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arXiv:2308.12005 · Physical review. B./Physical review. B

Delay update in determinant quantum Monte Carlo

Fanjie Sun, Xiao Yan Xu

Published Jun 20, 2024• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.109.235140

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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.
Determinant Quantum Monte Carlo (DQMC) provides numerically exact solutions for strongly correlated fermionic systems but faces significant computational challenges with increasing system size. While submatrix updates were originally developed for Hirsch-Fye QMC with onsite interactions at finite temperatures, their comprehensive application in DQMC has remained unexplored despite noted algorithmic similarities. We present the first comprehensive application of submatrix updates in DQMC, significantly extending beyond the original scope by enabling simulations with extended interactions and at zero temperature. Building upon conventional fast updates and delay updates, our generalized implementation achieves an order-of-magnitude improvement in computational efficiency, enabling simulations of the half-filled Hubbard model on lattices up to 8,000 sites - a scale previously challenging with standard DQMC implementations. This enhanced computational capability allows us to accurately determine the finite-temperature phase diagram of the 3D Hubbard model at half-filling. Our findings not only shed light on the phase transitions within these complex systems but also pave the way for more effective simulations of strongly correlated electrons, potentially guiding experimental efforts in cold atom simulations of the 3D Hubbard model.
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 introduce methodologies for highly scalable quantum Monte Carlo simulations of electron-phonon models, and we report benchmark results for the Holstein model on the square lattice. The determinant quantum Monte Carlo (DQMC) method is a widely used tool for simulating simple electron-phonon models at finite temperatures, but it incurs a computational cost that scales cubically with system size. Alternatively, near-linear scaling with system size can be achieved with the hybrid Monte Carlo (HMC) method and an integral representation of the Fermion determinant. Here, we introduce a collection of methodologies that make such simulations even faster. To combat "stiffness" arising from the bosonic action, we review how Fourier acceleration can be combined with time-step splitting. To overcome phonon sampling barriers associated with strongly bound bipolaron formation, we design global Monte Carlo updates that approximately respect particle-hole symmetry. To accelerate the iterative linear solver, we introduce a preconditioner that becomes exact in the adiabatic limit of infinite atomic mass. Finally, we demonstrate how stochastic measurements can be accelerated using fast Fourier transforms. These methods are all complementary and, combined, may produce multiple orders of magnitude speedup, depending on model details.
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.
At low temperatures TTT where 1/T=β≫11/T=β\gg11/T=β≫1 the naïve implementation of determinant quantum Monte Carlo (DQMC) methods suffers from loss of precision and numerical instabilities when evaluating the fermion determinant. This instability propagates into the calculation of observables that rely on the evaluation of the inverse of the fermion matrix, or the Greens function. For DQMC methods that rely on the Hamiltonian Monte Carlo (HMC) algorithm, an additional complication comes from evaluating the force terms required for integrating Hamilton's equations of motion, since here loss of precision and numerical instabilities are also prevalent. We show how to address all these issues using various choices of matrix decompositions, allowing us to simulate at β≳90β\gtrsim 90β≳90, which corresponds to room temperature for graphene structures. Furthermore, our implementation has numerical costs that scale similarly to the naïve implementation, namely as O(Nx3Nt)\mathcal{O}(N_x^3N_t)O(Nx3​Nt​), where NxN_xNx​ (NtN_tNt​) is the number of spatial (temporal) sites.
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.