Parameterized Quantum Circuits.

Parameterized quantum circuits (PQCs) consist of parameterized gates and offer a concrete way to implement quantum machine learning algorithms. Specifically, the common parameterized quantum gates contain:

$\text{R}\_\text{x}(\theta)=e^{-i \frac{\theta}{2} \sigma\_x} = \begin{bmatrix}\cos(\frac{{\theta}}{2}) & -\text{i}\sin(\frac{\theta}{2}) \\\\ -\text{i}\sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2}) \end{bmatrix},$ $\text{R}\_\text{y}(\theta) = e^{-i \frac{\theta}{2} \sigma\_y} =\begin{bmatrix} \cos(\frac{\theta}{2}) & -\sin(\frac{\theta}{2}) \\\\ \sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2}) \end{bmatrix},$ $\text{R}\_\text{z}(\theta)= e^{-i \frac{\theta}{2} \sigma\_z} = \begin{bmatrix} e^{-\text{i}\frac{\theta}{2}} & 0 \\\\ 0 & e^{\text{i}\frac{\theta}{2}} \end{bmatrix}.$

The parameters (e.g., $\theta$) in the quantum gate can be either learnable parameters for optimizers or classical information that we want to encode. A quantum machine learning model can be constructed using a sequence of parameterized quantum gates. The initial quantum states can be transformed into the output quantum states. By measuring the output of the quantum circuit, we can convert quantum information into classical information, which can be used to calculate the cost function of the optimization task. We can use classical optimizers to minimize the cost function by adjusting the parameters of quantum gates.

Thank you for acknowledging our work and your suggestions have been immensely helpful. Below is our detailed response.

W1: I think some background knowledge needs to be described in a bit more detail, at least it should be in the appendix (For some laymen like me).

Thank you for your suggestion. We will rewrite the quantum preliminary part and here we provide more details for you to make our paper more readable. Due to word count limitations, part of the answer are included in the Official Comment below..

Single-qubit quantum state. In quantum computing, the fundamental building blocks of computation are qubits (short for quantum bits), which are the quantum analog of classical bits. Unlike classical bits, which can only take on one of two possible values (0 or 1), a qubit can exist in a superposition of the two states, represented by the vector: $|\psi\rangle = \alpha_1|0\rangle + \alpha_2 |1\rangle,$ where $|0\rangle$ and $|1\rangle$represent the two basis states of one qubit, and $\alpha_1$ and $\alpha_2$ are complex numbers that satisfy the normalization condition $|\alpha_1|^2 + |\alpha_2|^2 = 1$. When $|\psi\rangle$ is measured, it will collapse to either the $|0\rangle$ or $|1\rangle$ state with a probability $|\alpha_1|^2$ or $|\alpha_2|^2$.

Mathematically, the quantum state of one qubit can be denoted as a complex 2-dimensional vector, e.g., $|0\rangle=[1,0]^{T}$, $|1\rangle=[0,1]^{T}$, and $|\psi\rangle=[\alpha_1, \alpha_2]^{T}$. The Bloch sphere is a sphere of radius 1, which is a useful tool for visualizing the state of a single qubit. Any other state of one qubit can be represented by a point on the surface of the sphere.

Multi-qubit quantum state. Multi-qubit quantum states are an extension of single-qubit quantum states, and a $N$-qubit quantum state can be represented as a complex $2^N$-dimensional vector in Hilbert space. This is why quantum systems are often described as living in a $2^N$-dimensional Hilbert space. More specifically, a two-qubit system can be represented as $|\phi⟩=\alpha_1|00\rangle+ \alpha_2|01\rangle+ \alpha_3 |10\rangle+ \alpha_4|11\rangle$, where $\sum_{i=1}^{2^2}|\alpha_i^2|=1$ and $∣00\rangle$ represent the tensor product $|0\rangle \otimes |0\rangle$.

Quantum circuits. Quantum circuits are constructed using quantum gates, which are analogous to classical logic gates. Some commonly used single-qubit gates include the Pauli-X gate, the Pauli-Y gate, and the Pauli-Z gate. They can be represented by the unitary matrix: $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. The Controlled-NOT (CNOT) gate is a two-qubit gate that flips the second qubit (target) if the first qubit (control) is in the $|1\rangle$ state. When a quantum gate acts on a quantum state $|\psi\rangle$, it transforms this state to another quantum state $|\psi'\rangle$, according to the mathematical operation $|\psi'\rangle = U|\psi\rangle$, where $U$ represents the unitary matrix associated with the quantum gate.

W2: The Atom stability and Molecule stability metric are compared in most 3D molecular generation paper, while this paper does not show this result.

