Quantum Neural Fields

Thank you for your insightful and valuable comments! We are delighted by your recognition of the technical novelty in combining quantum machine learning (QML) with neural implicit representations. We appreciate your acknowledgement of its significance to the rapidly advancing field of implicit neural representations and its potential to drive progress, particularly in visual computing. Below, we carefully address your questions and concerns.

question 1:

The computational complexity of this method running on a classical simulator is detailed between L320 and L323. When running a quantum circuit on physical hardware, analyzing its complexity is challenging as it depends on factors such as gate fidelity, hardware architecture, gate types and physical implementation etc. The fidelity and noise levels impact accuracy and may require additional corrections, while hardware-specific constraints like qubit connectivity can increase circuit depth. Different quantum systems (e.g., superconducting, trapped ions) impose varying demands on gate execution speed and reliability, and limited coherence times restrict the number of gates that can be executed. These factors make it hard to analyze the complexity of running quantum circuits on physical hardware.

question 2:

The computational resources required by this method depend on factors such as sampling density, model size, and others. In our setup, training the model on 2D images and 3D scenes takes approximately 3 hours and 8 hours, respectively, on an A100 GPU.

question 3:

The primary goal of this work is to explore an efficient model that can be applied to large-scale datasets while leveraging the advantages of our quantum circuit and processing, assuming we have a fault-tolerant quantum hardware. Therefore, we assumed no noise in this study. However, with presence of noise, we would anticipate a significant performance degradation, as demonstrated by other works which primarily investigate the influence of noise on machine learning; see Borras et al. Journal of Physics: Conference Series.Vol.2438.No.1.IOP Publishing, 2023.

question 4:

Given our focus on exploring the potential of these models and their applications to large-scale datasets, particularly dense 3D reconstructions, we have chosen to rely on high-end simulators rather than NISQ hardware. While there have been notable advancements in circuit-based quantum hardware, the availability of devices with a sufficient number of stable, high-quality qubits remains extremely limited. Besides, our attempt to use IBM’s circuit-based quantum computer was hindered by unrealistic queuing time. Despite the clear importance of testing on real quantum hardware, current limitations make classical simulations the most viable option to overcome the constraints of NISQ devices, even for recent research involving smaller datasets; see Rathi et al. "3D-QAE: Fully Quantum Auto-Encoding of 3D Point Clouds." (2023); Baek et al. "3D scalable quantum convolutional neural networks for point cloud data processing in classification applications." arXiv preprint arXiv:2210.09728 (2022).

Thank you for your time and for providing valuable comments. We are encouraged by your acknowledgement of our work as a good model in computer vision and graphics and its compellingness. As we did not find any concrete questions in the review, we carefully address the high-level concerns next.

Concern 1: confusing logic

Thank you for bringing this to our attention. We are somewhat surprised, as this concern has not been raised by other reviewers. Our contributions are clearly outlined in lines 89 to 99. We kindly encourage you to review this section thoroughly.

Concern 2: Insufficient literature review

We have enhanced the related work section by thoroughly discussing and analysing each referenced study. The distinctions between our work and existing research are now more clearly articulated.

Concern 3: unclear and undefined algorithm procedures

The algorithmic protocol (from L287 to L294) serves as a high-level guide for readers. Specifically, we use basic vanilla fully-connected multi-layer perceptrons to test the effectiveness of our hybrid model. The training procedure and loss function are clearly defined in Section 3.3. We kindly encourage you to review this section thoroughly.

Concern 4: Unfriendly paper writing

We have addressed the inconsistencies you mentioned and replaced "circuitry" with "circuit." Regarding the use of "neuro" and "neural," we kindly ask you to consider the different contexts in which each term should be used. Specifically, "neural" is an adjective, while "neuro" is a prefix. Regarding the undefined notations: As the details of the undefined notations were not specified, we assume you are referring to notations related to quantum mechanics. We have provided the necessary foundational information on quantum computing in the supplement, which covers all notations used in this paper. Please note that we assume some familiarity with quantum computing from our reviewers, as we cannot cover basic knowledge in this work. We have updated our work based on your suggestions, with the changes highlighted in red. We would greatly appreciate it if you could take a more detailed look at the paper and provide specific feedback on the proposed ideas, technical novelty, and experimental results, as well as further suggestions for improvement.

