The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

1- Enormous amount of work
2- and supporting graphics

1- Biased review and bad citation policy.
2- Several key contributions are not mentioned.
3- The average reader would struggle to disentangle the main concepts from the technical details of the implementation.
4- Lack of comparison with existing methods.

I have read the paper and have a mixed feeling, since it clearly contains a huge amount of work. However I find it very difficult in the present form to recommend it for publication. It looks to me at this stage more like an internal note, that misses some of the most important ideas, and fills several pages with irrelevant (to the general reader maybe important for a technical note inside the group) technical details.
I will try to give a constructive feedback in the case the authors want to try to reformulate the paper.
Based on the introduction (pag 5) the anthology should serve both as a review of known tensor network methods and as a basis for teaching a technical introductory course on Tensor Networks, both are very ambitious goals.

Let's start with the review part. A good review should indeed not only be the showcase of the proficiency of the authors on a given technique or subject but also a well equilibrated overview of the relevant results in the field supported by the appropriate references.
I feel that this last aspect is completely missing in the present form. The citation policy seems to me a bit random and clearly overstating the importance of the authors contributions . Key contributions are relegated to mere mentions after papers that build on their results.
A notable example is the fact that the original Vidal paper that mentioned the relation between entanglement and matrix product states is cited as references 46 after a later generalization of Vidal with Orus and a completely marginal paper by Vidal himself.
The foundational paper Physical review letters 93 (4), 040502 is not even cited, and the paper about the thermodynamic limit Physical review letters 98 (7), 070201 and the mixed state simulations, Physical review letters 93 (20), 207205.
Furthemore there are important omissions on the key out-of equilibrium contributions such as t-dmrg (White Feiguin, ...) and the approaches using MPS to compute directly structure factors (a good list of references can be found in the introduction of arxiv:1711.09207 for example), and on theoretical results obtained with tensor networks.
Another example is the mentioning of lattice gauge theories where the authors cite Wilson, the book by Creutz and their own PRX paper. I don’t doubt about the relevance of the paper but if I had to pick up 4 references on lattice gauge theories I am not sure I would list that paper.
Similarly when mentioning the importance of the many body problem in high energy they cite among a couple of references their own review on the topic. Any high-energy physicist would disagree on this choice. The fact that the QCD vacuum is a complex many body system is well known (and actually is at the origin of the invention of lattice QCD by Wilson).

The current self-citation policy makes of the current review a very biased piece of work.
I actually found few of the key papers on tensor network cited in an appropriate way in the anthology. There is an entire section on loop-free tensor networks that includes tree tensor network. All the relevant contribution in those field seem to come from the authors. The enormous developments of tree tensor network in studying relevant quantum many body systems are not even mentioned.

So would I suggest the reading of the anthology to young researcher starting in the field?
Possibly not in its current state. In the abstract of the anthology for example mention and innovative one site update for loop free tensor networks. I failed to understand what is the novelty compared to the traditional approaches presented originally in for example in Ref 60 but this fact is not mentioned or discussed.

This brings me to the section about Abelian symmetries. Here again, the presentation follows closely the results presented in a series of papers in the Vidal group. Despite this fact these papers are only marginally cited at pag 31 of the manuscript. Ref 138 is the basic reference that introduces the ideas presented in this anthology, and Ref 134 specializes it to the case of Abelian symmetries and the presentation here is a reformulation of that of Ref 134. This fact should be explicitly stated.

Furthermore the fact that symmetries and tensor networks have been used in order to characterize the phases of many body quantum systems is not even mentioned (see Polmann, Cirac, Verstraete etc etc).
There are fundamental theorems about the fact that acting with a symmetry operator on a state is equivalent to a gauge transformation on the elementary tensors of the network that are completely ignored.
Even saying that the anthology wants to focus on applications and hands on approaches,
the use of symmetries in numerical tensor networks is based on the main idea that rather than working on normal Hilbert spaces one wants to work on graded Hilbert spaces. I was very surprised that this fact (that is at the basis of the full construction) is not even mentioned. The discussion at Pag. 32 starts with mentioning Abelian symmetric links, without mentioning the structure of the vector spaces they act on. I find it very confusing, another example where the implementation get confused with the basic ideas.

Here I now get to the second part, using the Anthology as a set of lecture notes.
From my point of view, good lecture notes should consist in a good review and in addition a set of technical pieces that should be used as example to allow the student to work on more complicated material independently.
Neither good reviews nor good lectures notes can substitute the assimilation stage of someone that needs to learn the material. In a way the assimilation process cannot be transmitted but is something that one has to perform individually.
I have the impression that the authors have made an attempt to transmit their own assimilation process of the known tensor network material to the readership.
The anthology contains a lot of technical details that are from my point of view non-necessary and rather detrimental to the reader.
Why should the reader decide to implement symmetries in the way the authors have implemented them?

After all there are several open source implementation of symmetric tensor network libraries (think of Uni10, iTensor, TNT etc etc), so if someone is interested in writing his own implementation he/she will
1) need to read the original papers
2) understand the concepts
3) decide how to implement them in his/her own code or alternatively go for an open-source implementation

The present Anthology is somehow in between. It tries to explain the main ideas through an implementation, but it does not provide the code. So I would hardly recommend it in the current form to anyone starting on tensor networks, they would get a quite distorted picture, they would have an hard time to disentangle the basic ideas from the details of the implmentation.

Furthermore, by using this anthology in a course people would get a very biased picture focused on the Authors contributions (in order to justify the relevance of the many-body problem in the introduction they cite their own review for example).

Summarising, I think that in the present form the anthology misses its objectives to be a good review or a set of lecture notes. It is highly biased on the authors contributions ignoring the most relevant paper in the field and not mentioning explicitly what are the basic concepts and where the concepts that are being explained have been introduced originally. The fact that the authors have invested many hours of work in writing such an anthology does not justify the lack of accuracy in their citation policy nor their lack of comparison with existing methods/works.

1- Mention explicitly the original contributions, sort the references on similar topics by date and relevance. Add the key contributions to each of the field addressed.
2- Mention explicitly in the text what was obtained in each of the key references and explain what the current reformulation adds.
3- Compare the optimization methods described with those already available, discuss the differences (if any) and advantages.