It should be noted that the baselines consist of two categories. One category is the classic generation model for 3-D molecules. The other category includes the quantum model SQ-VAE and the hybrid model QGAN, which are limited to generating molecular graphs and cannot generate 3-D molecules. Therefore, we adopt the commonly used metrics Valid, Unique, Novel, which can be used for both 2-D and 3-D molecular generation, to evaluate the effectiveness of different types of molecular generation methods.

Moreover, Atom stability measures the proportion of atoms that have the right valency and Molecule stability measures the proportion of generated molecules for which all atoms are stable. In our case, all valid molecules are stable (which means here Molecule stability = Valid), thus here we only provide the atom stability further.

We hope this response could facilitate your understanding of our work and ease your concern, looking forward to receiving your feedback soon.

The current state of quantum machine learning

Due to the current limitations of quantum hardware, the number of layers and parameters in quantum methods are severely restricted. Quantum machine learning (QML) models, particularly quantum generative models, are still in their infancy compared to well-developed classical neural models with vast numbers of parameters. Consequently, the performance of current QML models may not yet match that of SOTA classical counterparts.

To support our above points, we collect the following facts:

1. The quantum versions of classical ML algorithms running on NISQ devices barely take SOTA classical algorithms as their baselines e.g. QCNN [1], QGAN [2], QLSTM [3], etc. On the one hand, conducting experiments on NISQ devices itself makes a significant contribution to the implementation of quantum algorithms. On the other hand, NISQ devices are difficult to obtain, and there is a significant gap between the physical qubit connectivity topology and quantum algorithm design, making it challenging to deploy quantum algorithms on NISQ devices. Additionally, running on NISQ devices faces the challenge of big quantum noise.

2. Quantum algorithms running on simulators included classical baselines, but rarely perform better than the classical baseline [4]. For example, [5] (NeurIPS 2020) proposed the quantum RNN for classification as evaluated on MNIST, with their results only achieving 94% accuracy (classical simple fully connected layer can achieve accuracy better than 95%). [6] (ICML 2023a) proposed the quantum molecular embedding algorithm and applied it to molecular property prediction, with their results showing nearly a 40% gap from the classical SOTA baselines. [7] (ICML 2023b) proposed a quantum Quadratic Assignment Problem (QAP) solver and applied it to the Traveling Salesman Problem (TSP), with their results falling short of the classical nearest insertion algorithm (A heuristic algorithm proposed in 1997).

The general limitation underscores the need for further research to address the deficiencies of NISQ devices, particularly regarding the practicality of real quantum computers and the challenges posed by quantum noise. Once these issues are resolved, we can increase the circuit depth and number of parameters in quantum models, potentially matching the performance of SOTA classical networks with vast numbers of parameters.

References

[1] Quantum convolutional neural networks. Nature Physics 2019.

[2] Experimental quantum generative adversarial networks for image generation. Physical Review Applied 2021.

[3] Quantum long short-term memory. ICASSP 2022.

[4] Better than classical? The subtle art of benchmarking quantum machine learning models, arXiv 2024.

[5] Recurrent quantum neural networks. NeurIPS 2020.

[6] Quantum 3D graph learning with applications to molecule embedding. ICML 2023.

[7] Towards quantum machine learning for constrained combinatorial optimization: a quantum QAP solver. ICML 2023

Thank you for acknowledging our work and your suggestions have been immensely helpful. Below is our detailed response.

W1: Formulas 8 and 9 can be represented using more rigorous mathematical notation.

Thanks for your suggestion and we will revise this.

The von Mises-Fisher (vMF) distribution is a probability distribution on the unit sphere in $\mathbb{R}^d$.

• Formula 8 is written as $q_{\phi}(z|x) = \text{vMF}(\mu, \kappa ) = C_{d,\kappa} e^{\kappa \langle \mu(x), z \rangle},$ where $\Vert \mu \Vert = 1$ denotes the mean direction and $\kappa$ denotes the concentration parameter, $\kappa$ is commonly set as a constant during training. The normalization constant $C_{d,\kappa}$ is equal to $1/ \int_{S^{d-1}}e^{\langle\xi,x\rangle} dS^{d-1}$, where $S^{d-1}$ is the sample space $\{x|x\in\mathbb{R}^d, \Vert x\Vert=1\}$.

• Formula 9 is written as $z \sim \text{vMF}(||\psi^E\rangle_A|, \kappa) = C_{d,\kappa} e^{\kappa \langle ||\psi^E\rangle_A|, z \rangle},$ which represents that the latent variable $z$ is sampled from the $\text{vMF}$ distribution with mean direction $\mu = \vert |\psi^E \rangle_{A}|$ in the latent space.

Q1: Please clarify the term "fully" and, if necessary, explain how it compares to its counterparts.

