Quantum many-body simulations with PauliStrings.jl

1. Efficient representation of operators that are sums of pauli strings
2. Clear presentation of benchmarks
3. Clear documentation of the package with several examples.

Notation in section II is a bit sloppy and partially unclear.

The manuscript presents the software library PauliStrings.jl which implements an efficient binary representation of opeartors which are sums of Pauli strings. They show clear benchmarks and demonstrations of its capabilities, comparing to MPS/MPO time evolution using ITensor both in terms of values and runtime.
With Fig. 4, they also clearly show that the sparsity of the Pauli string representation can help to reach significantly longer times with good precision especially in 2D, making the library a useful tool for exploring operator/hydro-dynamics at least at infinite temperatures. (Whether the sparsity is still preserved at finite temperatures, as suggested as a possible application in the conclusions, remains to be seen.)

1. The notation and equations in section II should be double-checked:
- In Equation 7 and 11, you have factors $(-1)^{\alpha_{1,2}}$ which you don't introduce and take into account for fthe phase of eq. 10. Further, to be consistent with $10$, the phase on the right hand side should be just $\alpha_{12}$, not $(-1)^{\alpha_{12}}$
- given that you write the explicit formula for $\alpha_{12}$ in (10), you could also explicitly write $\alpha = (-1)^{i \wedge l }$ just before equation (6).
- Equation 5 is missing a minus sign to be consitent with (7-10) (assuming that drop the phases on the left side of (7))
- Why not use $v_1, w_1, v_2, w_2$ for the bits in equation 6 as well?
As mentioned by the other referee already, there's also a typo in algorithm 1, with a v instead of w.

2. In algorithm 2, it should be made clear what $N$ is - if I'm not mistaken, it should be $min(\mathrm{StringLength}(A.\tau_i), \mathrm{StringLength}(B.\tau_i))$, where $\mathrm{StringLength}$.
Maybe also add a small comment that it is very straightforward to generalize Algorithm 2 to higher dimensions as well by iterating over the necessary shifts (x,y) instead of just x.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1- PauliStrings.jl provides an efficient implementation of operator time evolution based on binary representations of the Pauli group.
2- Efficiency is demonstrated by benchmarking against MPS simulations for the particularly challenging case of dynamics at infinite temperature and against MPO-based simulations for the computation of Lanczos coefficients.
3- The library provides a compact and intuitive interface to define the system and run simulations.

1- Computing a time-dependent two-point correlation function of a spin chain at infinite temperature is considered as a benchmark case. It is demonstrated, that the results qualitatively agree with the expectation of diffusive dynamics. A more quantitative assertion of the quality of the results could have been possible by extracting a diffusion constant for a model, where reference values exist.

2- Parts of the manuscript are hard to read, because they mostly consist of figures and code listings (especially section 5). It would be good to organize the arrangement such that the figures appear as close as possible to their reference in the main text and avoid their interleaving with code listings.

The manuscript describes a Julia library for quantum operator time evolution based on Pauli strings. This approach received quite some interest recently and it has been shown to yield competitive results when compared to other state-of-the-art methods in a number of use cases. The availability of a highly efficient implementation in the form of a Julia library is therefore very valuable for the community.

The manuscript reports two relevant example applications and runtime benchmarks in comparison with tensor network simulations. The source code is clean, well documented, and tested. The library is accompanied by clear documentation. Therefore, the acceptance criteria are met and I recommend publication, once the comments and requested changes below have been considered for a minor revision.

Comments:

- First sentence in second paragraph of "Tensor networks": The sentence mentions MPS and MPO, but only describes the shape of MPS tensors, which is not clear. It would be good to clarify, that the three-leg-tensor is the MPS building block, not MPO.
- In the manuscript, the inner product tr(A^\dagger B)/2^N is called "infinite temperature inner product". I think Frobenius inner product (up to normalization) would be a more appropriate term.

1- In Algorithm 1, there are two subsequent assignments to a variable 'v'. Should one of them be a 'w'? Moreover, the notation
if (u,v) is in C, then
C[(u,v)] <- C[(u,v)] + h
is not particularly clear. For example, above that, the operators A and B seem to store h-values in A.h or B.h member variables. But here the value is directly obtained from and assigned to C. It would be good to clarify this.
2- In Algorithm 2, the notation is chosen different than algorithm 1. For example, in Algorithm 1, double indices appear explicitly as tuples or independent member variables of the operators, whereas they are a single variable in Algorithm 2. Moreover, I assume, that the dot appearing in the argument of 'Shiftleft' is the product function from Algorithm 1, but that's not clear. It would be good to make the notation in the different Algorithms clear and consistent.

Ask for minor revision