References

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[2]Sim, Sukin, Peter D. Johnson, and Alán Aspuru‐Guzik. "Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum‐classical algorithms." Advanced Quantum Technologies 2.12 (2019): 1900070.

[3]Cerezo, Marco, et al. "Challenges and opportunities in quantum machine learning." Nature Computational Science 2.9 (2022): 567-576.

[4]Jeswal, S. K., and S. Chakraverty. "Recent developments and applications in quantum neural network: A review." Archives of Computational Methods in Engineering 26.4 (2019): 793-807.

[5]Abbas, Amira, et al. "The power of quantum neural networks." Nature Computational Science 1.6 (2021): 403-409.

[6]Beer, Kerstin, et al. "Training deep quantum neural networks." Nature communications 11.1 (2020): 808.

[7]McClean, Jarrod R., et al. "Barren plateaus in quantum neural network training landscapes." Nature communications 9.1 (2018): 4812.

[8]Das, Sreetama, and Filippo Caruso. "Permutation-equivariant quantum convolutional neural networks." Quantum Science and Technology (2024).

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[10]Larocca, Martín, et al. "Group-invariant quantum machine learning." PRX Quantum 3.3 (2022): 030341.

[11]Perdomo-Ortiz, Alejandro, et al. "Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers." Quantum Science and Technology 3.3 (2018): 030502.

[12]Houssein, Essam H., et al. "Machine learning in the quantum realm: The state-of-the-art, challenges, and future vision." Expert Systems with Applications 194 (2022): 116512.

[13] Zhang, Xiao-Ming, Tongyang Li, and Xiao Yuan. "Quantum state preparation with optimal circuit depth: Implementations and applications." Physical Review Letters 129.23 (2022): 230504.

[14] Sun, Xiaoming, et al. "Asymptotically optimal circuit depth for quantum state preparation and general unitary synthesis." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 42.10 (2023): 3301-3314.

[15] Cherrat, El Amine, et al. "Quantum vision transformers." Quantum 8.arXiv: 2209.08167 (2024): 1265.

[16] Kerenidis, Iordanis, et al. "Quantum vision transformers." Quantum 8 (2024): 1265.

[17] Unlu, Eyup B., et al. "Hybrid Quantum Vision Transformers for Event Classification in High Energy Physics." Axioms 13.3 (2024): 187.

[18] Rajesh, Varadi, and Umesh Parameshwar Naik. "Quantum convolutional neural networks (QCNN) using deep learning for computer vision applications." 2021 International conference on recent trends on electronics, information, communication & technology (RTEICT). IEEE, 2021.

[19]Biamonte, Jacob, et al. "Quantum machine learning." Nature 549.7671 (2017): 195-202.

[20] Sauvage, Frederic, et al. "Building spatial symmetries into parameterized quantum circuits for faster training." Quantum Science and Technology 9.1 (2024): 015029.

[21] Huang, Hsin-Yuan, et al. "Quantum advantage in learning from experiments." Science 376.6598 (2022): 1182-1186.

[22] Ristè, Diego, et al. "Demonstration of quantum advantage in machine learning." npj Quantum Information 3.1 (2017): 16.