Thank you. "Fully" means our method only uses quantum parameters. The counterparts are the hybrid quantum-classical methods, which use a mix of classical parameters from classical model layers and quantum parameters from quantum model layers. The hybrid method is hard to deploy to the real quantum computer because it needs to communicate frequently between quantum devices and classical devices, which is time-consuming. Generally speaking, designing a hybrid model is relatively trivial and does not clarify the role of the quantum layer within the overall model, whereas designing an effective fully quantum model is challenging and innovative.

Q2: Is the latent space defined as discrete? If so, please provide a rationale for this choice.

Thanks, the latent space of QVAE is NOT defined as discrete. We adopt von Mises-Fisher (vMF) distribution as latent prior, which lies in a hyperspherical space and is continuous.

L1: The emphasis on the QM9 dataset may constrain the generalizability of the results to other molecular datasets or different types of data.

Thanks, in line with baselines {E-NFs, G-SchNet, G-SphereNet, SQ-VAE, QGAN}, we only choose QM9 dataset as our evaluation benchmark. However, our approach can further extend to other bigger datasets since the number of qubits in our proposed framework comes to $O(C \log n)$ ($n$ denotes the number of atoms in one molecule).

Here we add experiments on a larger 3-D dataset named GEOM. Compared to QM9, GEOM stands out as a larger-scale dataset of molecular conformers, comprising 430,000 molecules, with up to 181 atoms and an average of 44.4 atoms per molecule $(n \approx 100)$. The molecules in this dataset exhibit larger sizes and more intricate structures.

In line with EDM, on GEOM, here we report the atom stability and the Wasserstein distance between the energy histograms of datasets and the generated molecules. Here we only report two baselines since the other baselines do not include experiments on GEOM dataset, we will leave replicating their models on the new dataset for future work due to the limited time for rebuttal.

It can be seen that our method outperforms MLP-VAE. Although our method falls short of EDM (the classical SOTA baseline ), this result demonstrates that our approach can achieve relatively reasonable generation results on datasets with larger and more complex data volumes. (For a discussion on the performance comparison between quantum algorithms and classical algorithms, see Supplement to L1 in the official comment below for details.)

We hope this response could address your concerns, looking forward to receiving your feedback soon.

References:

[1] Limitations on Quantum Dimensionality Reduction, ICALP, 2011

[2] Quantum computation and quantum-state engineering driven by dissipation, Nature Physics, 2009.

[3] Quantum state reduction: Generalized bipartitions from algebras of observables, Physical Review A, 2020.

[4] Nonlinear transformations in quantum computation, Physical Review R, 2023.

[5] Quantum Autoencoders for Efficient Compression of Quantum Data, Quantum Science and Technology, 2017.

[6] Structure-based drug design with equivariant diffusion models[J]. arXiv, 2022.

[7] Molecular generative model based on conditional variational autoencoder for de novo molecular design[J]. Journal of cheminformatics, 2018

[8] 3d equivariant diffusion for target-aware molecule generation and affinity prediction[J]. arXiv, 2023.

[9] Quantum convolutional neural networks. Nature Physics 2019.

[10] Experimental quantum generative adversarial networks for image generation. Physical Review Applied 2021.

[11] Quantum long short-term memory. ICASSP 2022.

[12] Better than classical? The subtle art of benchmarking quantum machine learning models, arXiv 2024.

[13] Recurrent quantum neural networks. NeurIPS 2020.

[14] Quantum 3D graph learning with applications to molecule embedding. ICML 2023.

[15] Towards quantum machine learning for constrained combinatorial optimization: a quantum QAP solver. ICML 2023

Supplement to W5: Detailed proof for "the KL term in the ELBO loss is constant"

The vMF distribution is defined on a $d−1$ dimensional hypersphere, with its sample space as $S^{d-1} = \{x | x \in \mathbb{R}^d, ||x|| = 1\}$, and its probability density function is given by: $p(x) = \frac{e^{\langle\xi,x\rangle}}{Z_{d, \Vert\xi\Vert}},\quad Z_{d, \Vert\xi\Vert}=\int_{S^{d-1}}e^{\langle\xi,x\rangle} dS^{d-1},$ where $\xi\in\mathbb{R}^d$ is a predefined parameter vector. As we can see, it is a distribution centered on $\xi$ across the space $S^{d-1}$. A more common notation for the vMF distribution is $p(x) = C_{d,\kappa} e^{\kappa\langle\mu,x\rangle},$ where $\mu=\xi/\Vert\xi\Vert, \kappa=\Vert\xi\Vert, C_{d,\kappa}=1/Z_{d, \Vert\xi\Vert}$. When $\kappa=0$, the vMF distribution is uniform on the sphere.

