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Journal of Chemical Theory and Computation

A High-Efficiency Delayed Update Algorithm for Evaluating Slater Determinants in Quantum Monte Carlo

Ye Luo, Jeongnim Kim, Paul R. C. Kent

Published Dec 1, 2025• 0 citations
Publisher

Abstract

For quantum Monte Carlo simulations of molecular systems or supercells with thousands of electrons, matrix operations related to Slater determinants lead the computational cost. McDaniel et al. [ J. Chem. Phys. 2017, 147, 174107] proposed a delayed update algorithm to increase computational efficiency by using matrix–matrix multiplication when updating the inverse matrices of Slater determinants. However, preparing intermediate matrices for applying the Sherman–Morrison–Woodbury formula remained a bottleneck. In this work, we introduce an improved algorithm for CPUs and GPUs that (1) reduces this bottleneck by iteratively updating the intermediate matrices and (2) is efficient at any acceptance ratio, with no cost for rejected moves on CPUs and minimal cost on GPUs. We show the full scheme of integrating the delayed update algorithm into a single-electron move. The high efficiency of our algorithm is demonstrated on CPUs and GPUs for a 512 atom/6144 valence electron calculation, with 12× and 2× overall speed-up compared to traditional rank-1 update schemes in diffusion quantum Monte Carlo, respectively.

DOI
10.1021/acs.jctc.5c01541

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For quantum Monte Carlo simulations of molecular systems or supercells with thousands of electrons, matrix operations related to Slater determinants lead the computational cost. McDaniel et al. [ J. Chem. Phys. 2017, 147, 174107] proposed a delayed update algorithm to increase computational efficiency by using matrix–matrix multiplication when updating the inverse matrices of Slater determinants. However, preparing intermediate matrices for applying the Sherman–Morrison–Woodbury formula remained a bottleneck. In this work, we introduce an improved algorithm for CPUs and GPUs that (1) reduces this bottleneck by iteratively updating the intermediate matrices and (2) is efficient at any acceptance ratio, with no cost for rejected moves on CPUs and minimal cost on GPUs. We show the full scheme of integrating the delayed update algorithm into a single-electron move. The high efficiency of our algorithm is demonstrated on CPUs and GPUs for a 512 atom/6144 valence electron calculation, with 12× and 2× overall speed-up compared to traditional rank-1 update schemes in diffusion quantum Monte Carlo, respectively.
We present an efficient low-rank updating algorithm for updating the trial wave functions used in quantum Monte Carlo (QMC) simulations. The algorithm is based on low-rank updating of the Slater determinants. In particular, the computational complexity of the algorithm is O(kN) during the kth step compared to traditional algorithms that require O(N(2)) computations, where N is the system size. For single determinant trial wave functions the new algorithm is faster than the traditional O(N(2)) Sherman-Morrison algorithm for up to O(N) updates. For multideterminant configuration-interaction-type trial wave functions of M+1 determinants, the new algorithm is significantly more efficient, saving both O(MN(2)) work and O(MN(2)) storage. The algorithm enables more accurate and significantly more efficient QMC calculations using configuration-interaction-type wave functions.
QMCPACK is open source scientific software designed to perform Quantum Monte Carlo (QMC) simulation, a first-principles method for describing many-body systems. The evaluation of each Monte Carlo move requires finding the determinant of a dense matrix in the wave functions. This calculation forms a key computational kernel in QMCPACK. After each accepted event, the wave function matrix undergoes a rank-one update to represent a single particle move within the system. The matrix inverse is updated via the Sherman-Morrison formula; occasionally, the explicit inverse must be recomputed to maintain numerical stability.; We propose an alternate approach to this kernel that utilizes QR factorization to maintain stability without refactorization. In addition, algorithms based on a novel delayed update scheme are explored in this effort. This strategy enables probability evaluation for multiple successive Monte Carlo moves, with application of accepted moves to the wave function matrix delayed until an event is denied, or a predetermined limit on acceptances p is reached. Accepted events grouped in this manner are then applied to the matrix en bloc with enhanced arithmetic intensity and computational efficiency.
Quantum Monte Carlo (QMC) methods are often used to calculate properties of many body quantum systems. The main cost of many QMC methods, for example, the variational Monte Carlo (VMC) method, is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators so that the cost of constructing Slater matrices in these systems is now linear in the number of particles, whereas computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing the determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. The main contribution of this paper is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of incomplete LU decomposition with threshold and pivoting (ILUTP) preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of QMC algorithms from O(n3)O(n^3)O(n3) per sweep to roughly O(n2)O(n^2)O(n2), where n is the number of particles, and a sweep is a sequence of n steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors. Our results show that preconditioned iterative solvers can dramatically reduce the cost of VMC for large(r) systems.
Determinant quantum Monte Carlo (DQMC) is a widely used unbiased numerical method for simulating strongly correlated electron systems. However, the update process in DQMC is often a bottleneck for its efficiency. To address this issue, we propose a generalized delay update scheme that can handle both on-site and extended interactions. Our delay update scheme can be implemented in both zero-temperature and finite-temperature versions of DQMC. We apply the delay update scheme to various strongly correlated electron models and evaluate its efficiency under different conditions. Our results demonstrate that the proposed delay update scheme significantly improves the efficiency of DQMC simulations, enabling it to simulate larger system sizes.
For atomic and molecular quantum Monte Carlo calculations, most of the computational effort is spent in the evaluation of the local energy. We describe a scheme for reducing the computational cost of the evaluation of the Slater determinants and correlation function for the correlated molecular orbital (CMO) ansatz. A sparse representation of the Slater determinants makes possible efficient evaluation of molecular orbitals. A modification to the scaled distance function facilitates a linear scaling implementation of the Schmidt- Moskowitz-Boys-Handy (SMBH) correlation function that preserves the efficient matrix multiplication structure of the SMBH function. For the evaluation of the local energy, these two methods lead to asymptotic linear scaling with respect to the molecule size.
Ground-state auxiliary-field quantum Monte Carlo is a powerful tool for interacting fermions but is often limited by the bottleneck of costly local update. Here, the authors develop two acceleration schemes: the delayed update, which replaces multiple vector outer products by one matrix-matrix multiplication, and the block force-bias update, which retains high efficiency while tuning acceptance via block size. Applied to 2D Hubbard models, they yield up to ~ 8 times speedup in 1600-site systems. These update schemes can have broad applicability to general correlated fermion systems.