TensorMixedStates: A Julia library for simulating pure and mixed quantum states using matrix product states

1. TMS package presented in this manuscript is based on Julia’s ITensor, which allows applying the existing optimized routines and solutions to the case of open quantum many-body systems.
2. The application examples are covering a wide set of physical systems and problems, showing the versatility of the developed instruments; the benchmarking tests are exceedingly provided.
3. The package addresses a demonstrable need for the community of open quantum many-body systems; the added value as compared to existing software is clear.
4. The documentation includes detailed instructions on downloading, installing and running the software. The userguide contextualizes the software, describes the logic of its workings.

1. Some parts of the text are written poorly with lots of jargon words, unclear formulations and grammar mistakes. The commented file contains the comments with improvement guidelines.
2. The introduction to the tensor network techniques for open quantum system is rather brief and unclear. In the dedicated part of the commented file the authors can find the suggestions of improvements. I would also additionally suggest to include tensor-network diagrams to improve the reader's understanding.
3. Some of the physics examples are there to merely demonstrate the scope of the TMS package. It would be much more interesting to include in-depth discussions on the limitations of the developed methods. In particular, it would be great to investigate the deviation of the numerics in the free fermion model (Fig. 4), in particular, studying the later dynamics and observe further deviations there (please see a comment to the corresponding figure content).
4. Some of the figures (in particular, Figs. 1 and 2) contain legends that interfere with the plot content. This should be reconsidered.
5. The steady-state solver is discussed very briefly in the manuscript, however it is not mentioned in the abstract. It is also demonstrated only for a single physics problem (Fig. 2). It leaves the reader puzzled whether this solver is implemented in the TMS package.
6. The quantum number conservation in mentioned in the abstract as if it is implemented in the package. However, in the conclusion to the manuscript the authors indicate this feature as the subject of the future work. This point should be revisited by the authors.

Please see the attached file with highlighted parts of the text and comments accompanying them. The comments contain the necessary changes as well as clarification requests to improve the quality of the submitted manuscript.

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1) Broadly, there should be some discussion addressing the details of how the TMS specific functions are implemented and optimized. The authors mention in the introduction that the novel functionality introduced by TMS are functions that create MPOs for the operations required in open systems such as Tr($O\rho$), $O \rho O^{\dagger}$, $[O, \rho]$, ${ O^{\dagger} O, \rho }$. However, the authors do not seem to further elaborate on how these functions are designed and optimized. In addition, it is not always clear which of the algorithms the authors have developed from scratch, and which are taken from ITensor, and there should be some details on the implementation for the novel ones (for example, TMS.approx_W?) 2) Section 1 (Introduction) should include a more comprehensive list of references supporting the statements (for instance references to relevant reviews in the fields mentioned, like cold atoms, trapped ions, development in quantum technologies, why is dissipation and decoherence often important). 3) At the end of Section 1, the reader would benefit from a brief content overview with a description of the organisation of the paper. 4) In section 2.2, the notation of equation (11) should be further clarified: what do the authors mean by "$T_i$ is a matrix whose elements are states of site $i$"? It would be useful to first define $\ket{\psi}$ as a sum of complex coefficients multiplying states, and then link the "simplified" notation of equation (11) to the usual/not simplified notation, where it becomes clear what are the indices of $T_i$ and how they are linked to the wave function. In general, the notation of this section could be more accurate. 5) Section 3.4: is it possible to specify states using generic linear combinations with complex coefficients? For example, to specify the state $a \ket{0} + b \ket{1}$ of a Spin(), with $a, b$ complex. For a system with 10 qubits (for example sys1), how can I specify different states for all the 10 qubits? It would be useful to expand the examples in this section to clarify these cases, possibly with an explanation of the function syntax in the general case. 6) In 3.5, it would be useful to specify that the dagger of an operator A can be obtained with dag(A)
7) In 3.5, for the example of $R_{xy}$, is there a simple way to type the tensor product symbol in Julia? If one should call a specific function, there should be the function in the code example. In general, this section could be more systematic, by explaining the syntax at a general level and then with examples (for the case of Hamiltonians, for example first write the equation for the Hamiltonian in a generic way), rather than by examples only, which can leave out more special cases. 8) The notion of OSEE should be defined in the introductory sections 9) Figure 2 should be made colour blind compatible 10) Would it be possible to show the data of Fig. 4 in Ref 37 in Fig. 2 as well? (I am not sure this can be done, if not, please ignore the comment) 11) Can the code snippets contained in the example sections be expanded? Alternatively, would it be possible to make the source code of the examples accessible from the documentation, with some text explaining the context and guiding the user through the implementation? I think it would be beneficial to be able to visualize in one place the code and the text explanation at the same time, to understand the details of how to write a comprehensive code running an explicit example.

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