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QGOpt: Riemannian optimization for quantum technologies

by I. A. Luchnikov, A. Ryzhov, S. N. Filippov, H. Ouerdane

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Ilia Luchnikov · Henni Ouerdane · Alexander Ryzhov
Submission information
Preprint Link:  (pdf)
Code repository:
Date accepted: 2021-03-24
Date submitted: 2021-02-05 13:30
Submitted by: Ouerdane, Henni
Submitted to: SciPost Physics Codebases
Ontological classification
Academic field: Physics
  • Quantum Physics
Approaches: Theoretical, Computational


Many theoretical problems in quantum technology can be formulated and addressed as constrained optimization problems. The most common quantum mechanical constraints such as, e.g., orthogonality of isometric and unitary matrices, CPTP property of quantum channels, and conditions on density matrices, can be seen as quotient or embedded Riemannian manifolds. This allows to use Riemannian optimization techniques for solving quantum-mechanical constrained optimization problems. In the present work, we introduce QGOpt, the library for constrained optimization in quantum technology. QGOpt relies on the underlying Riemannian structure of quantum-mechanical constraints and permits application of standard gradient based optimization methods while preserving quantum mechanical constraints. Moreover, QGOpt is written on top of TensorFlow, which enables automatic differentiation to calculate necessary gradients for optimization. We show two application examples: quantum gate decomposition and quantum tomography.

Author comments upon resubmission

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.


Henni Ouerdane, on behalf of all coauthors.

List of changes

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.


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.

Published as SciPost Phys. 10, 079 (2021)

Reports on this Submission

Report 3 by Michael Goerz on 2021-2-21 (Invited Report)

  • Cite as: Michael Goerz, Report on arXiv:2011.01894v2, delivered 2021-02-21, doi: 10.21468/SciPost.Report.2592


In their latest revision, the authors have sufficiently addressed all of my concerns. In terms of content and organization, I have no further comments and fully support publication of the manuscript in its current form.

A fair number of stylistic/grammatical/punctuation problems still exist and should be edited. I am attaching an annotated pdf with some suggestions.

Some points that stood out:

* For Eq. (12), I strongly recommend using the same notation that the first paragraph of Boumal's book's introduction (Ref [34]) defines.
* You usually don't have to write something like "in the following way" before an equation, e.g., Eq. (10), or a code example.
* Unlike with equations, I don't think you can put punctuation in code lines. Specifically with commas (second code block on p. 14), it changes the semantics: opt is not a tuple! The periods would be syntax errors, so at least they don't introduce a bug. I've struggled with this problem myself, but I can't think of any good solution apart from just omitting the punctuation, or using floats for longer code blocks.
* The typesetting of the newly added tables can be much improved: Generally, you shouldn't use vertical lines or horizontal lines between each row. See or for guidelines on typesetting tables. I might make an exception for Table 1 due to the amount of text. However, Table 1 should probably be rotated and made wider to fill the entire page in landscape. Tables 3-4 should use the full page width and definitely omit all vertical and most horizontal lines. If a table does not need the full page width, it should be centered on the page.


  • validity: -
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Author:  Henni Ouerdane  on 2021-03-02  [id 1275]

(in reply to Report 3 by Michael Goerz on 2021-02-21)

Dear Dr. Goerz,

Heartfelt thanks for your suggestions for improvement of the text, and the very useful annotated pdf you attached to your report. Thank you also for your positive recommendation.

Best regards,

H. Ouerdane

Anonymous Report 2 on 2021-2-19 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2011.01894v2, delivered 2021-02-19, doi: 10.21468/SciPost.Report.2588


The main remarks of my previous report have been taken into account.
I am not sure why the authors did not feel like accepting my advise not to write \mapsto to denote a function (see, e.g., line 7 of p.4 in the revised version). This is a plain misuse of mathematical notation, but I agree that your readers will understand anyhow.

  • validity: -
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Author:  Henni Ouerdane  on 2021-02-19  [id 1254]

(in reply to Report 2 on 2021-02-19)

Dear Reviewer,

You are correct. We overlooked the \mapsto issue as we had kept the correction of the minor mistakes and typos as the final task before resubmission, and we were far more concerned with all the new material added to the appendix and in the text. We noticed it after resubmission and decided to do the correction at the proof stage in case the manuscript is accepted for publication. Again, we thank you for your detailed review of our manuscript, which ensures that the it does not contain such a kind of minor mistakes.


H. Ouerdane

Anonymous Report 1 on 2021-2-7 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2011.01894v2, delivered 2021-02-07, doi: 10.21468/SciPost.Report.2516


Dear Editor,
since all the required changes have been made, I support the publication of this paper.
Best regards

  • validity: good
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: good

Author:  Henni Ouerdane  on 2021-03-02  [id 1276]

(in reply to Report 1 on 2021-02-07)

Dear Reviewer,

Thank you for your work, and your positive recommendation.

Best regards,

H. Ouerdane

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