QVAE-Mole: The Quantum VAE with Spherical Latent Variable Learning for 3-D Molecule Generation
Authors: Huaijin Wu
Xinyu Ye
Junchi Yan
Venue: NeurIPS 2024 poster (2024)
Keywords: quantum machine learning
quantum generative model
quantum VAE
3-D molecule generation
conditional generation
Primary area: machine_learning_for_physical_sciences
TL;DR: We propose the first fully (conditional) quantum VAE for 3-D data (molecule) generation with a von-Mises Fisher distributed latent space.

Abstract: Molecule generation ideally in its 3-D form has enjoyed wide applications in material, chemistry, life science, etc. We propose the first quantum parametric circuit for 3-D molecule generation for its potential quantum advantage especially considering the arrival of Noisy Intermediate-Scale Quantum (NISQ) era. We choose the Variational AutoEncoder (VAE) scheme for its simplicity and one-shot generation ability, which we believe is more quantum-friendly compared with the auto-regressive generative models or diffusion models as used in classic approaches. Specifically, we present a quantum encoding scheme designed for 3-D molecules with qubits complexity $\mathcal{O}(C\log n)$ ($n$ is the number of atoms) and adopt a von Mises-Fisher (vMF) distributed latent space to meet the inherent coherence of the quantum system. We further design to encode conditions into quantum circuits for property-specified generation. Experimentally, our model could generate plausible 3-D molecules and achieve competitive quantitative performance with significantly reduced circuit parameters compared with their classic counterparts. The source code will be released upon publication.

QVAE-Mole: The Quantum VAE with Spherical Latent Variable Learning for 3-D Molecule Generation
Huanjin Wut, Xinyu Ye', Junchi Yan'
Dept. of CSE & School of Al & MOE Key Lab of AI, Shanghai Jiao Tong University
edu.cn
Abstract
Molecule generation ideally in its 3-D form has enjoyed wide applications material, chemistry, life science, etc. We propose the first quantum parametric circuit for 3-D molecule generation for its potential quantum advantage especially considering the arrival of Noisy Intermediate-Scall Quantum (NISQ) era. We choose the Variational AutoEncoder (VAE) scheme for its simplicity and one-shot generation ability, which believe is auto-regressive generative models or diffusion models as used in classic approaches. Specifically, present quantum encoding scheme designed for 3-D molecules with qubits complexity (C log n)) (n is the number atoms) and adopt a von Mises-Fisher (vMF) distributed latent space meet the inherent coherence of the quantum system. We further design to encode conditions into quantum circuits for property-specified generation. Experimentally, our model could generate plausible 3-D molecules and achieve competitive quantitative performance with significantly reduced circuit parameters compared with their classic counterparts.
quantum-friendly compared with the
1 Introduction
Beyond molecule graph generation, 3-D molecule generation which can often be more challenging yet of practical value, e.g. for drug design, has received wide attention recent years []] On the one hand, the space of possible molecules and chemical compounds is vast, often described as a "chemical space" with an immense number dimensions, and data-driven methods relying machine learning (ML) have been introduced [2] 3.. On the other hand, quantum computing has demonstrated strong expressive capabilities in various learning and optimization applications, including solving [4, classification (5), and discovery [6]. Particularly, it has potential advantages in tasks related to the microphysical world, such as chemistry simulation 77. prediction of molecular properties [8) and approximation of ground-state energy [ Therefore, beyond classic ML, here we dive into the quantum world for 3-D molecule generation with the arrival the so-called NISQ era. However, limited by the current development of quantum hardware, quantum machine learning (QML) models, particularly quantum generative models, are still in their infancy stages, especially when compared to the well-developed classic neural models, thus the current performance of QML models may not match that SOTA classic counterparts [TC
Recent efforts have been made to introduce quantum methods for molecule design. QGAN-HG [1] is a hybrid model based on Generative Adversarial Network (GAN), consisting a classic discriminator and hybrid generator. However, hybrid models cannot be implemented NISQ devices, and
"Corresponding author. Equal contribution. This work was partly supported by NSFC (92370001, 62222607) and Shanghai Municipal Science and Technology Major Project under Grant 2021SHZDZX0102.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Statevoctor
Encoder
Decoder
Latent Space
Quantum
(Re)construct Molecule
Amplitude
Encoding
R-vMF(ju K).