5. “...with only empirical study on small-size quantum circuits.” While our circuits are moderate in size, we are able to represent 2D images and 3D shapes of resolutions previously unseen in the literature in the context of QML models. The scale of these experiments goes beyond the existing state of the art and provides fundamental evidence of the validity of our method and the design choices. While much of the current theoretical research in QML addresses small-scale problems, its applicability to real-world and large-scale scenarios remains unclear. This gap underscores the importance of designing quantum or quantum-inspired models that can handle real-world scenarios while demonstrating tangible advantages (i.e., faster convergence or a smaller number of parameters). With our work successfully reconstructing dense 3D fields for the first time, we believe it holds experimental significance and makes a meaningful contribution.
6. The high-level question of whether quantum machine learning (QML) genuinely offers advantages compared to the classical method is an open question in the field that would require many more years of research by multiple research groups. While we provide a tiny (on the scale of the entire field) but important and promising experimental evidence supporting the scalability of QML models, we believe answering the general question regarding QML is beyond our scope and the scope of any single paper). Both experimental and theoretical QML research is required to find the answers in the long term, as many assumptions in theoretical research are often invalid in practical settings.
7. Regarding the trainability issue: This work is not a study focused on trainability analysis or other open challenges in theoretical QML. Our primary objective is to develop a neural field representation model for practical applications storing data in the transformation parameters of the PQCs. While we acknowledge that barren plateaus in quantum circuits are often associated with expressivity and measures like Haar randomness, our integration of classical modules to prepare such circuits introduces additional complexity, and the trainability of this hybrid approach has not been fully explored. Nevertheless, in our experiments—aimed at designing quantum circuits capable of scaling to real-world datasets, such as those used in 3D scene reconstruction—we achieve promising results with competitive performance compared to classical (but far not toy or outdated) baselines. Note that many published papers in QML do not reach this level of experimental comparisons. Further theoretical investigations into trainability or related issues fall beyond the scope of this work but could be considered in future.
8. Amplitude encoding (AE). AE is chosen in this submission because it is a widely adopted technique across various applications and has been demonstrated to be effective in numerous contexts including 3D. While it is true that angle encoding could be easier to implement in real-world physical systems, it remains unclear how to use it efficiently to reach arbitrary quantum states. Additionally, there has been significant progress in developing efficient amplitude encoding methods; see, e.g., Ashhab et al., Phys. Rev. Research 4, 013091 (2022); Conde et al., Quantum 8, 1297 (2024); and Daimon et al., "Quantum Circuit Generation for Amplitude Encoding Using a Transformer Decoder," Physical Review Applied 22.4 (2024).

Concern 1: We have thoroughly reviewed the provided references. Regarding the novelty of our PQC design, we deliberately restrict quantum processing to the Bloch sphere around the Y-axis within our quantum circuit design. This design choice is supported by both theoretical analysis and empirical validation and preserves performance while significantly simplifying the optimization process. These insights are particularly impactful at the hardware level, where more complex quantum operations are prone to noise and lack efficient implementation methods. Regarding observable extraction, we believe that it is a valid quantum operation as it entails measuring a quantum system in an appropriate basis to extract meaningful information. We now address the concerns about the validity of our approach. Our work focuses on designing a model that leverages quantum circuits to learn neural field representations at the large scale shown in the literature for the first time while demonstrating an improvement to classical baselines. Although we provide some theoretical analysis of our model (see lines 215–230), its robustness is validated comprehensively through our experimental results (see Section 4 and visualized outcomes). Our quantum circuit is designed following the fully entangled pattern. Regarding the mitigation of the barren plateaus (BPs): we incorporated several techniques, including optimized initialization strategies and localized measurements, to mitigate BPs. Since our final model successfully learns large-scale field representations, we did not explore this problem further, as our primary emphasis is on the empirical design and observations, as stated at the outset of the paper. Lastly, regarding the citation of the Solovay-Kitaev theorem: We emphasize that this theorem is not invoked here to justify exponential gate growth. Instead, it establishes the theoretical guarantee that any unitary operation can be approximated to arbitrary precision using a universal gate set. This foundational principle also contextualizes the observed experimental performance improvements with deeper quantum circuits.

Conclusion. Given all that, we are hence kindly requesting the reviewer to adjust and rethink the evaluation criteria (or/and the confidence level), as we feel the reviewer applies too strong criteria and presumably criteria of physics journals when evaluating our work. Both theoretical and experimental QML enrich each other and can benefit from synergies. We believe that readers should decide the future of this work at this point, and we hope ICLR will allow us to present it and respectfully and openly exchange opinions.

I am delighted to put all the problems on the table. I have the following suggestions.

1. Directly using amplitude encoding requires more details and checks.

(1) There is no explicit way to encode arbitrary data into amplitude encoding, unless you use QRAM or some other simplifications. "We can then prepare our final quantum encoding...Eq. (4)" - No we can't. (This is partially why I said "missing details") The authors should indicate this in the paper with appropriate discussions.

(2) The description of Eq. (5) is wrong. First, when g(x)= exp(-iPx) where P is a Pauli operator, f(x) is a sine-wave instead of a Fourier sum. Second, I guess the authors want to say g(x) represents the encoding for all data, where g(x)=g_k(x) g_{k-1}(x) ... g_1(x), and each g_i(x) is a single-qubit rotation. However, this paper uses the amplitude encoding - this cannot derive Eq. (5).

