QGOpt: Riemannian optimization for quantum technologies

Dear Editor,

Thank you for giving us the opportunity to submit a revised manuscript for further consideration.
We have read the three reports carefully and with interest. We thank the Referees for their useful
and encouraging reviews that provided actual scope for improvement of our manuscript and QGOpt,
and provided also an engaging discussion. In our reply to the reports (see the SciPost Reply to Report Page)
we answer each of the points raised by the Referees, and we give a summary of the changes made to our manuscript.

Sincerely,

Henni Ouerdane, on behalf of all coauthors.

Requested changes:

1. Specify a package version to which the manuscript pertains. I noticed there has been unreleased development
on the master branch. If you want to follow "semantic versioning", you may want to declare the
package API "stable", release a 1.0, and write the manuscript specifically for that.

The package version is 1.0.0, and it is specified in the subsection 4.1 Manifolds API, page 8 of the revised manuscript.

2. The package needs to have *automatic* testing. The example notebooks are very nice, but not automatic
(ensuring that future development will not break the package, or make it easy for new contributors to
add code). I noticed that the repository includes some test code. I would recommend to set up continuous
integration (Github Actions) to run the tests automatically. Also, the documentation must include some
guidelines for contributors, most importantly how to run the tests manually.

All manifolds in the QGOpt 1.0.0 library are automatically tested. We also added automatic tests for the optimizers
for the example of the optimization problem on the Stiefel manifold.

3. Consider adding tutorials for *all* the supported manifolds. This is somewhat optional, and could be
done after publication. On the other hand, it might uncover bugs/problems that you’d want to fix before
publication or a 1.0 release. Rename "Quick Start" to "Quick Start: Quantum Gate decomposition".

We plan to include in the near future tutorials for the manifold of Hermitian matrices and the manifold of
complex positive definite matrices. At present, we offer a "Quick start tutorial" on the complex Stiefel manifold.
Four additional tutorials are provided: complex Stiefel manifold, density matrices of fixed rank manifold, Choi matrices
of fixed rank manifold, and POVMs manifold.

4. Include a discussion of the scaling of numerical cost, and briefly compare to second-order gradient optimization.

This discussion is the object of the new Appendix B; however, at present we could not find a fast way to compare the
first-order Riemannian optimization methods of QGOpt with second-order Riemannian optimization methods.
Indeed, second-order methods are not yet developed for the manifolds considered by QGOpt, and this would require
extensive work to develop such methods.

5. Correct the typos, and implement the change of mathematical notation pointed out in the reports.

Done.

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List of major additional changes:

1. The Introduction is revised.
2. Section 2 has been extended.
3. The new Section 6 is added.
4. Appendix A and Appendix B are added.