vMF-VAE selects the uniform distribution on the sphere ($\kappa = 0$) as the prior $q(z)$, and chooses the posterior distribution as the vMF distribution: $p(z|x) = C_{d,\kappa} e^{\kappa \langle \mu(x), z \rangle}.$ To simplify, here $\kappa$ is a hyperparameter (the larger the $\kappa$ , the more concentrated the distribution is around $\mu$), thus the only parameter of $p(z|x)$ comes from $\mu(x)$. Now we can calculate the KL divergence term:

$\int p(z|x) \log \frac{p(z|x)}{q(z)} dz = \int C\_{d,\kappa} e^{\kappa \langle \mu(x), z \rangle} (\kappa \langle \mu(x), z \rangle) + \log C_{d,\kappa} - \log C\_{d,0} dz \\ = \kappa \langle \mu(x), \mathbb{E}_{z \sim p(z|x)}[z] \rangle + \log C\_{d,\kappa} - \log C\_{d,0}.$

We know that the mean direction of the vMF distribution is aligned with $\mu(x)$, and its norm depends only on $d$ and $\kappa$. Substituting into the equation above, we know that the KL divergence term only depends on $d$ and $\kappa$. Once these two parameters (dimension $d$ and hyperparameter $\kappa$) are determined, the KL divergence term becomes a constant (when $\kappa \neq 0$, it is greater than 0).

Supplement to L1: The performance does not match that of classical methods.

Due to the current limitations of quantum hardware, the number of layers and parameters in quantum methods are severely restricted. Quantum machine learning (QML) models, particularly quantum generative models, are still in their infancy compared to well-developed classical neural models with vast numbers of parameters. Consequently, the performance of current QML models may not yet match that of SOTA classical counterparts.

To support our above points, we collect the following facts:

1. The quantum versions of classical ML algorithms running on NISQ devices barely take SOTA classical algorithms as their baselines e.g. QCNN [9], QGAN [10], QLSTM [11], etc. On the one hand, conducting experiments on NISQ devices itself makes a significant contribution to the implementation of quantum algorithms. On the other hand, NISQ devices are difficult to obtain, and there is a significant gap between the physical qubit connectivity topology and quantum algorithm design, making it challenging to deploy quantum algorithms on NISQ devices. Additionally, running on NISQ devices faces the challenge of big quantum noise.

2. Quantum algorithms running on simulators included classical baselines, but rarely perform better than the classical baseline [12]. For example, [13] (NeurIPS 2020) proposed the quantum RNN for classification as evaluated on MNIST, with their results only achieving 94% accuracy (classical simple fully connected layer can achieve accuracy better than 95%). [14] (ICML 2023a) proposed the quantum molecular embedding algorithm and applied it to molecular property prediction, with their results showing nearly a 40% gap from the classical SOTA baselines. [15] (ICML 2023b) proposed a quantum Quadratic Assignment Problem (QAP) solver and applied it to the Traveling Salesman Problem (TSP), with their results falling short of the classical nearest insertion algorithm (A heuristic algorithm proposed in 1997).

The general limitation underscores the need for further research to address the deficiencies of NISQ devices, particularly regarding the practicality of real quantum computers and the challenges posed by quantum noise. Once these issues are resolved, we can increase the circuit depth and number of parameters in quantum models, potentially matching the performance of SOTA classical networks with vast numbers of parameters.

Supplement to W2 & Q1: Baselines and Metrics "

Here we provide the detailed description of each baseline and we will further add this part to Appendix.

• MLP-VAE: We implement the original vanilla VAE(https://arxiv.org/abs/1312.6114) using a three-layer perceptron for both the encoder and decoder. While the original paper does not cover molecular or 3D data, we use the same data preprocessing scheme as our proposed QVAE, without normalization (see Section 3.1 and lines 202-209 in our paper for details). To ensure a fair comparison, we maintain the same input data dimensions, hidden state dimensions, and training configuration as in our QVAE.

• E-NFs,G-SchNet,G-SphereNet,EDM, QGAN: Each method provides its pretrained models in their respective Git repositories. We use their model checkpoints to generate molecule samples for evaluation. Specifically, since G-SchNet and EDM offer 10,000 generated molecules in their repositories, we directly use these samples for evaluation.

• SQ-VAE(https://arxiv.org/pdf/2112.12563): This method does not provide its code. Since it also addresses the molecule generation task, we choose to replicate it on TorchQuantum based on the description in the original paper, including the quantum circuits, the quantum measurement scheme, and the training configuration. For fairness, we use the same number of qubits and quantum parameters as in our method for the replication.

It should be noted that the baselines consist of two categories. One category is the classic generation model for 3-D molecules. The other category includes the quantum model SQ-VAE and the hybrid model QGAN, which are limited to generating molecular graphs and cannot generate 3-D molecules. Therefore, we adopt the commonly used metrics Valid, Unique, Novel, which can be used for both 2-D and 3-D molecular generation, to evaluate the effectiveness of different types of molecular generation methods. Here we provide the detailed definition of each metric:

• Valid: This metric measures the percentage of generated molecules that are chemically valid, which is defined as the percentage of molecular graphs which do not violate chemical valency rule.