Eq. (5) and Schuld et al. 2021 are not compatible with Eq. (4) and amplitude encoding. So it is questionable for the statement of "multi-dimensional frequency spectrum" as well as the "expressiveness of the model", negatively affecting the soundness of the theoretical part of the paper.

A minor issue: In Eq. (4), what is |\psi_i\rangle? (Typically this is the computational basis)

2. The complexity of the algorithm on a quantum computer should be discussed.

3. Using Theorem 1 to imply the "universal expressiveness" is inappropriate. The expressiveness should only be discussed when the circuit is of polynomial depth. Theorem 1 is only a textbook-level theorem that shows any unitary can be decomposed of a number of elementary gates, which is unrelated to this paper.

Thank you for the interesting opinions, points and observations! Thank you for the possibility to discuss. Our work will definitely benefit from them, despite not fully agreeing with all of them as outlined in the following. It is great that different views on the related fields of research exist. We find in the detailed review even stronger evidence that our work will be a valuable contribution to ICLR and could ignite follow-up works. Along with that, we feel that rev. DCFN is somewhat generally sceptical towards QML and applies criteria of physics journals when evaluating our work (but not ML conferences). We believe this circumstance could bias the evaluation and should not justify paper rejection. As these differences in opinions concern some fundamental questions, they could serve as material for an open panel discussion with a respectful and open exchange. We are happy to address the raised concerns below. Concern (2,3):

1. First of all, our paper is not a theoretical machine learning paper (there might have been a misunderstanding). While large and fault-tolerant quantum computers are not there yet, it is understandable that a lot of work in the field is done relying on proofs (usual standard: worst-case assumptions). In contrast, we are doing things differently from the theoretical QML community. We rather target the general machine learning (ML) and computer vision (CV) communities. We follow the path of open exploration in contrast to the vast majority of QML works. We are motivated by addressing important and practical CV problems. Sometimes research follows the path from experiments to the theory: A working solution is demonstrated and afterwards, research is done to understand how and why it works from the theoretical perspective. Consider the stream engine invented long before the laws of thermodynamics have been formalised. Another prominent example in the context of classical techniques without guarantees is Simulated Annealing. It does not have provable guarantees for finding global optima. Nevertheless, it works remarkably well in practice. Moreover, consider the following evidence/argument: Classical state-of-the-art ML currently works beyond the provable regime. This could also be an argument in support of the experimental QML research. It is important to test different design choices. Experiments with QML models (also involving simulations) will very probably become increasingly more important in future. There is no final conclusion on QML yet, and there are many open research questions.

2. There is not only a single criterion for a paper to be valuable and relevant. We agree that not all theoretical QML researchers might necessarily find our work interesting (as reviewer DCFN is suggesting), which is fine and valid. Our contribution is addressing a new problem using QML. We demonstrate design choices that work empirically on a simulator of fault-tolerant quantum hardware, and we propose the first method addressing an important problem (esp. for the ML and CV communities) using the gate-based quantum computational paradigm. This alone makes the work valuable and worthy for these communities to become familiar with the results. This opinion—that empirical works are becoming increasingly important—is also shared by many researchers in the QML community.

3. Citing Rev. DCFN: “Much more work should be done to show the scalability and the potential advantage”. We are happy to hear that our work would have a strong potential to ignite follow-ups for further investigation. We find it as evidence of its relevance and timeliness. Demonstrating scalability and the potential advantage in the context of QNF would require not a single but a sequel of follow-up works. We believe the current submission reaches the level of strong technical innovation on the ICLR level. We also would like to encourage the ML and CV communities to think about QNFs and their drawbacks and help improve them. We are by no means claiming that all design choices are optimal. We are initiating a new research direction at the intersection of quantum computing, machine learning and computer vision with multiple possible follow-ups (as DCFN agrees with us).

