Fast and stable determinant quantum Monte Carlo

Abstract

We assess numerical stabilization methods employed in fermion many-body quantum Monte Carlo simulations. In particular, we empirically compare various matrix decomposition and inversion schemes to gain control over numerical instabilities arising in the computation of equal-time and time-displaced Green's functions within the determinant quantum Monte Carlo (DQMC) framework. Based on this comparison, we identify a procedure based on pivoted QR decompositions which is both efficient and accurate to machine precision. The Julia programming language is used for the assessment and implementations of all discussed algorithms are provided in the open-source software library StableDQMC.jl.

The ALF (Algorithms for Lattice Fermions) project release 2.3. Documentation for the auxiliary-field quantum Monte Carlo code

“…In order to circumvent this, more sophisticated methods have to be employed. In the realm of the BSS algorithm there has been a long history [4,93,[122][123][124][125] of using various matrix factorization techniques. The predominant techniques are either based on the singular value decomposition (SVD) or on techniques using the QR decomposition.…”

SmoQyDQMC.jl: A flexible implementation of determinant quantum Monte Carlo for Hubbard and electron-phonon interactions

“…A straightforward approach for computing these quantities is outlined in Algorithm 2. Unfortunately, this naive approach fails due to well-documented [2,[113][114][115][116] numerical instabilities associated with evaluating the ill-conditioned products of B σ,l matrices. Specifically, repeated matrix multiplication by the propagator matrices B σ,l accumulates numerical errors that quickly become severe.…”

“…Practical implementations of the DQMC algorithm have to overcome these numerical instabilities by introducing stable matrix factorizations [2,[113][114][115][116]. The SmoQyDQMC.jl package uses stabilization procedures based on those introduced in Ref.…”

The ALF (Algorithms for Lattice Fermions) project release 2.0. Documentation for the auxiliary-field quantum Monte Carlo code

“…In order to circumvent this, more sophisticated methods have to be employed. In the realm of the BSS algorithm there has been a long history [4,97,[127][128][129][130] of using various matrix factorization techniques. The predominant techniques are either based on the singular value decomposition (SVD) or on techniques using the QR decomposition.…”