## SciPost Submission Page

# Tensor network representations from the geometry of entangled states

### by Matthias Christandl, Angelo Lucia, Péter Vrana, Albert H. Werner

### Submission summary

As Contributors: | Angelo Lucia · Albert H. Werner |

Arxiv Link: | https://arxiv.org/abs/1809.08185v2 (pdf) |

Date submitted: | 2020-06-09 |

Submitted by: | Werner, Albert H. |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Quantum Physics |

Approach: | Theoretical |

### Abstract

Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 2 on 2020-7-10 Invited Report

### Strengths

1. nontrivial application of some of the iconic results of algebraic geometry to tensor networks

2. paper is written in a very nice way, also understandable to people working on tensor networks but not experts in algebraic geometry

### Weaknesses

1. not so clear whether this leads to any practical gain in actual algorithms

### Report

One of the more fascinating concepts in the mathematical treatment of tensors is the notion of border rank - for the purpose of this paper, the authors demonstrate that this means that certain tensors can be well approximated with tensors with a smaller rank. The rank of the tensors used in tensor network algorithms is the main factor in the cost of contracting them, so this is possibly a very valuable insight. The authors give some interesting examples, such as the description of RVB states on the Kagome lattice, which is certainly of interest to the tensor network community.

This is certainly a very interesting and original contribution to the field, and may potentially be very useful in the numerical treatment of tensor networks. However, I think that the main value of the paper is the fact that it opens up a dialogue between two different fields.

### Anonymous Report 1 on 2020-7-9 Invited Report

### Strengths

1. The authors describe a relevant physics application for ideas from algebraic complexity theory, and show how theoretical advances in that field are relevant in the context of tensor network representations of quantum states.

2. Some bounds on conversions and degenerations of multi-party states that are interesting in their own right.

2. Very explicitly and carefully explained, working out an interesting example in full detail.

### Weaknesses

1. From this work it is not yet very clear (at least to this reviewer) how large the practical computational benefits could be in general (one explicit example is worked out in detail), and in what physical classes of states one may expect an advantage.

2. In this work it is not explained how a variational method could be implemented that actually benefits from these representations. However, this has been addressed very recently by a subset of the authors in https://arxiv.org/abs/2006.16963.

### Report

This is a very nice and original work. It works out the idea to consider states 'at the boundary' of the set of fixed bond dimension tensor network states. It translates concepts and results from the field of algebraic complexity to be relevant for tensor network states. A specific example is worked out rigorously. The paper is well written and satisfies all acceptance criteria. The results are fairly technical and, even though carefully explained, especially the plaquette conversion proofs may be hard to understand for the general physics reader. However, the possibility of introducing these ideas and techniques to a physics audience merits the publication in a physics journal.

### Requested changes

1. A few small typos: pg 10, line 2 "representable with by" -> "representable by", pg 14, last paragraph of section 4 "back to RVB state" -> "back to the RVB state", pg 19, below 3rd displayed equation a , should be omitted and logs -> log, pg 25 line 2 of section 6 "provide" -> "provides".

2. The discussion in section 5.2 was hard to understand (at least to this reviewer). Perhaps the structure of the argument can be made a bit more clear?

3. Something I wondered while reading is whether the construction in Theorem 14 is optimal in any sense, or whether for combining multiple plaquettes there could be better degenerations than the tensor product degeneration of single plaquette degenerations. If this question makes sense, perhaps the authors could make a remark on the possibilities for this.