Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 3: reject, not good enough
review_content: summary_of_the_paper: The authors present two algorithms for quantum optimization and machine learning based on the Koopman operator. They test their algorithm with simulations and a small hardware experiment.
strength_and_weaknesses: + the Koopam operator optimization method may turn out to be interesting for learning
- there is hardly any strong evidence in this work of the performance of the methods
- the experiments are trivial (3 qubits) and no conclusions can be drawn from them
- no theoretical reasons why this methods should outperform other gradient-free optimization methods are given
clarity,_quality,_novelty_and_reproducibility: The paper is clear and well written. The ide is a simple extension of classical work to the quantum case. The results are reproducible
summary_of_the_review: The paper has an interesting idea of using Koopman operator theory as a way to avoid gradient calculations in training quantum neural networks. More work and evidence is needed for the impact of this methods
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The author has discussed the issue of backpropagation and obtaining the gradient of a quantum computer. The gradient computation during backpropagation in a quantum computer is not similar to a classical computer because the complexity increase with the number of parameters and measurements. The author discusses the Koopman operator theory for predicting nonlinear dynamics with the natural gradient method in quantum optimization to address this issue. The author proposed a data-driven approach using the Koopman operator to accelerate quantum optimization and Quantum machine learning. The author proposed Sliding window dynamic mode decomposition (DMD) and Neural DMD for efficiently updating parameters in quantum computers. The author shows that the method can predict gradient dynamics and accelerate the quantum variational eigensolver in quantum optimization and quantum ML.  
 

strength_and_weaknesses: Strength:

The paper discusses an essential problem in the literature.
  
Also, the author provides an excellent solution to tackle the problem, but the analysis and intuition part is very poorly represented in this paper.
  
The paper is easy to follow and understand. The authors introduce several quantum machine learning strategies, advantages, and weaknesses. Based on this, the motivation of this paper is reasonable.


Weaknesses:

The author has discussed accelerating quantum mechanics and quantum optimization algorithms compared to classical operators. Did the author quantify or qualify this using some experiment? 

How did the Koopman operator help to accelerate the gradient calculation in quantum computers compared to classical computers? Can the author provide a brief explanation for this?

In section5.2, the author has discussed the sequence of the quantum gate. Is this similar to layers? Can the author discuss the objective of each block in Figure 4a (like the objective of block "ansatz" and its input and output)? Similarly need to discuss other blocks. Can the author describe all the blocks and their functionality in the figure-1 for MNIST dataset example (Like, for an image sample)? The author should describe the complete algorithm and discuss which part is classic machine learning and which is quantum machine learning.

Along with MNIST, the author also discuss Figure 1 with the time series task. Did the author report any comparative results on time series?

The author should report the performance and analysis of results on the MNIST dataset.

The author can add a small preliminary section that describes the steps for the quantum ML algorithm (Training and Inference)  

The experiment section is not designed correctly. The author should compare the results with SOTA and do an ablation analysis on the Koopman operation. The analysis section of the paper needs to improve with more analysis results.

Take home: Koopman operator  for stability and robustness and 
MNIST on QML

clarity,_quality,_novelty_and_reproducibility: Please refer to the weaknesses section.
summary_of_the_review: The overall writing quality is ok, and the proposed method is simple and beneficial. However, the experiment comparison lacks justification, and the technical contribution is plain.
review_title: null

final_score: rating: 3: reject, not good enough
review_content: summary_of_the_paper: This paper proposes to model the variational quantum eigensolver optimization as a dynamical system, and approximate its iterative updates via Koopman-based approaches. In particular, the authors propose a sliding window DMD method, and a neural DMD approach. The authors evaluate their approach on quantum machine learning tasks, and compare it to standard DMD. 
strength_and_weaknesses: In my opinion, the strongest point of the paper is the application domain of quantum machine learning (QML) and the incorporation of Koopman-based methodologies in it. Having said that, it should be noted that the technical problem solved in this paper is essentially the ability to approximate complex dynamics with Koopman methods, and the relation to QML is only by application.

Unfortunately, there are several weak points. First, the same observation and application was already considered in Dogra et al. (and other papers, some of which cited in this paper). In this context, the main difference between this paper and the paper of Dogra et al. is in the evaluation on QML tasks vs. vision tasks. While this difference may be significant in certain applications and algorithms, I do not believe this is the case here, as they essentially optimize a standard neural network, similar to those considered in Dogra et al.