Condition
Condition
Condition
Condition
Vector
Layer
Vectoo
Layer
Figure I: Pipeline of QVAE-Mole and QCVAE-Mole with a vMF distributed latent space. We first use amplitude encoding get the initial quantum state vectors from classic data of molecules. Then the quantum encoder learns the mean direction in the latent space, which is used to sample latent variable 2; ~ (MF(u, se). A subsequent quantum decoder decodes z trying to match the input state vector. Then molecule is reconstructed from the output state vector of decoder. For conditional generation, we have condition vector for both input data and latent space, then we use condition qubits as well as condition layers to encode the given conditions into the quantum circuit. Solid and dashed arrows represent the training and inference phase, respectively.
QGAN-HG only utilizes quantum circuits learn features with low dimensions, which actually decreases the performance of classic neural networks. Another work SQ-VAE [15], a quantum Variational AutoEncoder (VAE) approach. represents the molecular graph as a flattened adjacency matrix and converts it into quantum state through amplitude encoding. However, it is difficult for its quantum circuit to reconstruct the topology structure by the inputted flattened adjacency matrices [6]. Furthermore. SQE adopts normal distributed latent space, where latent variables remain unconstrained by normalization requirements. Nevertheless, the quantum state outputs of VAE are subject to normalization, necessitating linear mapping with classic parameters to map the output into non-noneen normalized latent variables. This indicates the incompatibility of the normal distribution with quantum systems. In all, these two quantum methods are limited to generating molecular graphs and fall short of generating 3-D molecules.
To address the above issues and explore a quantum version of VAE for 3-D data generation (which is the first time literature), we develop full quantum VAE framework. especially for 3-D molecule generation. Firstly, we introduce a quantum encoding scheme designed for 3-D molecules to fulfill the normalization constraints by quantum states. This involves normalizing the 3-D coordinates, atom types, and an auxiliary value associated with each atom. These normalized atom vectors are then concatenated to form the initial quantum state. Secondly, we adopt von Mises- Fisher (vMF) distribution, which lies in hyperspherical space and can inherently meet the restrictions of quantum systems (the norm of quantum state is 1).
In addition, the ultimate objective is to generate molecules directly with desired properties in various domains [17]. For instance, there is growing demand for molecules with a low HOMO-LUMO gap in the field of organic semiconductor development [18]. To this end, we further present conditional version of quantum VAE named QCVAE-Mole which have the ability generate molecules with multiple desired properties. Specifically, we introduce condition qubits as well as condition parametric layers to encode given conditions into the proposed quantum ansatz, thus we can generate molecules with desired properties by giving specific condition vectors. The contributions are:
1) We propose the first fully (to best knowledge) quantum VAE for 3-D data generation and detailed quantum circuits compatible with NISQ devices. For generated quantum states, we fulfill its inherent normalization requirement via the vMF distribution in a spherical latent space.
2) Our conditional VAE version manages to encode the conditions into the quantum circuit, which is, to our best knowledge, the first fully quantum circuit capable of conditional VAE-based generation.
3) We conduct all the experiments many QML works [ [9 20) Extensive results on the QM9 benchmark show that our model outperforms all other quantum (or hybrid) methods [14] and delivers comparable results several classic methods 21 with significantly reduced parameters.
TorchQuantum- based simulation environment in line with

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2 Preliminaries
Quantum machine learning. A quantum bit (qubit) is the unit of quantum information and com- puting, which exists in the superposition state of both 0 and I, as denoted by (15) = al)) + 311). where a and / are complex coefficients. la]² and |3]² are the probability amplitudes for the qubit being in basis states (10) or [1)). For n-qubit system, there are 2" basis states. Quantum gates are the building blocks that manipulate quantum states. These gates are represented by unitary matrices. Parameterized quantum gates are quantum gates with one or more parameters. See Appendix
Bannndxxxxoo for
the single-qubit and double-qubit parameterized quantum gates used in this paper.
