# Fast and stable determinant quantum Monte Carlo

### Submission summary

 As Contributors: Carsten Bauer Arxiv Link: https://arxiv.org/abs/2003.05286v6 (pdf) Date accepted: 2020-06-26 Date submitted: 2020-06-04 02:00 Submitted by: Bauer, Carsten Submitted to: SciPost Physics Discipline: Physics Subject area: Condensed Matter Physics - Computational Approach: Computational

### 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 [http://github.com/crstnbr/StableDQMC.jl].

### Ontology / Topics

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Published as SciPost Phys. Core 2, 011 (2020)

We thank the referees for their positive comments and their recommendation for publication. Below we list all the changes made to the manuscript to take into account the points raised by the referees.

### List of changes

* On page 5, we clarified the system size and number of (visible) singular values in the caption of Fig. 1.

* On page 6, in the first paragraph, we added a note that the matrix decomposition based stabilisation idea has already been raised by Loh et al. in 1989.

* On page 7, in the SVD section, we now cite Dongarra et al. (Ref. 32) as a reference for research on SVD algorithms.

* On page 7, in the SVD section, we indicate that the gesvd variant is a bidiagonal QR iteration scheme.

* On page 7, in the QR section, we added a note that the popular DQMC implementations ALF and QUEST (which we included as references) use the pivoted QR.

* On page 14, we rephrased two sentences in first paragraph of section 6.2.1 to clarify that we are comparing the QR to the failing SVD variants.

* On page 16, in the second to last paragraph of the discussion section, we weakened the previous statement about the QR being "optimal" and "fast and stable".

* On page 16, we added a note to the acknowledgments that this work has received partial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).