronic systems. However, the quality of its predictions depends critically on the choice of trial wavefunction, and it is not obvious how to make an optimal choice especially for strongly correlated states of large systems. Mean-field wavefunctions are compelling trial wavefunction candidates as they map directly to chemical concepts and can be obtained with O(N4) cost. Yet in the strongly correlated regime one faces a symmetry dilemma and the existence of multiple nearly-degenerate solutions. In this work we investigate active space models of [2Fe-2S]2+, mixed-valent [4Fe-4S]2+, and [4Fe-4S]4+ and explore the sensitivity of ph-AFQMC to the choice of unrestricted Hartree-Fock trial wavefunction. We find that chemical properties and physical symmetries, rather than the variational energy, ought to guide the choice of mean-field trial for ph-AFQMC (or reference state for coupled cluster models), and show that surprisingly accurate ground-state energies for these systems can be obtained. However, in all cases we find a rapidly vanishing overlap between the stochastic wavefunction and the UHF trial, indicating that the trials are suboptimal importance functions. By analogy to a similar situation in the stretched helium dimer cation, we show how this sampling bias pushes ph-AFQMC towards artificially negative energies, which evidently can be compensated for by the phaseless bias in certain cases.
Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has emerged as a promising electronic structure method for correlated electronic systems. However, the quality of its predictions depends critically on the choice of trial wave function, and it is not obvious how to make an optimal choice, especially for strongly correlated states of large systems. Mean-field wave functions are compelling trial wave function candidates, as they map directly to chemical concepts and can be obtained withO(N4) cost. Yet in the strongly correlated regime, one faces a symmetry dilemma and the existence of multiple nearly degenerate solutions. In this work, we investigate active space models of [2Fe-2S]2+, mixed-valent [4Fe-4S]2+, and [4Fe-4S]4+ and explore the sensitivity of ph-AFQMC to the choice of unrestricted Hartree-Fock trial wave function. We find that surprisingly accurate ground-state energies for these systems can be obtained when trial selection is guided by chemical properties and physical symmetries, rather than the variational energy. However, in all cases, we find a rapidly decaying overlap between the stochastic wave function and the UHF trial, indicating that the trials are acting as suboptimal importance functions. By analogy to a similar situation in the stretched helium dimer cation, we show how this sampling bias pushes ph-AFQMC with UHF trials toward artificially negative energies, which evidently can be compensated for by the phaseless bias in certain cases.
Phaseless auxiliary-field quantum Monte Carlo (AFQMC) has in several cases been found to perform well on strongly correlated systems. Here, we benchmark the method for three iron-sulfur clusters ([2Fe-2S], [4Fe-4S], and the FeMo cofactor) using a hierarchy of trial states derived from coupled cluster (CC) theory, including up to quadruple excitations, as well as multi-Slater trial states derived from the density matrix renormalization group. Our results reveal for these systems that, as the symmetry-broken trial is improved, the phaseless AFQMC energy can become less accurate, and in some cases even less accurate than the underlying trial projected energy, displaying an inverted energy pattern that is only corrected once the trial fidelity is sufficiently high. For [2Fe-2S], we show that this can coincide with a simultaneous improvement in the trial state and the walker ensemble. We further find that this is not solely due to the use of spin-unrestricted trial states, as the inversion persists in [2Fe-2S] when we explicitly break the symmetry of the Hamiltonian by applying a fictitious spin-Zeeman field. Instead, we find that the energy inversion is related to the choice of measurement trial, where using a high-order CC trial state for measurements may introduce errors that are suppressed when the measurement wave function is restricted to lower excitation subspaces. In particular, measuring the energy with the mean-field reference while guiding the walkers with a CC trial improves the overall accuracy across the iron-sulfur clusters, with a possible exception for [4Fe-4S]. Taken together, our findings suggest that the relatively accurate energies obtained with an HF trial state in these systems arise from favorable error cancellation, warranting significant caution about the reliability of phaseless AFQMC with such trials for strongly correlated transition-metal systems of this kind.
The phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) method is a stochastic imaginary-time projection method for computing ground-state properties of strongly correlated quantum systems, whose accuracy depends critically on the trial wavefunction. We present a benchmarking study of trial wavefunctions prepared using parameterized quantum circuits, spanning unitary coupled cluster (UCC), Hamiltonian-informed, Jastrow-inspired, and adaptively constructed ansatzes, and evaluate their accuracy, expressibility, and resource requirements within the quantum-computing-enhanced AFQMC (QC-AFQMC) framework. Trial states are prepared with GPU-accelerated statevector simulations using CUDA-Q and tested on stretched linear hydrogen chains and nine closed-shell molecules at equilibrium with varying static correlation. We find that at similar parameter counts, ph-AFQMC largely removes differences between families for weakly and moderately correlated systems: every trial yields a ph-AFQMC energy within chemical accuracy despite trial energy errors spanning nearly three orders of magnitude, whereas ansatz choice becomes consequential as static correlation grows. Our results indicate that trial variational energy is not always a reliable indicator of ph-AFQMC quality, and that adding variational parameters can lower the trial energy while degrading the ph-AFQMC energy. Across our benchmark systems, trial fidelity ranks trials more reliably than variational energy, and for spin-contaminated trials, the singlet-projected fidelity is more reliable than unprojected fidelity. The UCCSD ansatz initialized from unrestricted CCSD amplitudes provides a robust and inexpensive-to-optimize baseline across benchmark systems, with circuit depth as its principal limitation, while adaptive and compact layered ansatzes remain competitive in accuracy under strong correlation with substantially shallower circuits.
Generalized Hartree Fock (GHF) is a long-established electronic structure method that can lower the energy (compared to spin-restricted variants) by breaking physical wavefunction symmetries, namely S^2 and Sz. After an exposition of GHF theory, we assess the use of GHF trial wavefunctions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC-G) calculations of strongly correlated molecular systems including symmetrically stretched hydrogen rings, carbon dioxide, and dioxygen. We find comparable or better accuracy than CCSD(T) with unrestricted HF and GHF references, and consistently smooth dissociation curves -- a remarkable result given the relative scalability of ph-AFQMC-G to larger system sizes. The present exploration of model strongly correlated systems marks a promising starting point for future studies of more chemically-relevant molecules, and demonstrates that ph-AFQMC-G provides a highly accurate (and, in contrast to active-space-based trials, relatively black box and always size-consistent) description of challenging systems exhibiting, e.g., antiferromagnetic coupling and/or geometric spin frustration.
We assess the utility of Hartree-Fock (HF) trial wavefunctions in performing phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) on the uniform electron gas (UEG) model. The combination of ph-AFQMC with spin-restricted HF (RHF+ph-AFQMC) was found to be highly accurate and efficient for systems containing up to 114 electrons in 2109 orbitals, particularly for rs ≤ 2.0. Compared to spin-restricted coupled-cluster (RCC) methods, we found that RHF+ph-AFQMC performs better than CC with singles, doubles, and triples (RCCSDT) and similarly to or slightly worse than CC with singles, doubles, triples, and quadruples (RCCSDTQ) for rs ≤ 3.0 in the 14-electron UEG model. With the 54-electron, we found RHF+ph-AFQMC to be nearly exact for rs ≤ 2.0 and pointed out potential biases in existing benchmarks. Encouraged by these, we performed RHF+ph-AFQMC on the 114-electron UEG model for rs ≤ 2.0 and provided new benchmark data for future method development. We found that the UEG models with rs = 5.0 remain to be challenging for RHF+ph-AFQMC. Employing nonorthogonal configuration expansions or unrestricted HF states as trial wavefunctions was also found to be ineffective in the case of the 14-electron UEG model with rs = 5.0. We emphasize the need for a better trial wavefunction for ph-AFQMC in simulating strongly correlated systems. With the 54-electron and 114-electron UEG models, we stress the potential utility of RHF+ph-AFQMC for simulating dense solids.
We assess the utility of spin-restricted Hartree-Fock (RHF) trial wavefunctions in performing phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) on the uniform electron gas (UEG) model. This particular combination, RHF+ph-AFQMC, was found to be highly accurate and efficient for systems containing up to 114 electrons in 2109 orbitals, particularly for rs ≤ 2.0. Compared to spin-restricted coupled-cluster (RCC) methods, we found that RHF+ph-AFQMC performs better than CC with singles, doubles, and triples (RCCSDT) and similarly to or slightly worse than CC with singles, doubles, triples, and quadruples (RCCSDTQ) for rs ≤ 3.0 in the 14-electron UEG model. With the 54-electron, we found RHF+ph-AFQMC to be nearly exact for rs ≤ 2.0 and pointed out potential biases in existing benchmarks. Encouraged by these, we performed RHF+ph-AFQMC on the 114-electron UEG model for rs ≤ 2.0 and provided new benchmark data for future method development. As the UEG models with rs = 5.0 remain to be challenging for the RHF+ph-AFQMC, we emphasize the need for a better trial wavefunction for ph-AFQMC in simulating strongly correlated systems. With the 54-electron and 114-electron UEG models, we stress the potential utility of RHF+ph-AFQMC for simulating dense solids.