• Unique: This metric measures the ratio of unique molecules among the generated set. This metric ensures that the model is not generating the same molecule multiple times, promoting a variety of different structures.

• Novel: This metric assesses the fraction of generated molecules that do not appear in the training data. A higher novelty score indicates that the model can generate new, previously unseen molecules, which is crucial for discovering new compounds.

Note that it is unreasonable to only consider novelty and uniqueness without validity ((https://arxiv.org/pdf/2203.17003) also points out this issue): like in the extreme case if the model’s validity is only 1%, but these valid molecules are all unique from each other and different from the training set, resulting in 100% for both uniqueness and novelty. Thus, we adopt Unique×Valid and Novel×Valid as metrics instead.

Supplement to W3-a: How to calculate properties like logP.

We use rdkit to compute the property SA, QED, and logP of each generated molecule. Specifically, here we use

from rdkit.Contrib.SA_Score import sascorer
from rdkit.Chem.QED import qed
from rdkit.Chem import AllChem, Crippen

SAscore = sascorer.calculateScore(mol)
SAscore = round((10-SAscore)/9,2)
QEDscore = qed(mol)
logP = Crippen.MolLogP(mol)

For homo-lumo gap, we use the method in https://github.com/divelab/DIG/blob/dig-stable/dig/ggraph3D/utils/eval_prop_utils.py.

Thanks for your review and feedback, below is our detailed response. Due to word count limitations, part of the answer and references are included in the Official Comment below.

W1: About novelty (SQ-VAE (https://arxiv.org/abs/2205.07547) and 3D-QAE).

Thanks, and we would like to humbly point out that you may have some misunderstandings that may affect the interpretation of our work.

• First, the SQ-VAE mentioned in our paper, which also serves as one of our baselines, is the one that proposes a scalable quantum generative autoencoder (SQ-VAE) (https://arxiv.org/pdf/2112.12563). It is not the same paper as you referred to. The paper you referred to neither has any quantum components nor targets on molecule generation, so we neither include it as our related work nor baseline.

• Second, the dimension reduction approach is not from paper 3D-QAE. Instead, tracing out a subsystem is a common technique used to reduce the dimensionality of a quantum state [1~3]. Moreover, it has already been applied in many fields. For example, [4] uses it to implement nonlinear transformations of quantum states, and [5] utilizes it to efficiently compress data.

For the position and contribution of our work, please see general response above.

W2 and Q1: Questions of baselines.

As we mentioned above, the SQ-VAE referenced in our paper is not the same as the paper you referred. And we give detailed description of the setting of baselines, see Supplement to W2 & Q1 in official comment below.

W3-a: How to calculate properties like logP.

In line with many recent works in AI4drug [6～8], We utilize external chemical libraries, especially rdkit.Chem, see Supplement to W3-a in official comment below for details.

W3-b: How equality of continuous properties is defined.

We believe there are some misunderstandings regarding the experimental setup, and we apologize for any confusion. In the section Single Condition, we trained four different models, each targeting SA, QED, LogP, and the homo-lumo gap, respectively. Each column in Table 2 indicates the percentage of molecules whose properties, when rounded up, match the specified condition. For instance, for the condition logP = 0.0, the results show that 57.8% of generated molecules have logP values within the range [-0.5, 0.5), rather than 0 for negative values. Similarly, the data in the logP = 1.0 column reflects that using logP = 1.0 as the input condition, 45.6% of generated molecules have logP values in the range [0.5, 1.5). Table 2 aims to demonstrate that the QCVAE model can increase the proportion of generated molecules with desired properties when given a single condition. We will include details of the experimental setup in our revised paper.

W3-c: The violin plot does not support the results presented for logP in Table 2.

We believe there are still some misunderstandings. First, we want to clarify that Figure 5 and Table 2 report results from two different experiments. Table 2 presents the results under a single condition, while Figure 5 shows results for multiple conditions given simultaneously. In the section Multiple Conditions, we train and evaluate the proposed QCVAE-Mole under multiple conditions, meaning we simultaneously provide the four properties: SA, QED, LogP, and gap. Details can be found in lines 335-341 of our paper.

W4: It is not reported explicitly how the metrics validity, novelty and uniqueness are calculated.

Thanks for the suggestion, we will add this part, see Supplement to W2 & Q1 in official comment below.

W5:: It is stated without proof or explanation that the KL term in the ELBO loss is constant.

Thanks, here we provide the detailed proof, see Supplement to W5 in official comment below.

W6: Writing suggestion for Line 180.