The concept of Variational Quantum Algorithms (VQA) was proposed by (24), which utilizes quantum advantages to solve ML problems on NISQ devices. Then, Parameterized Quantum Circuits (PQCs) serve the specific implementations of these VQAs, with the parameterized quantum gates mentioned above being key components of PQCs. The parameters 0 can be optimized by a classic optimizer to minimize loss function
((0), which evaluates the dissimilarity between the output of QPC and the target result. Even if the measurement itself does not provide gradient information, the gradients of 0 can be directly estimated by perturbing slightly. For instance, the Parameter Shift Rule (25] is popular technique used to achieve the gradients in many QML models [19] [2]. Also using gradient backpropagation, classic learning models are adapted into their quantum version, e.g.
QCNN (27]. QRNN [19], QGAN 12), QLSTM [28]. and etc, They are highly recognized for their intellectual novelty as well as their potential in the NISQ era.
Variational autoencoder. Variational autoencoders (VAEs) [21] are a class of generative models that combine autoencoders with variational inference techniques. VAEs provide a principled approach to learning latent representations and generating new data samples. Formally, VAE is represented by an inference network (i.e., encoder) (2(x)) and a generator network (i.e.. decoder) pe (z]z), where denotes the latent variables. The intractable true posterior p0(z|z) is approximated by the inference network qo (z(2z)). which is parameterized by a neural network and outputs a probability distribution for each data point x. Given the training set D { J and the prior p(z). the final objective is to minimize the negative of Evidence Lower Bound (ELBO):
min LELBO (0, x) = - -Esogo (=(u) [log (po (x|[=) + DrL. (=<())))l)(=)) 0.0
serves as proxy for the ligelike of the data with a regularizer. The first term is known as reconstruction loss, and the second is the Kullback-Leibe (KL) divergence.
3 Methodology
We leave a detailed discussion on related works in Appendix A including classic methods for molecule generation and quantum generative learning. In this section, we first introduce how to encode classic data into a quantum state and then elaborate on the quantum architecture of QVAE- Mole and QCVAE-Mole. Fig. [ shows the pipeline.
3.1 Encoding 3-D Molecule to Initial Quantum State
We use an attributed point cloud {(1; a to represent a 3-D molecule with n atoms, where each point represents an atom. Here v. E R denotes the 3-D coordinates of the atom and
a, E (0, 1} denotes the atom type (k is the number of atom types). To encode the molecules into quantum states, we normalize them in two aspects: 1) Each 3-D molecule can undergo ar- bitrary translations and rotations, and there is no inherent ordering among atoms, which requires us to fix the order of atoms and the molecular conformation to obtain unique encoding. 2) The amplitudes of quantum states require us to keep the input data of the molecule within the positive octant. See details of normalization in Appendix
[xo Yo. 2n
Arpliaaa
2x Byr 2,4
Canca
29
[0, ... 1,0%%
1 Eye
x - -
Figure 2: Classic data encoding. We encode 3-D molecules with 3-D coordinates and atom types through three steps: Dintroduce an auxiliary value, 2) normalize to a norm 1.0, and 3) convert into a quantum state vector via amplitude encoding.
3

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We will introduce how to encode the normalized 3-D molecules into quantum states. It converts the given classic data sample z into its corresponding quantum state ( Here, we choose amplitude encoding, which allows encoding. The data of length N is encoded into the amplitudes of q = flog, N qubits. In this case, we need to encode each atom with its 3-D coordinates as well as the one-hot form of atom types.
