Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: In this paper the authors analyze the single qubit neural networks by using the re-uploading QNN formalism. The contributions of the paper are the proofs that a single qubit neural network, simply a single qubit quantum circuits with half of parameterizable gates can approximate any single bit uni-variate function. In additions the authors provide analysis of the re-uploading networks for simulating multi-variate functions. The authors show that the re-uploading networks are not universal in approximating multi-variate functions. 
strengths_and_weaknesses: The paper analysis of single qubit networks in the re-uploading format, limited by the Fourier-transform blocks is a nice extension to the original re-uploading NN.The proof of the univariate function approximation is a simpler form of the universal circuit Solovay-Kitaev  approximation theorem. The principle of the proof starting from eq.4 can be simplified by assuming that if the target function is expressed simply by V(\theta_0) then the sum over S(x)V(\theta_j) = I. That is each of the encoding block can be nilled by finding exactly the inverse transform by learning. For the multi-variate approximation, it is true that the studied setting is not universal. 

While both of the studied problems are correctly analyzed I do wonder if this is really related to QNN. In particular, now when most of the algorithms are being built as VQE I wonder if this is the correct venue. 
questions: What is a reasonable approximation of multi-variate function as to the dimension d in Omega and error e?
limitations: No ethical concerns have been determined
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper aims to explore the data re-uploading expressive ability of quantum neural networks (QNNs) by providing rigorous theoretical proofs for native single-qubit QNNs. They point out that native QNNs demonstrate the expressivity of the quantum part of hybrid QNNs, decoupled from the classical part of hybrid QNNs. In a series of proofs, they show that single-qubit native QNNs can approximate any univariate square-integrable function arbitrarily close by an exact mapping to the partial Fourier series. They then discuss how the native QNNs lack the expressivity in approximating multivariate functions and hybrid single qubit QNNs can approximate arbitrary multivariate functions with the help of classical structures. They further show that increasing the number of QNN laybers decreases the mean squared error for approximating a function.
strengths_and_weaknesses: The paper provides rigorous proofs that build up the foundational work for native single-qubit QNNs in a purely quantum context. It presents, in a logical and consistent manner, how univariate square-integrable functions can be loaded into single-qubit QNNs up to arbitrary precision using the partial Fourier series. The numerical experiments show clear evidence for the correlation between layer number and loss convergence, and the inclusion of the square wave is excellent as it requires a large number of fourier terms to approximate. It also does a good job explaining with theory and demonstrating with an experiment how multivariate functions can not be approximated well with single qubit QNNs.

The paper talks about the limitations of single-qubit QNNs and the importance of “investigating QNNs with universal approximation properties for multivariate functions”. It also provides a potential design for multi-qubit QNNs in Appendix C. However, it is not immediately clear how the results of this paper, especially theoretical proofs which are exclusively on single-qubit QNNs, can be generalized for future studies of multivariate function approximation in QNNs.

questions: It is mentioned that hybrid QNNs provide no quantum advantage over classical neural networks. So do the results of this paper prove, or show a potential of quantum advantage for native QNNs? If not, will a multivariate function approximating native QNN provide quantum advantage?
limitations: The authors have adequately addressed the limitations of native single-qubit QNNs as a trivial case for data loading into QNNs using a partial fourier series.
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: The paper proves that single-qubit quantum neural networks can approximate any univariate function. Both the expressivity and limitation of single-qubit QNNs are validated by numerical simulations involving also multivariate functions. 
strengths_and_weaknesses: Strength: a controlled setup to study the approximation capabilities of a single-qubit QNN

Weakness: the setup seems really too simple to capture any of the complexity of representing multi variate functions and, especially, the benefits of using entangling gates. This single qubit analysis is not really illuminating in this sense. The power of the universal approximation theorem in the classical case is that it can be extended to prove an analogous result for multi variate functions, instead in this case it is not clear how and if the proof carried in this work can be used in the more general (and interesting) case. 
questions: What would happen for the multi qubit case? Can you establish a representation theorem there? 
limitations: yes
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: The authors show that a data reuploading QNN with only $R_Z(x)$ encoding gates and $R_Y(\cdot)$ as trainable gates can be represented as a truncated Fourier series with real coefficients. Thus QNNs of this form can approximate even functions arbitrarily well. The authors extend the QNN model with trainable gates of the from $W(\theta_j, \phi_j):= R_Y(\theta_j) R_Z(\phi_j)$ to represent complex Fourier coefficients. Using established results from Fourier analysis the authors can show that the extended QNN model is a universal function approximator in the class of square-integrable functions. Within the Fourie series framework the authors provides arguments for the inadequacy of their QNN ansatz to universally approximate multivariate functions. Numerical experiments were performed to show the increasing approximation capability with increasing number of layers of the QNN model and also the extrapolation ability of the QNN was demonstrated for a simple periodic function. In the end the single qubit QNN is tested on a bivariate function, where the performance is as expected not optimal.

strengths_and_weaknesses: Pro
- the foundational presentation of the topic was clearly structured

Cons:
- the universal approximations theorem for QNNs was already proven in M.Schuld et 2021
- the concept of data reuploading was also already established in A.Perez-Salinas et al 2020
- thus the originality of the paper is not clear 
- though the prior work was cited, the results were not discussed and were not set into perspective with this work
questions: - What is the difference between Lemma 1 and 3 except for the additionals parameters?
- Why do you use only 200+100 datapoints for training/testing of your 1000 point dataset?
- Why do you use the damping function? Have you considered a more diverse set of functions?
- will the code be available to reproduce the findings?

limitations: The authors adequately show the limitation of their model in the bivariate case. No societal impact was discussed.
review_title: null

Meta-review: metareview: This submission is borderline.  Reviewers generally agreed that its theoretical contribution is sound and non-trivial, but at the same time pointed to a major limitation,, which is that the theory applies only to single Qubit QNNs, thus disregards the aspect of entanglement, which is one of the most important factors in quantum computational systems.  This point is clearly acknowledged by the text (including the title), so I do not think much more can be done on the authors' part.  I recommend that the paper be accepted, while highly encouraging the authors to at least provide some directions as to how their theory might be relevant to the multi Qubit case.
confidence: Less certain
Decision: Accept