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Fast and stable determinant quantum Monte Carlo
by Carsten Bauer
This is not the current version.
Submission summary
As Contributors:  Carsten Bauer 
Arxiv Link:  https://arxiv.org/abs/2003.05286v2 (pdf) 
Date submitted:  20200314 01:00 
Submitted by:  Bauer, Carsten 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
We assess numerical stabilization methods employed in fermion manybody 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 equaltime and timedisplaced 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 opensource software library StableDQMC.jl [http://github.com/crstnbr/StableDQMC.jl].
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Reports on this Submission
Anonymous Report 1 on 202048 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2003.05286v2, delivered 20200408, doi: 10.21468/SciPost.Report.1615
Strengths
1) Provides a systematic analysis of stabilization problems, and how to mitigate them, related to the determinant QMC method.
Weaknesses
1) It is not clear how general the results are.
Report
The paper considers the propagation of errors du to finite numerical precision in computing products of a large number of matrices, as appears in determinant QMC calculations. The stabilization problem and how to mitigate it has a long history, and here the author attempts to test several stabilization technique and compare them, both as to how well they work and as to their computational costs. The main conclusion is that the QR method is overall much better than the three SVD methods considered.
A potential limitation of the approach is that only a product of the same matrix is considered, which corresponds to a timeindependent bosonic field in the determinant QMC simulation. In reality, the field of course fluctuates, and it is quite likely that the stabilization problems will be even worse for the typical contributing configurations, at least for some models (e.g., the Hubbard model with large U).
The paper is very technical, and it is not clear how general the results are. Nevertheless, the results and testing methodology may be interesting to researchers working on the technical aspects of the determinant QMC method. I recommend the paper to be published after revisions as listed below:
Requested changes
The illustration of the beta dependence of the singular values in Fig. 1 and some later figures requires some more explanation. I think the upper green line and lower orange line in Fig. 1 represents the smallest and largest singular values. But exactly what is the jagged data set that suddenly starts to increase with beta, taking off from the lower bound, e.g., around beta=10 in Fig. 1(a). Is that "curve" showing that the actual lowest singular value in the unstable numerical calculation? If so, it should be explained why there are no longer any small singular values. Perhaps it is somehow obvious that the destabilization causes the singular values to grow, but at least to me its not completely clear why.
As I mentioned above, the limitation to a product of the same matrix may give misleading results. Ideally, the author should at least show some results from an actual simulation, e.g., for the Hubbard model. otherwise it is not at all clear what the value is of this work. For instance, one could imagine that the QR method could work less well if the stabilization problem is even worse, as it may be for large U in typical Hubbard configurations.
Author: Carsten Bauer on 20200420
(in reply to Report 1 on 20200408)We thank the referee for his/her positive comments and his recommendation for publication. In the following, we reply to the questions raised by the referee.
Why are small singular values lost and why do they "grow" in Fig. 1? The fact that small singular values are lost and start to "grow" can be explained by the mixing with large ones: The finite machine precision limits the accuracy of a singular value relative to the largest singular value. When the difference surpasses floating point precision roundoff errors occur and the smallest singular values will try to be "as small as they can be" relative to the largest one. Note that the increase of the smallest singular values is precisely parallel to the growth of the largest one (both the lowest and topmost green lines have the same slope in Fig. 1) supporting this argument. A similar point holds for the loss of the smallest singular values, in Fig. 2: The LAPACK documentation, cited in the manuscript, explicitly mentions for the SVD error bounds: "Thus large singular values [...] are computed to high relative accuracy and small ones may not be." We will improve the caption text for the mentioned figures and will explain this point in more detail in a revised version of the manuscript. We thank the referee for pointing out this unclarity.
How general are the results? While this is an inherently difficult question, which asks for a systematic theoretical study beyond the scope of this manuscript, we agree that it would be beneficial to also present results for a more "realworld" model. To that end, we will add a new section to the appendix in which we consider a spinfermion model for a metallic antiferromagnetic quantum critical point  concretely, the model studied in Ref. [1]. This appendix will contain the analogue of Fig. 1 for this strongly coupled system near quantum criticality, confirming our main finding that the QR decomposition is superior in terms of combined performance and accuracy.
We would again like to thank the referee for his/her comments and recommendation.