Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has recently emerged as a promising method for the production of benchmark-level simulations of medium- to large-sized molecules because of its accuracy and favorable polynomial scaling with system size. Unfortunately, the memory footprints of standard energy evaluation algorithms are nontrivial, which can significantly impact timings on graphical processing units (GPUs) where memory is limited. Previous attempts to reduce scaling by taking advantage of the low-rank structure of the Coulombic integrals have been successful but exhibit high prefactors, making their utility limited to very large systems. Here we present a complementary cubic-scaling route to reduce memory and computational scaling based on the low rank of the Coulombic interactions between localized orbitals, focusing on the application to ph-AFQMC. We show that the error due to this approximation, which we term localized-orbital AFQMC (LO-AFQMC), is systematic and controllable via a single variable and that the method is computationally favorable even for small systems. We present results demonstrating robust retention of accuracy versus both experiment and full ph-AFQMC for a variety of test cases chosen for their potential difficulty for localized-orbital-based methods, including the singlet–triplet gaps of the polyacenes benzene through pentacene, the heats of formation for a set of Platonic hydrocarbon cages, and the total energy of ferrocene, Fe(Cp) 2 . Finally, we reproduce our previous result for the gas-phase ionization energy of Ni(Cp) 2, agreeing with full ph-AFQMC to within statistical error while using less than 1/15th of the computer time.
Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has recently emerged as a promising method for the production of benchmark-level simulations of medium- to large-sized molecules because of its accuracy and favorable polynomial scaling with system size. Unfortunately, the memory footprints of standard energy evaluation algorithms are nontrivial, which can significantly impact timings on graphical processing units (GPUs) where memory is limited. Previous attempts to reduce scaling by taking advantage of the low-rank structure of the Coulombic integrals have been successful but exhibit high prefactors, making their utility limited to very large systems. Here we present a complementary cubic-scaling route to reduce memory and computational scaling based on the low rank of the Coulombic interactions between localized orbitals, focusing on the application to ph-AFQMC. We show that the error due to this approximation, which we term localized-orbital AFQMC (LO-AFQMC), is systematic and controllable via a single variable and that the method is computationally favorable even for small systems. We present results demonstrating robust retention of accuracy versus both experiment and full ph-AFQMC for a variety of test cases chosen for their potential difficulty for localized-orbital-based methods, including the singlet–triplet gaps of the polyacenes benzene through pentacene, the heats of formation for a set of Platonic hydrocarbon cages, and the total energy of ferrocene, Fe(Cp) 2 . Finally, we reproduce our previous result for the gas-phase ionization energy of Ni(Cp) 2, agreeing with full ph-AFQMC to within statistical error while using less than 1/15th of the computer time.
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.
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.
We explore the use of the stochastic resolution-of-the-identity (sRI) with the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) method. sRI is combined with four existing local energy evaluation strategies in ph-AFQMC, namely, (1) the half-rotated electron repulsion integral tensor (HR), (2) Cholesky decomposition (CD), (3) tensor hypercontraction (THC), or (4) low-rank factorization (LR). We demonstrate that HR–sRI achieves no scaling reduction, CD–sRI scales as O(N3), and THC–sRI and LR–sRI scale as O(N2), albeit with a potentially large prefactor. Furthermore, the walker-specific extra memory requirement in CD is reduced from O(N3) to O(N2) with sRI, while sRI-based THC and LR algorithms lead to a reduction from O(N2) extra memory to O(N). Based on numerical results for one-dimensional hydrogen chains and water clusters, we demonstrated that, along with the use of a variance reduction technique, CD–sRI achieves cubic-scaling without overhead. In particular, we find that for the systems studied, the observed scaling of standard CD is O(N3–4), while for CD–sRI, it is reduced to O(N2–3). Once a memory bottleneck is reached, we expect THC–sRI and LR–sRI to be preferred methods due to their quadratic-scaling memory requirements and their quadratic-scaling of the local energy evaluation (with a potentially large prefactor). The theoretical framework developed here should facilitate large-scale ph-AFQMC applications that were previously difficult or impossible to carry out with standard computational resources.
The ground state electronic energy of benzene is the focus of a recent blind test by Eriksen and co-workers [arXiv:2008.02678]. In this note, we report the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) results on the identical problem. We examine trial wavefunctions of the spin-restricted Hartree-Fock (RHF), and complete active space self-consistent field (CASSCF) with an active space of 6-electron and 6-orbital (i.e., CAS(6,6)) form for the cc-pVDZ basis set. ph-AFQMC+RHF deviates from the value where many methods agreed on (i.e., -863 mEh) by -3.1(3) mEh whereas ph-AFQMC+CAS(6,6) deviates from the same value by -1.3(4) mEh. In addition to this, we report the frozen core correlation energy of ph-AFQMC+RHF in the cc-pVTZ and cc-pVQZ basis sets as well as their complete basis set limit. Our findings highlight the accuracy, flexibility, and scalability of ph-AFQMC with simple trial wavefunctions.
We present efficient algorithms for using selected configuration interaction (sCI) trial wave functions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC). These advances, geared toward optimizing computational performance for longer configuration interaction expansions, allow us to use up to a million configurations in the trial state for ph-AFQMC. In one example, we found the cost of ph-AFQMC per sample to increase only by a factor of about 3 for a calculation with 104 configurations compared to that with a single one, demonstrating the tiny computational overhead due to a longer expansion. This favorable scaling allows us to study the systematic convergence of the phaseless bias in auxiliary field quantum Monte Carlo calculations with an increasing number of configurations and provides a means to gauge the accuracy of ph-AFQMC with other trial states. We also show how the scalability issues of sCI trial states for large system sizes could be mitigated by restricting them to a moderately sized orbital active space and leveraging the near-cancellation of out of active space phaseless errors.
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.