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State Diagrams to determine Tree Tensor Network Operators
by Richard M. Milbradt, Qunsheng Huang, Christian B. Mendl
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Submission summary
Authors (as registered SciPost users):  Richard Milbradt 
Submission information  

Preprint Link:  https://arxiv.org/abs/2311.13433v3 (pdf) 
Code repository:  https://github.com/Drachier/state_diagrams_for_ttno 
Date submitted:  20240307 17:09 
Submitted by:  Milbradt, Richard 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
This work is concerned with tree tensor network operators (TTNOs) for representing quantum Hamiltonians. We first establish a mathematical framework connecting tree topologies with state diagrams. Based on these, we devise an algorithm for constructing a TTNO given a Hamiltonian. The algorithm exploits the tensor product structure of the Hamiltonian to add paths to a state diagram, while combining local operators if possible. We test the capabilities of our algorithm on random Hamiltonians for a given tree structure. Additionally, we construct explicit TTNOs for nearest neighbour interactions on a tree topology. Furthermore, we derive a bound on the bond dimension of tensor operators representing arbitrary interactions on trees. Finally, we consider an open quantum system in the form of a Heisenberg spin chain coupled to bosonic bath sites as a concrete example. We find that tree structures allow for lower bond dimensions of the Hamiltonian tensor network representation compared to a matrix product operator structure. This reduction is large enough to reduce the number of total tensor elements required as soon as the number of baths per spin reaches 3.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2024517 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2311.13433v3, delivered 20240517, doi: 10.21468/SciPost.Report.9069
Strengths
1 This paper describes a general method to form almostoptimal treetensornetwork operators (TTNOs) for a general treetensor network (TTN). TTNs, which are generalizations of matrixproduct states, are useful variational ansatz states for a variety of strongly correlated quantum problems including, but not limited to, quantum impurity problems (Kondo, Anderson, DMFT), postHartreeFock quantum chemistry, and open quantum systems. It is therefore presents a conceptual and methodic advance in TTN technology, in particular, making it possible to use TTNOs in a general and efficient way.
2 The method, adapting state machines to the geometry and topology of TTNs so that general TTNOs that can also be longranged can be efficiently treated, is original and clever.
3 The paper is carefully and clearly written, and the method is described in detail, in words, in pseudocode, and in state diagrams. As such, it makes both the conceptual aspects of the method and the implementation aspects accessible to the reader.
Weaknesses
1 While the paper describes a methodic advance and indicates areas of application in which the use of efficient TTNOs will be useful, in particular, in the area of open quantum systems, no physical results are presented in the paper. The example treated for open quantum systems is the spin1/2 Heisenberg chain coupled to a bath modelled by bosonic sites. While this is an interesting test system, it is not necessarily a system of particulary high experimental and physical interest. In addition, the model is only used as example to explore the efficacy of the TTNO representation; a calculation of the physical properties of the model, not even to show what the method is capable of, is not carried out.
Report
The general acceptance criteria for SciPost articles are certainly met by this paper. The SciPost Corespecific expectations are that the work:
1 Address an important (set of) problem(s) in the field using appropriate methods with an abovethenorm degree of originality.
I find that the authors address the somewhat technical problem of how best to form TTNOs for TTNs in an original, insightful and general way. In doing so, they make an import technical advance for the set of TTNbased methods. Tensornetwork methods in general (in particular, MPS and iPEPSbased method) are a highly useful and fast developing set of methods. TTN methods are an interesting and variant of TNS methods, but not do not (at least yet) have the same degree of usefulness and wide range of applicability as other TNS methods.
2. Detail one or more new research results significantly advancing current knowledge and understanding of the field.
The new research results are of a methodic, technical nature. A such, they do make a significant advance in one technical aspect of a specific kind of tensor network. However, as detailed above under weaknesses, this work does not contain any new physical results; the authors make clear that this is not their intention.
Requested changes
1 I find that the notation in Eqs. (17), (18), and (19) is not completely defined. The authors should define all symbols and notation that are used.
2 The paper should be proofread for punctuation, especially commas and hyphenation.
Recommendation
Ask for minor revision