Thanks, we will modify this according to your suggestion.

W7: Line 190: “where we take the absolute value of to transform it from the complex domain to the real domain.” How is this operation defined?

We perform the transformation by computing the absolute value of the complex number, which is specifically implemented as $\sqrt{a^2 + b^2}$ for any complex number of the form $a+bi$. This operation ensures that the norm of the amplitude vector is still unity.

W8: Question with Line 348:

Thanks for the the suggestions. We admit that using "prove" here is not appropriate, and we will modify this.

Q2: Amplitude encoding vs. angle encoding.

• Angle encoding represents classical data through rotation quantum gates on individual qubits. For a vector $\mathbb x = (x_1, x_2, ... x_n)$, each component $x_i$ is used to parameterize a rotation, such as $|\psi\rangle = \bigotimes_{i=0}^{n-1} R_y(x_i)|0\rangle$.
• Amplitude encoding can take full advantage of the exponential size of the Hilbert space associated with quantum states. Given a vector $\mathbb x = (x_1, x_2, ... x_N)$, where $N = 2^n$, it can be encoded as the quantum state $|\psi\rangle = \sum_{i=0}^{N-1}x_i|i\rangle$. Here, each $x_i$ represents the amplitude of the corresponding basis state $|i\rangle$.
• As we can see, for a $n$-qubit quantum system, using amplitude encoding can encode $2^n$ dimensional classical data, while using angle encoding can only encode $O(n)$ dimensional classical data.

L1: The performance does not match that of classical methods.

Please refer Supplement to L1 for detailed discussion.

We hope this response could clear up your misunderstanding and address your concerns. We believe that this work contributes to the quantum ML community and also marks a step towards integrating quantum ML with science. We would sincerely appreciate it if you could reconsider your rating and wish to receive your further feedback soon.

Dear Reviewer CAJp,

Thank you very much again for your feedback and the efforts you have put into reviewing our work.

As the discussion window nears its end, we eagerly anticipate your further feedback on our submission. We have already received positive responses from the other three reviewers. In particular, Reviewer hKnP and 5Ebq have raised their score to 7. We hope that our responses can also address your concerns and clarify potential misunderstandings. If there is any more clarity you seek or further discussion you would like to have, we are here to respond in the time remaining. Should our clarifications meet your expectations, we would be truly grateful for your reconsideration of the score.

With gratitude,

Authors

W3-b: add the precise definition of the range of each condition, explain why this range was chosen and report the novelty and diversity.

Thanks, the choices of the range are in line with "MGCVAE: Multi-Objective Inverse Design via Molecular Graph Conditional Variational Autoencoder"[1], where they also set the condition range of logP as 1 ([-0.5, 0.5), [0.5,1.5)), the range of SA and QED as 0.1 ([0.25,0.35), [0.35,0.45) ...). Although they did not provide a detailed explanation for such a setting, we think a reasonable explanation is: that the condition range is determined by the total range of the property. For most molecules in QM9, LogP property ranging from [-6,5], the SA score and QED properties are both ranging from [0,1), with the gap property ranging from [2,12]. The condition range could be about 1/10 of the entire range. We will add the precise definition of the range in the revised version according to your suggestion.

In addition, We further report the Valid, Unique, and Novel metrics in the condition generation. Here Unique* and Novel* means Unique×Valid and Novel×Valid respectively, we have mentioned the reason above.

It can be observed that when given conditions, the valid metric remains relatively stable, fluctuating between 70% and 80%. The unique* metric shows a slight decline, but overall, it aligns with expectations, which may also be due to the fluctuations in the valid metric. However, we observe that when the proportion of generated molecules with desired properties increases significantly, there is a noticeable drop in the novel* metric. This may be because the model is overly reliant on the conditional input, causing it to generate molecules that closely match typical examples in the training data, thereby neglecting its ability to generate novel structures.

[1] Lee M, Min K. MGCVAE: multi-objective inverse design via molecular graph conditional variational autoencoder[J]. Journal of chemical information and modeling, 2022.

In mathematics, the dot often denotes a function argument, thus it would be important to clarify this notation.

Thanks for the suggestion, we will clarify this notation in the revised version.

Thank you for the advice and the opportunity to engage in this valuable discussion. We understand your concerns, and we have made our best effort to address these issues and provide additional ablation studies. We hope our response could ease your concerns and we will include all these experimental results and discussions in our final version. We would sincerely appreciate it if you could reconsider your rating. As the discussion period is coming to a close, we may not have enough time to respond to your further questions, sorry for that.

Thanks for acknowledging our efforts and the advice for more extensive ablations. Below we respond to your comments.

