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Compactness of quantics tensor train representations of local imaginarytime propagators
by Haruto Takahashi, Rihito Sakurai, Hiroshi Shinaoka
Submission summary
Authors (as registered SciPost users):  Hiroshi Shinaoka 
Submission information  

Preprint Link:  https://arxiv.org/abs/2403.09161v1 (pdf) 
Date submitted:  20240318 05:33 
Submitted by:  Shinaoka, Hiroshi 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Spacetime dependence of imaginarytime propagators, vital for \textit{ab initio} and manybody calculations based on quantum field theories, has been revealed to be compressible using Quantum Tensor Trains (QTTs) [Phys. Rev. X {\bf 13}, 021015 (2023)]. However, the impact of system parameters, like temperature, on data size remains underexplored. This paper provides a comprehensive numerical analysis of the compactness of local imaginarytime propagators in QTT for onetime/frequency objects and twotime/frequency objects, considering truncation in terms of the Frobenius and maximum norms. We employ random pole models to study worstcase scenarios. The numerical analysis reveals several key findings. For onetime/frequency objects and twotime/frequency objects, the bond dimensions saturate at low temperatures, especially for truncation in terms of the Frobenius norm. We provide countingnumber arguments for the saturation of bond dimensions for the onetime/frequency objects, while the origin of this saturation for twotime/frequency objects remains to be clarified. This paper's findings highlight the critical need for further research on the selection of truncation methods, tolerance levels, and the choice between imaginarytime and imaginaryfrequency representations in practical applications.
Current status:
Reports on this Submission
Strengths
1. Of possible broad interest
Weaknesses
1. Requires additional details about range of applicability and physical rationale
Report
The manuscript provides valuable insight into the compactness of local imaginarytime propagators in quantics tensor train format. The work is written with the quantum field theory community in mind, but given its focus on tensor trains could have relatively broad readership in mathematics, physics, chemistry, etc. with changes. It would also improve the paper to clarify not only the findings, but the key breakthrough they wish to highlight (with regards to literature in SVD/TCI, etc.).
Requested changes
1. It would be helpful to provide more information (and physical intuition) to explain why the numerical results for the models expected here are expected to be more broadly applicable.
2. The paper could have a broader impact by further referencing the extensive work being done outside of quantum field theory with imaginarytime propagators. Connection to MPS would also help.
3. Can a statement be made about how the reshaping order is chosen in the general case (i.e., in applications where there are concerns that QTT rank depends significantly on ordering)? Are there additional calculations to support whether reordering significantly affects the presented results?
4. It would be beneficial to provide a rationale for why the BetheSalpeter equation is an important future direction.
5. When Figure 9 is mentioned, it would help to provide an explanation for the dip near 1000, which is distinct from all other models presented.
Recommendation
Ask for minor revision
Report
The authors investigate the quantics tensor train (QTT) representation of local imaginarytime and frequency propagators. They consider one and twotime/frequency objects. They find that  in certain cases  the considered objects are compressible with a bond dimension that remains finite even for inverse temperature going to infinity.
At the present stage I do not believe that the paper has enough substance to warrant publication in SciPost Physics. There are two main reasons:
(1) The paper remains on a superficial level. Whenever an interesting observation is made a thorough analysis thereof is delegated to future work. Particularly for twofrequency objects, Figs. 7 and 8, it seems that the Frobeniusnorm approach has small bond dimension but large errors (the color plot for the absolute error looks similar to that of the absolute values) while the maximumnorm approach gives small errors in the color plots but has bond dimensions growing indefinitely. The question whether twofrequency objects are QTT compressible or not  crucial for the scope of the paper  remains unclear.
(2) Generally it seems insufficient to me to consider random pole models to judge the compressibility of manybody functions. Particularly in the field of analytic continuation it is a common problem that new methods are presented with excellent results on pole models and later turn out not remotely as successful for models encountered in 'real life'. Therefore it is advisable to study realistic models too. Here these could be obtained e.g. in dynamical meanfield theory with quantum Monte Carlo solvers. Moreover one should also consider non Fermi liquid scenarios, possibly using simple but typical analytic expression.