to utilize the exponentially large Hilbert space compared with angle
As amplitude encoding demands a state vector with the unit norm, inspired by [2 here we introduce an auxiliary value /3 z yr 2 for each normalized atom 0 (zz: 3/s 2;) as constant normalization factor. Therefore, we need 4 + k entries for each atom (3 for 3-D coordinates, & for one hot embedding, for auxiliary value). and the total number of entries is n (4 k).. See Fig. 2 for our encoding. Furthermore, the state vector size is always a power of 2, so we fill the remaining entries with zeros (padding entries) and obtain state vector of size 29. The norm of the state vector is 1, thus we need to normalize all values by 2/n.. We get the initial quantum state of molecule by:
1
(100)
(ndrs)) + wrr + 1) 2,/r: + +2 1/r 3 42)
n
2
+ 3 - x - v2 r. + t + 33 +
Σ
Olj),
(=(((1++))
where (k + 4) * i and t; is the atom type of i-th atom. The discussion of the initial state preparation can be found in Appendix C
Qubits complexity analysis. The number of qubits in our proposed framework comes to q = [loga(Mmaz (4 + k)] O log n), where Rmax represents the maximum number of atoms. The qubits complexity analysis of other quantum methods can be seen from Appendix C
3.2 Full Quantum Architecture
We propose QVAE-Mole, fully quantum circuit-based VAE for 3-D molecule generation. Like many works on QML, e.g. QCNN QGAN [12, QLSTM (28). we follow the architecture of its classic design, the VAE in our case, which generally includes encoder, decoder, and latent space. In fact, proposing a quantum counterpart as well as its detailed quantum circuits compatible with NISQ devices is nontrivial, especially for 3-D data (molecule) generation.
3.2.1 Encoder
We present the encoder network of our QVAE-Mole. Similar to classic neural networks, the PQC is built layer by layer, where each layer consists of the same arrangement of quantum gates with different trainable pa- rameters. Fig, 3 depicts the general framework of the quan- tum encoder ansatz. Denote U.,))), (ene(0' (f'r) as the (-th single-qubit layer and entangle- ment layer with trainable param- eters, respectively. The unitary matrix of the proposed encoder can be formulated as follows, where L denotes the total number of layers.
Singlqphii Layer (,(0))
Entonglamon Layer Ua m
R.(o
R,(0))
R.(0
Ry(0")
CR2 CR (P)))
[,(0
R,(0
CR, (o
R,00%)
R,cor")
Figure 3: The ansatz of our quantum encoder. It takes the initial quantum state (vo) as input and outputs the mean direction of vMF distribution. Each layer includes a single-qubit layer U, and an entanglement layer Uent- In the end, we trace out the state of the last qB qubits: qB = qA-
I.
U(0) = U.
U (U..)(((..)).,(())))
(3)
E
Specifically, the I-the trainable single-qubit layer is formulated as:
U,(0'))
R₂ (p(p.))
Ry (o(x.)) R2 (0(z..)
(4)

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where ( ((r,)) denote the parameter of the first and second R₂ gate at the 1-th layer on the p-th qubit, respectively. And the I-th entanglement layer can be formulated as:
Uent(0
(On ent
II (CRx (p. + 1)%4, ( )) p=1
(5)
where CRx the controlled Rx gate, p denotes the control qubit, (p + 1)%o denotes the target qubit, and (fP.)) denotes the corresponding trainable paramenter. The quantum state after the encoder ansatz is 1 ('o). Detailed discussions and the theoretical analysis of the expressive power of the designed quantum circuit are shown in Appendix E
At the end of the encoder circuit, we introduce measure reduce the dimension of the output quantum state. Inspired by 30), we divide the encoder circuit into two subsystems, namely subsystem A with 9A qubits and subsystem B with dB qubits, and 9A QB q. Therefore, the output quantum state the encoder ansatz becomes pb))) AB = ((0)
3.2.2 Latent space
To convert to latent space, here we discard the information contained in the subsystem B via tracing the state of qa qubits. This approach integrates quantum non-linearity into the encoder, thereby enriching the transformation process with layer of complexity beyond that of a mere unitary transformation. Formally, the partial trace is:
PA Tro(PAB) E( (110(1l)))) PAB (LA (AB)
(IA
m))
(6)
where I is the identity matrix, li) (C2) Eyp are all basis states of subsystem B and pAB AB The diagonal of PA contains the squared amplitudes of (A), as can be converted into the output quantum state of encoder [v = E, VPA,, (j). Then, we perform quantum tomography (31) rather than random measurements on the encoder. The output is the vector of the latent space of dimension n More discussions about tracing out and quantum state tomography are given Appendix [F)
In common VAFS, normal distributions are assumed for the distribution of the latent variables during training. However, normal distributions unsuitable for data with a latent hyperspherical structure. The quantum state vector requires its L.2 norm to be equal to 1, which lies hypersphere space. Instead adopt using von Mises-Fisher (yMF) distribution, leading to hyperspherical latent space. Formally, the distribution is:
(2/2)) vMF(,, x)) = Ca
(7)
where ||m||| denotes the mean direction. K denotes the concentration parameter, which is commonly set as constant during training 32 The normalization constant Ca, is equal to 1/ 1/Ssa-1 e(..x) a5d-1, where & IRd is predefined parameter vector and 5d-1 is the sample space {a|æ Rdd ||=||| = 13. We set that the quantum state (1) seamlessly functions as μ for learning the vMF distribution. In addition, rejection sampling [33.32] 32 is utilized to efficiently sample latent
variable from the vMF distribution the latent space:
( A)) (lae*(1l )alie)
(
which represents that the latent variable sampled from the vMF distribution with mean direction 5) in the latent space.
