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arXiv:2005.11779 · The Journal of Chemical Physics

Stochastic resolution-of-the-identity auxiliary-field quantum Monte Carlo: Scaling reduction without overhead

Joonho Lee, David R. Reichman

Published Jul 28, 2020• 28 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0015077

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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.
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.
We present a reduced-scaling auxiliary-field quantum Monte Carlo (AFQMC) framework designed for large molecular systems and ensembles, with or without coupling to optical cavities. Our approach leverages the natural block sparsity of the Cholesky decomposition (CD) of electron repulsion integrals in molecular ensembles and employs tensor hypercontraction (THC) to efficiently compress low-rank Cholesky blocks. By representing the Cholesky vectors in a mixed format, keeping high-rank blocks in block-sparse form and compressing low-rank blocks with THC, we reduce the scaling of exchange-energy evaluation from quartic to robust cubic in the number of molecular orbitals N, while lowering memory from cubic toward quadratic. Benchmark analyses on one-, two-, and three-dimensional molecular ensembles (up to ∼1,200 orbitals) show that (a) the number of nonzeros in Cholesky tensors grows linearly with system size across dimensions; (b) the average numerical rank increases sublinearly and does not saturate at these sizes; and (c) rank heterogeneity─some blocks nearly full rank and many low rank, naturally motivates the proposed mixed block sparsity and THC scheme for efficient calculation of exchange energy. We demonstrate that the mixed scheme yields cubic wall-time scaling with favorable prefactors and preserves AFQMC accuracy.
Auxiliary Field Quantum Monte Carlo (AFQMC) has emerged as a powerful framework for treating strongly correlated electronic systems, offering a favorable balance between computational cost and accuracy. In this paper, we present a novel AFQMC method that uses the isometric tensor hypercontraction (ITHC) technique to diagonalize the two-body Coulomb interaction of molecular electronic Hamiltonians by introducing additional fictitious fermionic modes. Our method shows reduced theoretical complexity and better practical performance for both propagation and local energy evaluation compared to the standard AFQMC method. We demonstrate the efficacy of this approach by computing the ground-state energies of a linear $\ce{H10}$-chain and the benzene molecule. Our results show that the extended-basis AFQMC recovers many-body correlations with a precision comparable to that of high-level wavefunction methods such as Coupled Clusters (CC) or Density Matrix Renormalization Group (DMRG), while offering significantly improved scaling.
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.
auxiliary-field quantum Monte Carlo (AFQMC) is a systematically improvable many-body method, but its application to extended solids has been severely limited by unfavorable computational scaling and memory requirements that obstruct rigorous access to the thermodynamic and complete-basis-set limits. By combining tensor hypercontraction with k -point symmetry, we reduce the computational and memory scaling of AFQMC for solids to O ( N 3 ) and O ( N 2 ) , respectively, with an arbitrary basis, comparable to diffusion Monte Carlo. This enables simultaneous, controlled extrapolation of full-system AFQMC calculations to the thermodynamic and complete-basis-set limits across insulating, metallic, and strongly correlated solids, without embedding, local approximations, empirical finite-size corrections, or composite schemes. Our results establish AFQMC as a general-purpose, systematically improvable alternative to diffusion Monte Carlo and coupled-cluster methods for predictive simulations of solids, enabling accurate energies and magnetic observables within a unified framework.