4. Regarding NISQ and the sensitivity of amplitude encoding to noise: We assume fault-tolerant quantum computers to address the fundamental challenges of designing a neural field with PQCs. We have adjusted the beginning of Sec. 3 accordingly. We strongly believe it is a valid assumption that is worth investigating. A follow-up could investigate the influence of the noise occurring in NISQ devices but first, the results for fault-tolerant machines have to be known and understood.

concern 1.2:

We believe Eq. (5) is correct in the current form after a careful examination upon the reviewer’s request. We agree that its interpretation and connection to AE can be improved. Hence, we updated the corresponding paper section. While Eq. (5) originally applies to angle encoding explicitly, i.e. g(x)= exp(-iPx), angle embedding can approximate amplitude encoding by iteratively encoding data in the angles of quantum gates and leveraging the expressive power of parameterized quantum circuits. The theoretical basis for this approximation can be partially explained through the Solovay-Kitaev theorem, stating that any unitary operation can be approximated to arbitrary precision using a finite set of universal gates given sufficient circuit depth. In this context, angle embedding can be seen as constructing an approximation of the transformations required for AE by composing parameterized rotation gates. With the increasing circuit depth, the fidelity of this approximation improves, allowing angle embedding to effectively simulate the state preparation required for AE. In this work, we use Eq.(5) as a general representation of the circuit structure that is an ansatz–encoding–ansatz framework. The encoding module g(x) in Eq. (5) is versatile and covers a wide range of encoding strategies (it is not limited to Pauli encoding, i.e. angular encoding with Pauli gates, and can include multiple data-reuploading operations).

The discussion of the connection between AE and angle embedding also highlights their inherent links to the Fourier domain. In this context, the multi-dimensional spatial field is first abstracted into an energy field representation in a learnable manner. Consequently, the Fourier spectrum becomes intrinsically related to this abstracted energy field and is specifically determined by the eigenvalues of the quantum circuit gates required to encode the data. This relationship allows us to naturally associate the inferred energy levels with the multi-dimensional frequency spectra, as detailed in ll. 218–230.

concern 2:

In Eq.4, |\psi_i\rangle refers to a general and local basis. Notably, the computational basis serves as a valid example of such a basis to represent our prepared quantum state. We have updated the draft accordingly (see L214 in the latest updated draft).

Analyzing the computational complexity of executing quantum circuits on physical hardware is a challenging task due to various factors. We are hesitant we can provide a precise estimate in this work. Gate fidelity, hardware architecture, types of gates, and the specifics of physical implementation all play critical roles. Fidelity and noise levels directly affect accuracy and may necessitate additional error correction, while hardware constraints like qubit connectivity can increase the required circuit depth. Furthermore, different quantum platforms, such as superconducting qubits or trapped ions, come with unique requirements for gate execution speed and reliability. These combined factors make assessing the practical complexity of running quantum circuits on hardware a non-trivial problem.

While it is uncommon to discuss computational complexity for practical QML work, we try our best to address your question from an algorithmic perspective. The complexity of running a parametrized quantum circuit can be estimated by the number of gates in the circuit (assuming physical factors are ignored). In our method design, we optimized the circuit design and partially constrained its expressivity without compromising performance with given theoretical evidence and empirical verification. This approach reduces the number of gates required compared to other QML works that fully exploit rotations across the entire Bloch sphere.

concern 3:

We agree and do not claim that the Solovay-Kitaev theorem directly establishes expressiveness in circuits of polynomial depth. We reference it to highlight the general principle that any desired unitary transformation can be approximated, albeit with circuit depth bounded—at worst—by an exponential factor. This also accounts for the experimental observation that with increasing circuit depth, the performance improves due to the circuit’s ability to approximate the target measurements with greater precision. We would like to emphasize that, as this is an ML conference, we assume some of our readers are less familiar with certain quantum computing principles. Therefore, we included and explained some of the relevant concepts in the draft.

Concern: Quantum circuit learning and its significance:

Thank you for your detailed observation and for highlighting that smaller MLPs can serve as effective alternatives for fitting CIFAR-10 images, raising important questions about the significance of each model component. Since our experiments span both 2D images and dense 3D scene reconstructions, we carefully scaled our networks to ensure consistent parameter counts across tasks, providing clarity and comparability. While smaller parameter counts may suffice for CIFAR-10, 3D scene learning typically demands significantly more parameters; see Park et al. "Deepsdf: Learning continuous signed distance functions for shape representation." . As shown in Table 1, incorporating a medium-scaled quantum circuit under strictly controlled conditions enabled us to achieve better performance using only 70% of the original parameters. Since the quantum circuit was the only variable introduced, we believe this validates its effectiveness in enhancing the overall model. Additionally, we fully agree that visualizing the evolution of parameters in the quantum circuit is insightful and therefore have included such a figure in the appendix of our revised draft, with updates highlighted in red.