I think that the work combines the two approaches. It would benefit the paper to mention this more explicitly.*

Thank you for your suggestion. We admit our work draws inspiration from these related works, and we have already mentioned and cited them in line 34-44 and line 124. Here, we further highlight the connections and differences between the 3D-QAE and the SQ-VAE, and incorporate these points into our revised paper.

Firstly, regarding the 3D-QAE, the similarity lies in the use of amplitude encoding to encode 3D information into quantum states. We acknowledge that but it is important to note that amplitude encoding is also a common operation in quantum information processing. The fundamental difference between our work and 3D-QAE is that the quantum AE is primarily designed for information compression, whereas our QCVAE possesses generative capabilities, which are attributed to the design of our intermediate latent space and the resampling module. In addition, the Parameterized Quantum Circuits(PQC) we used are entirely different. In 3D-QAE, a simple PQC and its inverse form the encoder and decoder. However, we found that directly using the inverse parameters of encoder as decoder may even harm the performance of our quantum VAE. Thus, in our method, we designed a different hardware-efficient PQC, where even though the circuit structures of the encoder and decoder are the same, the quantum parameters they contain are independently optimized.

Secondly, regarding the SQ-VAE, we share a similar conceptual goal utilizing quantum VAEs for molecular generation, but our methodological framework is fundamentally different. Specifically, our input and output are tailored for 3D molecular structures, reflected in the encoding of input information. And the sampling of latent space variables and the final measurement method of the quantum circuit are also different. Moreover, we propose a full quantum neural network capable of multi-conditional control as encoder/decoder while SQ-VAE uses a hybrid quantum-classical layer. In defining the latent variable space, we employ the von Mises-Fisher (vMF) distribution to harness the inherent properties of quantum states, while they simply Imitate a classical VAE using a Gaussian distribution without providing any additional insight. We would like to emphasize the importance of our quantum neural network with multi-conditional control and the vMF latent space tailored to quantum characteristics.

We will add the above-detailed comparison with the existing 3D-QAE and SQ-VAE to our revised manuscript. Thank you again for your suggestion.

What has to happen concretely such that the QML approach will actually be better than classical ML?

We think the current factors hindering the further improvement of QML performance are: (1) The limited resources of simulators based on classical computers. (2) The significant quantum noise present in the available quantum hardware. Based on (1)(2), the depth of quantum circuits and the number of quantum parameters in current QML methods are significantly constrained. Once these issues are resolved, we can increase the circuit depth and number of parameters in quantum models, potentially matching the performance of SOTA classical networks with vast numbers of parameters.

Q5: What role do the four loss functions and fidelity loss in the Appendix D section play in the design? What is the contribution of each loss? Are there any hyperparameters to balance the losses?

Thanks, our classic loss function consists of four parts: 3-D coordinate loss $L_1$, atomic classification loss $L_2$, constraint loss $L_3$, and auxiliary loss $L_4$. Together, they form the reconstruction loss, which indicates the true physical meaning of the information. Specifically, $L_1$ supervises the reconstruction of the molecule 3-D position by geometric distance error, and $L_2$ supervises the reconstruction of atom types by weighted cross entropy. $L_3$ is used to constrain the sum of the probability of all atom types for each atom to be the same, and we use MSE loss here. Since we add padding entries to the input data, so $L_4$ is designed to supervise the reconstruction of these entries of zero.

On the other hand, the design of fidelity loss does not consider the real physical meaning of the output quantum vector, but instead treats the input and decoder output of the encoder as two quantum states, and then designs the loss by calculating the fidelity between them.

In our paper, we experimentally compared using only classical loss or fidelity loss. Regarding the four components of the classical loss, there are indeed hyperparameters to balance them. We can set $\alpha, \beta, \gamma$, and the final loss becomes $L = L_1 + \alpha L_2 + \beta L_3 + \gamma L_4$. The best hyperparameter configuration can be determined through grid search. Due to limited time during the rebuttal period and the time-consuming nature of grid search, we plan to include this part in the appendix as an ablation study in the future.

L1: 1.How to improve the performance to obtain similar performance to classical methods. 2. How to accelerate the method to make it as fast as other quantum-based methods.

• The general limitation underscores the need for further research to address the deficiencies of NISQ devices, particularly regarding the practicality of real quantum computers and the challenges posed by quantum noise. Once these issues are resolved, we can increase the circuit depth and number of parameters in quantum models, potentially matching the performance of SOTA classical networks with vast numbers of parameters.

• As previously mentioned, the faster simulation speed of other quantum methods on classical computers does not necessarily indicate better performance. (see the answer to W2 for details)

We hope this response could answer your questions and address your concerns, looking forward to receive your further feedback soon.

Reference:

[1]. Quantum autoencoders for efficient compression of quantum data

[2]. Chapter 14 of Deep learning[M]. MIT press, 2016.