3.2.3 Decoder
The decoder takes z sampled from the vMF distribution input and outputs the reconstructed quantum state, which can be further converted to a molecule. The input = has dimension of 244, and the reconstructed quantum state should have the same dimension as the initial quantum state, which is 29, This means that the quantum decoder needs to map from a lower-dimensional space to higher-dimensional one using a unitary transform. We achieve this by first turning z into the state
5

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( via amplitude encoding and then expanding | with qubits of B, which are reset to (0) B. Now we get the initial quantum state of the decoder:
w 1 )BB 1// @ 10)
(9)
We design the quantum ansatz of the decoder the same that of the encoder, then the reconstructed quantum state results in (x+)) U(0 ( here 0' denotes the learnable parameters decoder.
We can get the output quantum state ( denoted as (av.... a2g-1). Now, we convert this vector back into classic data, representing a molecule, which serves as the inverse process of encoding:
(x', 1 2 = (akr: Or,+ (kr++)),
(10)
t. argmax (ar.43 , (1,+3++(-1))++
(11)
Tv = (It + 4) + i= 1=
(12)
where x', y', z'' are the reconstructed coordinates of each atom, & is the reconstructed atom type, and n is the number of atoms. Note that n is known in training but can be arbitrary at inference. Therefore, we need to infer the number of atoms in the generated molecule from the output state vector. We set the following criteria: if (ay+3 +... + Br+3-(k-1) < T for certain i, we consider all subsequent entries be padding items instead of carrying valid information, and the number of atoms in this generated molecule comes to n = i. The hyperparameter I denotes the threshold.
Remark. We further discuss why choosing amplitude encoding instead of angle encoding from the perspective of encoding and reconstruction. For angle encoding, we obtain the initial quantum state by converting input information into the rotation angles of qubits. Although is friendly initial state preparation, it becomes intractable to reconstruct the input angles from the entangled quantum state, which is the output of the decoder. On the contrary, if we encode the input into the amplitudes of qubits, we can directly reconstruct the input information from the output quantum state vector.
3.2.4 Training
Recall the objective function in Eq. space. The vMF prior prevents the KT. collapse typically observed in Gaussian VAE settings (34). In fact, the KI. term in our loss term is constant and only depends the chosen variance K, which is hyperparameter in our model. Thus, we can simplify the Eq. to:
uniform vMF prior p(2) vMF(. 0) on the latent
min CELBO (0, o; x) = - -Bz (=|=) [lo(()(<]2))..
(13)
In other words, we only need to calculate the reconstruction loss. Here, we can calculate the loss of (40) and |ę5 in two ways. One design the reconstruction loss based on converted classic data, while the other is to construct the loss using the fidelity of the quantum state (we denoted as fidelity loss experiments). The former can achieve better model performance but requires transferring the data to classic computer for computation, while the latter can be computed by the quantum circuit, e.g. the swap test circuit. Details of loss function design can be found in Appendix [
3.3 Conditional Generation
We further propose a frame- work named QCVAE-Mole, ex- tending upon our above QVAE- Mole by adding certain con- ditions. Unlike the classic
Cooditon Layer U:(05)
(R.(o!
CR.