More detailed criticism is as follows:
 In the first paragraph of 3.2 it is unclear how the grid of Matsubara frequencies is mapped to the bit representation. Which values does n take? How is the grid from infinity to infinity mapped to an interval from 0 to 1?
 In Sec. 4 I do not understand the motivation for analyzing Eq. (12) and Eq. (14) separately. According to the Lehmann representation a generic twofrequency propagator is a superposition of Eq. (12) and Eq. (14). Moreover the authors did not address the presence of anomalous terms proportional to delta_{iOmega,0} where Omega is a bosonic Matsubara frequency.
 In the last paragraph of Sec. 4.3.2 the authors state that the twofrequency object G^FB has less information than G^FF but their explanation is based on onefrequency objects. It is unclear why properties of the latter carry over to the former.
 The color plots in Figs. 58 are not well formatted. It is unclear how to extract from these plots whether the QTT representation works or does not work. For instance in Fig. 6a the absolute values seem to have the same size as the absolute errors since both color bars go up to 2 (or even 1).
 In the last paragraph of Sec. 5 the authors mention the presence of constant terms in vertex functions. They state that the single/multiboson exchange (SBE/MBE) framework does not involve constant terms. I cannot follow here for two reasons: First the SBE/MBE framework does contain constant terms, namely the bosonic propagator contains the bare interaction and the Hedin vertex contains a term that equals unity. Second why is the SBE/MBE framework singled out here? There are many other frameworks that similarly have constant terms which similarly can be subtracted.
Author: Hiroshi Shinaoka on 20240413 [id 4416]
(in reply to Report 1 on 20240408)We greatly appreciate your efforts on reviewing the manuscript. Your comments are very useful to improve the readability of the manuscript, although we disagree with the conclusions that the paper has enough substance to warrant publication in SciPost Physics.
We are preparing a revised manuscript, but the SciPost platform allows the authors to communicate with a referee online interactively. Please let us address the main criticism before an extensive revision of the manuscript.
The scaling plots in Fig. 68 strongly indicate that the twofrequency objects are compressible. We agree that the colormaps are to be improved. We attached a revised version of Fig. 6 for a smaller cutoff of 1e8 in this reply. This cutoff corresponds to 4digit accuracy [Eq. (8)]. The new figure demonstrates the compressibility of the twotime object more clearly.
We believe that our nontrivial numerical finding will stimulate further (mathematical) studies. The bond dimensions remain constant at low T in the Frobenius norm case, leading to the data size scaling O(chi^2 log beta) = O(log beta). The results surprisingly indicate that the QTT approach superpasses the scaling of the conventional approach (i.e.,IR, O((log beta)^2).
The maxnorm case is very singular; since the function is diverging, we should avoid using the max norm. Since TCI has been attracting more attention, this result is useful to other manybody theorists as a red flag.
Thank you for raising this important point. We should have explained the motivation of using the randompole model in more detail.
We used the randompole model to generate random imaginarytime data. In our analysis, we deal with only the imaginarytime data but never reconstruct the spectral functions. If the pole positions are chosen appropriately, we could reproduce an imaginarytime function generated by an any continuous spectral function precisely, according to previous studies on IR and DLR. This setup is clearly different from the numerical analytic continuation, where whether or not the original spectral function is discrete or continuous can make significant differences in the quality of the reconstruction of the spectral function.
Our randompole model is more challenging than “reallife” data. We have already done extensive numerical analysis of reallife data from various numerical simulations such as, realfrequency spectral functions of Kondo impurity models, vertex functions from DFT+DMFT, nonequilibrium DMFT and more in PRX 13, 021015 (2023). However, analyzing data from numerical simulations is not enough to understand the intrinsic properties of the QTT representation. For example, QMC data include statistical noise, hindering analysis at low temperature. The Kondo spectral function is too simple and contains only a few peaks (Fig. 13 of the PRX). In contrast, our random pole model contains many poles, whose number increases at low temperature. This allows us to encode more and more information into the random pole model through many random coefficients. We will update the manuscript to stress these points, i.e., our test cases are much more challenging to QTT than reallife cases.
Attachment:
two_svd_tau_rotated_cut864_8_5.pdf