Efficient higher-order matrix product operators for time evolution

The manuscript presents new ideas on how to approximate the time evolution operator of a quantum many-body Hamiltonian using tensor networks. Building on previous works, the authors identify a systematic way of constructing compact MPO representations of the mentioned operator that are exact up to a given order. Furthermore, by cleverly grouping the terms included in the expansion they construct an algorithm that is suitable to work on the thermodynamic limit.

While the subject itself is technical, the paper is well-writen, contains practical benchmarks and is of interest to the computational physics community, so I recommend its publication.

1- It would be interesting to compare the computational times needed to obtain a given fidelity in Fig. 2 with the scheme presented in this paper to present methods in the TN literature. In the introduction, the authors argue that the methods that rely on integrating Schrödinger's equation using small time steps require investing many CPU hours. The authors could compare the times needed with the scheme presented here to the times that would be needed with TDVP, as they use that method to obtain the benchmark state.

2- Similarly, the authors could add a line with the (analytically or numerically) exact results in Fig. 3 to increase the clarity of the presentation.

-provides a novel efficient algorithm for representing powers of operators as compact MPOs

In spite of significant progress over the past few years, the study of the time evolution of quantum systems using matrix product states still holds many formidable challenges. An important element in this context is the search of efficient representations of time-evolution Hamiltonians in terms of matrix product operators, a task which could in principle become extremely costly if one wishes to go beyond the lowest-order algorithms.

Through some careful inspection of the block structure of the resulting MPOs, the authors manage to obtain a recipe for systematically codifying higher orders of powers of generic model Hamiltonians in an efficient way, allowing for higher-order calculations of the time evolution operators ~exp(tH).
The key of this algorithm is a clever grouping of terms, collecting iteratively non-overlapping (or, when going to higher orders, with limited overlap) operators to each order, which can then be encoded efficiently.

Interestingly enough, the authors find that for a concrete example they consider (the transverse-field Ising model on a cylinder), the second-order expansion turns out to be equivalent to the third-order one. While this is somewhat of a nice surprise, it seems to be in some way at odds with the authors’s statement that evidence from “numerical” compression, obtained by performing singular-value decompositions of the MPO tensors, suggests that this encoding is the most efficient possible: in this case, it seems that the third-order prescription would not really be the most efficient one.
I understand that this feature, as the authors clearly state, seems to be model-specific, I simply wonder if this property is shared by a family of models/geometries, and if so, whether for this kind of family an even more efficient representation could be derived.

It is also not entirely clear to me from the presentation in Fig.1 what is the individual effect of the “approximate” compression presented in Sec.6: the authors claim that the result is more accurate with less bond dimension required, but from what I understand, the results presented combine also another resummation together with it, so it’s hard for me to tell what is the individual effect of this approximation - perhaps the authors could comment a bit more on that.

Another minor point is that I don’t understand the meaning of the open light blue circles in the bottom-left corner of Fig.1, which seem to deviate completely from the other trends.

Aside from these remarks, I think the paper is well-written and of interest to the community, so I recommend its publication.

-add a few comments on the issues discussed in the report