CR.(0
Halt
CVAE [5| 36), which achieves conditional generation by sim- ply adding specific condition vectors to input data and la- tent space. We design en- code conditions into the quan- tum circuits of the encoder and decoder. Specifically, suppose there a multi-condition vec- tor [e1, C2, ..., Cml, where each
CR.
CR.10-00
Figure 4: The ansatz of QCVAE-Mole: it incorporates m ad- ditional condition qubits for m target properties and additional condition layers. At the end of the circuit, further trace out the state of m condition qubits.

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item corresponds to the normalized value (ranging from 0 to 1) of one specific property. Then, we use angle encoding encode one property into the parameter of one quantum gate, and the initial quantum state becomes:
M Ry ((G: - 0.5) x 2r)) x 10)-
(14)
As shown in Fig 4 we extend the scale of the previous quantum circuit and encode the conditions in the last qubits construct our condition layer. The initial quantum state the encoder becomes:
Ivo! |/bo & (Va,) 60 (V...).
(15)
Similarly, the input of the decoder undergoes the same change. The unitary matrix of QCVAE is:
U = U (0) II (U"m) (O)U'
(16)
where the I-th condition layer can be formulated as:
( = II IT R (CR i,,, p(.
(17)
Note that every target property requires one more qubit for QCVAE-Moke which means we have more qubits than QVAE-Mole, but the output dimension of the encoder and decoder should remain the same as vanilla QVAE-Mole, which can be achieved by tracing oul the condition qubits at the end of the quantum circuit. The process of training is similar to QVAE-Mole and the decoder of the trained model generates molecules with the desired properties according to the given condition vector along with the latent vector.
Remark. QCVAE-Mole can be regarded as solving the inverse problem of property prediction, as particularly done in g Indeed, generation could be much more challenging than prediction.
Experiments
In line with other QML works 8 19 20] our experiments are conducted in a simulation environment. Specifically, we use machine with an i9-10920X CPU, RTX 3090 GPU, and 128G RAM. The source code is written by PyTorch and TorchQuantum (37 is used as the quantum simulator. In addition, the ansatz of QVAE and QCVAE are all hardware efficient, thus the model can be directly trained on NISQ devices theoretically. Although our approach may involve initial state preparation and quantum state tomography like [15 29), there are many efficient solutions to address these two challenges, such as 42), and this not the focus of our paper. Detailed discussions initial state preparation and tomography can be found in Appendix C and respectively.
4.1 Setting
Data. We evaluate QVAE-Mole and QCVAE-Molc on QM9 43), which is a popular dataset that contains molecular properties and atom coordinates for 130k small molecules with up to heavy atoms. We train our model to either randomly or conditionally generate molecules with 3-D coordinates and atom types (C, N, O, F). We use the train/val partitions introduced in 44 which consist of 100K/18K samples, respectively.
Actual number of qubits. When evaluating QM9, we need 7 qubits for random generation, 8 qubits for single-condition generation, and 11 qubits for multiple conditions, respectively.
Metric. In line with 23 45 all the generated 3-D molecules are converted to molecular graphs by the method in [46], and the molecular graphs can be converted to SMILES deterministically with the rdkit [47] toolkit. We use the chemical validity percentage (Valid), uniqueness (Unique), and novelty (Novel) evaluate the generation quality of QVAE-Mole. Validity measures the percentage of molecules that comply with chemical valency rules, ensuring chemical plausibility. Uniqueness assesses the proportion of distinct molecules generated, promoting structural diversity. Novelty evaluates the fraction of molecules not found in the training data, indicating the model's 's capacity to generate new compounds. To evaluate the molecular geometry, we use the average Maximum Mean Discrepancy (Avg. MMD) 48 distances of bond length distributions (see details in Appendix G Note that it unreasonable to only consider novelty and uniqueness without validity [49]: like the extreme case if the model's validity is only 1 16, but these valid molecules are all unique from each
7

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Table I: Comparison of different methods QM9. All metrics were evaluated 10K randomly generated molecules. The method with i denotes generates molecular graphs, not 3-D molecules, and * denotes it involves quantum computing. #C/Q denotes the number of classic and quantum parameters respectively, and time denotes the inference time to generate one molecule.