Second, the proposed approaches to approximate the Koopman operator generally ignore many similar if not identical ideas in the literature. Both sliding windows with time delay coordinates and neural DMD have been heavily studied before. Properly discussing these approaches and positioning the proposed method in this context is essential.
clarity,_quality,_novelty_and_reproducibility: Overall, the paper is very clear and can be probably reproduced. Based on the discussion above, I believe the novelty of the paper is somewhat limited.
summary_of_the_review: In summary, while the application domain of this paper is interesting and refreshing, it is essentially a different application of an existing paper. Moreover, the proposed methods are similar or equivalent to existing work in the field. I suggest the authors to conduct a proper literature search in Koopman and DMD communities (see e.g., Hankel DMD and Koopman Autoencoder), and to position their contribution accordingly. Due to these reasons, I think the paper should not be published in its current form.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The paper propose using Koopman operator to speed up the optimization of hybrid quantum computing algorithm. The motivation is gradient step in quantum computer takes many forward evaluations (linear in number of parameters). Using Koopman operator I the classical computing sense, to predict future optimal quantum machine learning parameters. then pick the best, optimal time, and then do more predictions from there.

strength_and_weaknesses: The motivation of the paper is good. It is one plausible solution to an important problem. The method of Koopman operator learning using sliding window and neural network to learn dictionary, can potentially be use for optimising other classical computing algorithm. its advantage is most in quantum computing.

I have not thought through thoroughly if using the sliding window approach is theoretically sound. Certainly did not sit down, write the required equations and convince myself this is a theoretically sound approach. Loss function of Eq. (8) (or similar) can be constructed in many ways, sliding window way etc, seems arbitrary to me. I understand several other works in the area of Koopman operator also uses similar approach of constructing some measurable / observable and then do optimization. In the limit of infinite observables, one can agree that the trained Koopman operator can represent the underlying dynamics accurately. There is also an issue of how to invert the "g" function to get back the underlying "x" (Eq. (1)). In this paper, it seems g is the identity function.



clarity,_quality,_novelty_and_reproducibility: Clarity of the paper is good.

The main novelty should be finding a good application in quantum computing. Sliding window method offers some novelty, but to my ignorant, I do not know of any other similar approach.

summary_of_the_review: Good motivation for this work. Not absolutely sure if the equations are theoretically sound due to my ignorant.
My sense is this paper is in between (6) Marginally above acceptance threshold and (8) accept, good paper.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: 
Summary:

The paper discusses backpropagation and gradient computation in a quantum computer. The gradient computation in a quantum computer has a complexity that increases with the number of parameters and measurements. The paper discusses the Koopman operator theory for predicting nonlinear dynamics with the natural gradient method in quantum optimization to address this issue. The paper proposes a data-driven approach using the Koopman operator to accelerate quantum optimization and Quantum machine learning. The paper shows that the method can predict gradient dynamics and accelerate the quantum variational eigensolver in quantum optimization and quantum ML.

Strengths:

- the Koopam operator optimization method may turn out to be interesting for learning
- the paper discusses an essential problem in the literature.
- the author provides an excellent solution to tackle the problem, but the analysis and intuition part is very poorly represented in this paper.
- The paper is easy to follow and understand. The authors introduce several quantum machine learning strategies, advantages, and weaknesses. Based on this, the motivation of this paper is reasonable.
- the application domain of quantum machine learning (QML) and the incorporation of Koopman-based methodologies in it

Weaknesses:

- there is hardly any strong evidence in this work of the performance of the methods
- the experiments are trivial (3 qubits) and no conclusions can be drawn from them
- no theoretical reasons why this methods should outperform other gradient-free optimization methods are given
- the technical problem solved in this paper is essentially the ability to approximate complex dynamics with Koopman methods, and the relation to QML is only by application
- the same observation and application was already considered in Dogra et al.
- the proposed approaches to approximate the Koopman operator generally ignore many similar if not identical ideas in the literature

Recommendation:

Some reviewers vote slightly toward acceptance and others vote strongly for rejection. The reviewers with the most confident decisions vote for rejection. In a comment submitted during the time to write the meta-review, the authors point out concerns regarding the reviews from reviewers 5VYL and TgmW, indicating that they have misunderstood the theoretical and algorithmic contributions of the paper. If such were the case, that would indicate that the paper would still need more work to clearly present such theoretical and algorithmic contributions in a better way. Based on all this, I have decided to recommend rejection and encourage the authors to use the feedback provided to improve the paper and resubmit to another venue.
justification_for_why_not_higher_score: Two reviewers vote strongly for rejection with high confidence.
Decision: Reject