[3]. Wikipedia of autoencoder https://en.wikipedia.org/wiki/Autoencoder#cite_note-:12-1

[4]. Nonlinear principal component analysis using autoassociative neural networks[J]. AIChE journal, 1991.

Thank you for acknowledging our work. Your rating has provided us with great encouragement, and your detailed feedback and suggestions have been immensely helpful. Below is our detailed response. Due to word count limitations, part of the answer and references are included in the Official Comment below.

W1: Compared with classical molecule generation methods, the performance of the proposed method is still lag behind.

Thanks. Since quantum machine learning technology is still in its infancy and quantum computers have not yet reached a mature stage, most quantum machine learning methods currently cannot surpass the existing SOTA classic methods. See (1) Current state of Quantum machine learning in general response for detailed discussion.

W2: Compared with quantum methods QGAN-HG and P2-QGAN-HG, the speed of the proposed method is still lag behind.

Table 1 lists the runtime of all methods on classical computers, where quantum methods are tested using a classical quantum simulator. It is well known that quantum algorithms cannot be efficiently simulated by classical computers, so the runtime comparison on a classical computer for different types of quantum algorithms may be unfair in some sense. We here report the speed to just provide a reference and to demonstrate that, even on simulators, our methods have a speed advantage compared with the SOTA classic model.

QGAN-HG and P2-QGAN-HG indeed show higher efficiency on classical computers, this is because they are hybrid classical-quantum methods, and they contain only a small number of quantum parameters. However, our approach is fully quantum circuits with fully quantum parameters, which need more time to simulate the execution of quantum circuits. We designed fully quantum parameters to enable deployment on real quantum computers, as incorporating classical parameters would result in significant time consumption due to communication between quantum and classical devices.

W3: It is unclear what Valid, Unique and Novel mean and how to define these metrics.

Thanks for the suggestion, we will add the detailed definition of each metric to our paper.

• Valid: This metric measures the percentage of generated molecules that are chemically valid, which is defined as the percentage of molecular graphs which do not violate chemical valency rule.

• Unique: This metric measures the ratio of unique molecules among the generated set. This metric ensures that the model is not generating the same molecule multiple times, promoting a variety of different structures.

• Novel: This metric assesses the fraction of generated molecules that do not appear in the training data. A higher novelty score indicates that the model can generate new, previously unseen molecules, which is crucial for discovering new compounds.

Q1: Why do the methods proposed in Table 1 have zero classical parameters? Would moving some quantum parameters to classical parameters speed up the model?

As we mentioned earlier, using classical parameters would hinder the deployment of algorithms on quantum computers because the hybrid model requires frequent communication between quantum and classical devices, resulting in significant time costs. Generally speaking, designing a hybrid model is relatively trivial and does not clarify the role of the quantum layer within the overall model, whereas designing an effective fully quantum model is challenging and innovative.

Q2: In line 170:" To convert to latent space, here we discard the information contained in the subsystem $B$ via tracing out the state of $q_B$ quits. ". Why would you do this? What's the intuition behind the compression method mentioned in [1]?

Thanks, [1] aims to propose a quantum AE， and traditionally, AEs are used for dimension reduction or feature learning [2,3,4]. In classical neural networks, dimensionality reduction is straightforward, as it can be achieved with a linear layer. However, dimensionality reduction can not be achieved by Parameterized Quantum Circuits, since these operations involve unitary matrix multiplication, which preserves dimensionality. In quantum systems, a common method for dimensionality reduction is through the loss of information by discarding certain qubits, achieved via the tracing-out operation. This approach allows us to effectively extract information about the quantum subsystem.

Q3: In line 348: " ..., which proves classical data tends to follow a normal distribution.". Do VAE and QVAE use different datasets? Please clarify this.

Thanks for pointing this out. Indeed, VAE and QVAE use the same dataset, but the input data for QVAE undergoes additional normalization to meet the requirements of a quantum system. Here, we want to convey that when dealing with data without normalization, imposing a normal distribution as the latent prior, compared to the von Mises-Fisher distribution, is beneficial for classical variational autoencoders. We acknowledge that using the word 'prove' here is not appropriate and might cause some misunderstanding. We will revise this.

Q4: In Figure 6 a), what is the difference between N-VAE and N-QVAE besides the quantum implementation? Why can N-QVAE generate more effective molecules than classical methods?

Thank you. In N-QVAE, the input data undergoes additional normalization to meet the requirements of a quantum system. Additionally, as you mentioned, N-QVAE uses Parameterized Quantum Circuits instead of classical neural networks for the encoder and decoder. The input dimension, latent dimension, latent prior, loss function, and training strategy are consistent with those used in N-VAE. N-QVAE can generate more effective molecules, possibly due to the advantages of Parameterized Quantum Circuits over a simple multi-layer perceptron.