Methods
Model
Class
Valid
Unique Valid
Novelx Valid
C/Q Params
Time(s)
MLP-VAE
P1
VAE
One-shot
51.26%
3.43%
48.66%
360,448 0
0.04
B-NFs 22
Flow
One-shot
41.14%
40.83%
34.91%
647,117/0
0.27
G-SchNet
Sumpling
Autoreg.
82.35%
73.29S.
67.08%
902,111/
0.41
NeSphereNet 23
Flow
Autoreg.
82.63%
13
29.75%
37.77%
3,148,095 /0
0.55
EDM [49
Diffusion
One-shot
91.93%
90.72%
75.32%
5,340,921 70
0.86
SQ-VAE-1
[5
VAE
One shot
44.23% L.
7.24% 1.2
16.32% 088
128 / 224
0.15
QGAN-HG'*
14
GAN
One-shot
66.64% = 0.S
8.08% +: a.
18.48% 1.0
453,644/ 38
0.04
P2-QGAN-HG-1
GAN
One-shot
17.64% 122
12.38%
9.54% 1 08
112,524/
0.02
QVAE-Mole*
VAE
One-shot 78.13%
27.39%
57.38%
0/ 224
0.08
QVAE-Mole (fidelity loss)"
VAE
One-shot
74.39% us
26.93% a
31.50%
0/224
0.08
other and different from the training set, resulting in 100% for both uniqueness and novelty. Thus, we adopt Unique x Valid and Novel Valid as metrics.
Baseline. We adopt two kinds of methods our baselines. One category is the classic generation
model for 3-D molecules, including MLP-VAE (2T] E-NFs [22). G-SehNet [50], G-sphereNet [23] and EDM [49]. Another category contains quantum model SQ- SQ-VAE [S] and hybrid model QGAN- HG [44 (P2-QGAN-HG is a variant of QGAN-HG) for molccular graph generation. Note that SQ-VA VAE still introduces several classic parameters since needs a linear mapping in latent space. As for QGAN-HG, it simply utilizes quantum circuits to output a feature vector for the classic generator of MOLGAN [51] To the best of our knowledge, we are the first full quantum model without any classic parameters for 3-D molecule generation.
4.2 Random 3-D Molecule Generation
The results are shown in Table [ reported as the mean with standard deviation across three runs. training details and results of the MMD distance comparison shown in Appendix G
The
Compare with classic methods. The results show that QVAE-Mole surpasses classic MLP-VAB with notable margin in all metrics, indicating the potential advantages of quantum circuits over classic MLPs. However, there is still a performance gap between our method and the other SOTA baselines, especially EDM, method based on the diffusion model. On the other hand, when assessing the efficiency of the model, it is also necessary to consider the number of parameters utilized. We can that our results are very close to the auto-regressive method G-SphereNet and even superior to E-NFs to some extent, which uses 3, 1,880995 and 647.117 parameters, respectively. In contrast, model only uses 224 quantum parameters instead. Furthermore, our model has advantage in inference speed over all the classic models except for the simple VAE. In terms of training cost, our model (even when executed on a simulator) achieves convergence within 2 hours with only a few epochs, which is significantly faster compared to classic methods (according to the original paper on EMD 49), it takes approximately 3.2 days on a 1080Ti GPU to complete 1100 epochs).
Compare with quantum methods. Our model outperforms all other quantum or hybrid approaches by significant margin in all metrics. SQ-VAE uses amplitude encoding for molecular graphs (attributed topology), while the input of our model is a 3-D molecular structure (attributed 3-D point cloud). This indicates that amplitude encoding cannot model topology as effectively as for 3-D point clouds. Furthermore, we utilize a larger latent space compared with SAE and adopt vMF distribution instead of normal one, with vMF distribution naturally fitting the inherent and strict normalization requirement of output vectors. The performance of QGAN-HG is poor, probably because the quantum circuits in the hybrid model are unable to fully leverage potential advantages or due to the complex and unstable training process of GAN itself.
4.3 Conditional 3-D Molecule Generation
The objective here to generate molecules with specific desired properties. We train QCVAE- Mole (as discussed in Sec. 3.3) conditioning on four properties: synthetic accessibility score (SA),
8

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Table The results of QCVAE-Mole under single condition given. Each result is the percentage of the number of molecules that, when rounded up. have the same value as the given condition.