Thank you for your insightful and valuable comments! We are encouraged by your acknowledgement of our technical novelty in combining QML with neural implicit representations and its significance to the rapidly growing implicit neural representation research. We would also like to thank you for appraising the clarity of our writing and paper structure. Below, we carefully address your questions and concerns.

Impact of loosening the constraint:

We agree that relaxing the constraint would be an interesting direction, which we have identified as potential future work. One approach could involve replacing the hard constraint with a soft constraint implemented through penalization. We plan to explore this in our future research

Running on real gate-based quantum computers:

We also agree that running quantum circuits on real quantum computers is an exciting avenue. However, despite recent advancements in circuit-based quantum hardware, access to a sufficient number of qubits with the required stability and quality remains highly restricted. Moreover, our attempt to use a real circuit-based quantum computer provided by IBM was hindered by unrealistic queuing time.

Given our focus on exploring the potential of these models and their applications to large-scale datasets, we have opted to use high-end simulators as an alternative. While we acknowledge the importance of conducting experiments on real hardware, the current constraints necessitate simulation on classical computers even for those work that use small dataset; see Rathi et al. "3D-QAE: Fully Quantum Auto-Encoding of 3D Point Clouds." (2023); Baek et al. "3D scalable quantum convolutional neural networks for point cloud data processing in classification applications." arXiv preprint arXiv:2210.09728 (2022); Zhao et al. "Quantum Implicit Neural Representations." Forty-first International Conference on Machine Learning.

Memory depletion issues:

Regarding memory depletion, we encountered issues when extending our experiments to dense 3D reconstructions. Specifically, for 3D scenes with a sampling density of 500k points per scene (the original setting where we faced memory issues), we were unable to train the model on our A100 GPU with 80 GB of memory. In terms of training time, the model takes approximately 3 hours for 2D image data and 8 hours for 3D scene data, though this also depends on factors such as sampling density, learning rate, model size or even the type of simulator used.

I thank the authors for providing more detail about the model architectures, and suggest including these encoder ablations in the supplementary materials for completeness. While I do have reservations about the baseline comparisons, I believe the main strength of this paper is in its novel application of QML techniques to INRs, which will be of interest and worth accepting to ICLR.

We greatly appreciate your suggestion regarding encoder ablations. We launched additional systematic experiments and will include the results in the supplementary material. Thank you once again for your feedback and recognition of the novelty of our work in applying QML techniques to INRs for the first time.

Concern 1: Baseline comparison

With quantum machine learning still in its infancy, MLP baselines remain a robust choice and have yet to be surpassed in many CV/CG studies; see Rathi et al. in "3D-QAE: Fully Quantum Auto-Encoding of 3D Point Clouds." Our model enhances fully connected neural networks with quantum circuits, offering fundamentally different theoretical expressivity. Consequently, we chose to benchmark it against a similarly structured network with enhanced performance—Siren, an MLP architecture that leverages periodic activation functions for greater robustness compared to standard ReLU. In our experiments, we demonstrated that our model not only achieved better metric performance and used fewer parameters than standard MLPs but also outperformed Siren, a comparison that, to our knowledge, has not yet been explored in quantum machine learning research. While many recent advancements in neural field representations, such as Mildenhall et al.'s NeRF, Chen et al.'s MobileNeRF, and Shue et al.'s 3D Neural Field Generation Using Triplane Diffusion, focus on optimizing applications and efficiency, they rely on architectures beyond fully connected networks and often involve massive parameter counts. For these reasons, we did not consider them for comparison in our work.

Concern 2: Quantum circuit scale

We acknowledge your concerns, however, simulating deep quantum circuits with many qubits is computationally intensive, especially for large-scale datasets. In this work, as we focus on developing a model capable of handling large-scale datasets, such as dense 3D shape reconstructions, we need to consider minimizing quantum resource usage while preserving quantum advantage.

