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arXiv:2605.02056

Benchmarking quantum trial wavefunctions for phaseless auxiliary-field quantum Monte Carlo

Rod Rofougaran, Neil Mehta, Katherine Klymko, Pooja Rao, J. Wayne Mullinax, Sam Stein, Norm M. Tubman, Ermal Rrapaj

quant-phSubmitted May 3, 2026• Announced May 5, 2026• 0 citations

v1 · May 3, 2026 v2 · May 5, 2026 v3 · Oct 27, 2026

Abstract

The phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) method is a stochastic imaginary-time projection technique for computing ground-state properties of strongly correlated quantum systems, with accuracy that depends critically on the choice of trial wavefunction. Here, we investigate ph-AFQMC with trial states prepared using parameterized quantum circuits. In this work, we present a comprehensive benchmarking study of quantum trial wavefunctions spanning unitary coupled-cluster, Hamiltonian-informed, Jastrow-inspired, and adaptively constructed ansatze. The benchmarking evaluates accuracy, expressibility, and scalability of these ansatze within the QC-AFQMC framework. We test these ansatze on linear hydrogen chains under bond stretching and find that several ansatz families produce chemically accurate ph-AFQMC energies across the dissociation curve. We have performed simulations using the CUDA-Q quantum development platform on the GPU partition of the Perlmutter supercomputer. When comparing ansatze at similar numbers of variational parameters, we find that different ansatz families yield comparable ph-AFQMC results despite exhibiting substantially different variational energies, optimization costs, and circuit depths. Our results indicate that the variational energy of an ansatz is not always a reliable indicator of its quality for ph-AFQMC and reveal instances of over-parameterization. In the strongly correlated regime, trial wavefunctions obtained from adaptive ansatze, exemplified here by ADAPT-VQE with the UCCSD operator pool, can outperform their fixed-ansatz counterparts (UCCSD) in terms of projected energies while using substantially more compact circuits, providing a flexible route to optimize quantum resources within the ph-AFQMC framework.

Comments: 20 pages, 7 figures

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The phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) method is a stochastic imaginary-time projection technique for computing ground-state properties of strongly correlated quantum systems, with accuracy that depends critically on the choice of trial wavefunction. Here, we investigate ph-AFQMC with trial states prepared using parameterized quantum circuits. In this work, we present a comprehensive benchmarking study of quantum trial wavefunctions spanning unitary coupled-cluster, Hamiltonian-informed, Jastrow-inspired, and adaptively constructed ansatze. The benchmarking evaluates accuracy, expressibility, and scalability of these ansatze within the QC-AFQMC framework. We test these ansatze on linear hydrogen chains under bond stretching and find that several ansatz families produce chemically accurate ph-AFQMC energies across the dissociation curve. We have performed simulations using the CUDA-Q quantum development platform on the GPU partition of the Perlmutter supercomputer. When comparing ansatze at similar numbers of variational parameters, we find that different ansatz families yield comparable ph-AFQMC results despite exhibiting substantially different variational energies, optimization costs, and circuit depths. Our results indicate that the variational energy of an ansatz is not always a reliable indicator of its quality for ph-AFQMC and reveal instances of over-parameterization. In the strongly correlated regime, trial wavefunctions obtained from adaptive ansatze, exemplified here by ADAPT-VQE with the UCCSD operator pool, can outperform their fixed-ansatz counterparts (UCCSD) in terms of projected energies while using substantially more compact circuits, providing a flexible route to optimize quantum resources within the ph-AFQMC framework.

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 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.

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.

High Resolution Image Download MS PowerPoint Slide We report a scalable Fortran implementation of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) and demonstrate its excellent performance and beneficial scaling with respect to system size. Furthermore, we investigate modifications of the phaseless approximation that can help to reduce the overcorrelation problems common to the ph-AFQMC. We apply the method to the 26 molecules in the HEAT set, the benzene molecule, and water clusters. We observe a mean absolute deviation of the total energy of 1.15 kcal/mol for the molecules in the HEAT set, close to chemical accuracy. For the benzene molecule, the modified algorithm despite using a single-Slater-determinant trial wavefunction yields the same accuracy as the original phaseless scheme with 400 Slater determinants. Despite these improvements, we find systematic errors for the CN, CO 2, and O 2 molecules that need to be addressed with more accurate trial wavefunctions. For water clusters, we find that the ph-AFQMC yields excellent binding energies that differ from CCSD(T) by typically less than 0.5 kcal/mol.

To make useful connections with experimental measurements, correlated electronic structure theories must accurately predict chemical properties in addition to energies. We present a finite-difference based algorithm to compute first-order response properties with phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) that relies on a branching correlated sampling approach. Focusing on electric dipole moments, we show that mean-field trial wave functions are sufficient to obtain high accuracy relative to CCSD(T) and experimental measurements for a set of 21 molecules, ranging in size from 2 to 18 atoms in their equilibrium geometries. As with energies, the quality of predicted dipole moments can be systematically improved with the use of correlated trial wave functions, even (or especially) in strongly correlated regimes. We show that the challenges faced by low-order perturbation theories in predicting the dipole moment of hydrogen fluoride across its dissociation coordinate are overcome with ph-AFQMC when using relatively simple trials. The key advantage of our approach over those previously reported for ph-AFQMC is its scalability to large system sizes with a phaseless bias no worse than that of a typical ground-state energy calculation; we routinely converge dipole moments for systems with more than 1000 basis functions.

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.