Concern 3: Quantum hardware issues such as noise and decoherence:

As there has not been a model capable of scaling up to 3D reconstruction before our work, the primary goal of this study is to develop a model that can handle large datasets while maintaining quantum advantage, assuming the availability of noise-tolerant quantum computers. Therefore, we did not place significant emphasis on the effect of noise or its potential influence on our experiments. However, in the presence of noise, we would anticipate substantial performance degradation, as shown by other works that primarily investigate the impact of noise on machine learning; see Borras et al., Journal of Physics: Conference Series.Vol.2438.No.1.IOP Publishing, 2023.

Concern 4: Structural layout

We thank you for pointing it out. We will work to incorporate it into the broader experimental results as suggested.

Thank you for your insightful and valuable comments. We are encouraged by your acknowledgement of our motivation, technical soundness, experimental results, and recognition of our work as a compelling indication that quantum computing can benefit data-intensive fields such as visual computing. Below, we carefully address your questions and concerns.

question 1:

Similarities:

Both works draw inspiration from the universal approximation capabilities of neural networks, leveraging them to tackle problems across different domains. Carleo and Troyer, Science (2017) instead of numerically evaluating the interaction of quantum many-body systems and simulating the evolution of wave functions with time, propose an approach to learning many-body wave functions and their evolution.

Differences:

Our work introduces latent vectors for conditional learning, enabling simultaneous training on diverse data and generating unseen samples via latent space interpolations, positioning our approach as a generative network—a direction unexplored by Carleo et al. Unlike their focus on using AI to simulate quantum wavefunction evolution without encoding classical data as quantum states, we enhance classical machine learning with quantum computing through a novel learning-based amplitude encoding, linking classical data to energy inference with neural networks. Additionally, we propose optimized quantum circuit modules with a simplified parameter search space, improving efficiency without sacrificing performance. Targeting field representation in visual computing, our model extends the solvable problem scale to reconstruct multiple high-quality 3D shapes, contrasting with Carleo et al.'s focus on many-body wavefunctions, reflecting fundamentally distinct data processing needs.

question 2:

In our experiment under the specified setting, we observed that increasing the number of qubits provides limited performance improvement. However, as the system scales with an increasing number of qubits n, summing over an exponentially growing number of terms quickly becomes computationally infeasible. A practical solution to this challenge is to encourage sparsity during learning through regularization and to prune values below a predefined small threshold.

question 3:

While our energy inference scheme is executed on classical hardware (see the method figure), it is entirely feasible to prepare the quantum states on quantum hardware. For example, quantum states with amplitudes following a Gibbs-Boltzmann energy distribution can be prepared on a quantum annealer by implementing a specific Hamiltonian evolution schedule, leveraging the annealer's inherent function as a quantum Boltzmann machine (Amin et al., Phys. Rev. X 8, 021050 (2018)). Moreover, as the proposed energy-based encoding aligns with the broader category of amplitude encoding, various implementation techniques have been explored in the literature (e.g., Conde et al., Quantum 8, 1297 (2024); Ashhab et al., Phys. Rev. Research 4, 013091 (2022)).

question 4:

The spikes observed in the training curve during our experiments may be attributed to the coordination challenges between the classical and quantum modules. Since the energies and features generated and inferred by the classical network module and the quantum circuit have inherently different interpretations, we think it is reasonable to observe such behavior.

question 5:

We thank the reviewer for pointing this out. We will update the figures/tables/sections for better readability in the revised version.

We thank you for asking this question and we agree it is important to clarify it. The observation that the current model works and can be executed on a simulator of a fault-tolerant quantum computer is correct. Quantum gate sets such as the Clifford allow for efficient classical simulation. Unfortunately, most QML models rely on gates outside these efficiently simulatable sets, rendering their classical simulation computationally expensive. This highlights the potential for circuit-based quantum computers to offer significant advantages in future. Next, even with more qubits involved in our experiment and stricter control of other conditions, we did not observe significant improvements in model convergence, at least in the tested settings. The scalability of a quantum circuit depends not only on the number of qubits but also on the circuit depth and the types of gates employed—both of which directly influence execution runtime on classical simulators. Currently, we prioritize the depth of the PQC instead of the qubit count in our design. On real quantum hardware, one can in future expect scaling QNF-Net to a higher number of qubits (>20) thus further increasing the supported data sample resolution and even faster training and convergence compared to a classical baseline. Note that the model with more qubits currently trains slower but supports a higher resolution of the learned signals (2D-pixel grids and sampled surfaces for SDFs).