SA f
QED
logP
gap
Condition
0.4
0.5
0.3
0.4
0.0
1.0
3.0
4.0
QVAE-Mole 29.8
19.8
40.2
52.5
49.8
2.6
0.1
3.1
QCVAF-Mole
44.1
23.4
42.8
75.2
57.8
45.6
6.4
22.7
AQCVAE-QVAR
14.3
3.6
2.6
22.7
8.01
43.0
6.3
19.6
Comparison QVAE-Mole and QCVAE-Mod Mole with Multi condition Generution
0.4
HUMO-LUMO GAP
0.5
9.35
0.30
0.25
0.2
QVAE
0.30
QOVAE
OVAE-Male
QCVAE
Mole
Method
Method
Method
Method
Figure 5: Distribution of four properties of generated molecules under multi-condition. Dashed lines represent the given condition values.
80
Valid
Avg. MMD
80
Valad
Avg. MMD
Valig
Unique- Valid
Movei-Valid
Novel-Valdd
Percattage
60
Avg MDMD
Percentaga
60-
22 122
Ang. MMDD
40
40
1
20
A.QVAE
SOVAE
Methods
Log
(b) Figure a) Comparison of a normal distribution (A-) or a vMF one (S-) in the latent space of MLP-VAE (VAE in short) and QVAE-Mole (QVAE in short). b) Different in vMF distribution.
(a)
quantitative estimation of drug-likeness (QED), octanol-water partition coefficient (logP), and HOMO-LUMO gap (gap). See details in Appendix
Single condition. We train our model with SA, QED, logP, and gap, respectively, resulting in four models. For each model, we compare the conditional generation (QCVAE-Mole) with random generation (QVAE-Mole) and the corresponding results are shown in Table 2 Each column in Table@iiniccates the percentage 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). The results demonstrate that QCVAE-Mole improve the proportion of generated molecules with desired propertics for all given conditions, the improvement even reaches 43% when we give the condition with logP= 1.
Multiple conditions. We train the proposed QCVAE-Mole under multiple conditions, which means we give SA, QED, logP, and gap these four properties simultaneously. evaluate under combination of multiple conditions, following [52], we select reference molecule and adopt its property values as our multi-condition. Here we choose "CCI=C-C-CCN-OOCI-00 and the property values are: (SA 0.52, QED 0.38, logP- = 0.92, gap = 4.45). Fig. illustrates the comparison with random generation, demonstrating that QCVAF-Mole capable of achieving multi-condition generation, albeit with less pronounced improvement compared to single-condition generation.

---

4.4
Ablation Study
Normal distribution vMF distribution. We study adopting normal distribution and von Mises- Fisher (vMF) distribution in classic VAE and quantum VAE, corresponding to Euclidean space and hyperspherical space, respectively. Here we conduct experiments under random generation setting. Fig. [[a] shows QVAE with hyperspherical latent space performs the best validity, which reveals the advantages of using vMF distribution in QVAE. In addition, we found that compared to A-VAE, N-QVAE can generate relatively reasonable 3-D coordinates, ensuring that the distances between atoms fall within the range of chemical bond lengths. However, the atom numbers and atom types generated by N-QVAE are relatively homogeneous, resulting in lower scores for novel and unique compared to V-EE We also observe that utilizing normal distribution leads to better performance in classic VAE than using the
-isse-FFisher distribution.
Compact of In Sec. 3.2.2 we have mentioned the concentration parameter A in vMF distribution is commonly set as constant during training. To further explore its impact on the quality of random generation, here we vary x from 0.1 to 100 and the results are shown in Fig. 66b) We observe that the smaller the & value, the higher the validity, but the uniqueness and novelty will decrease correspondingly. This indicates trade-off in the selection of w: when we need high accuracy, we
should choose smaller while diverse molecules are in need, we should opt for a larger
5 Conclusion, Limitation, and Outlook
We have proposed the first fully quantum VAE for 3-D molecule generation, the best of our knowledge, featuring a von Mises-Fisher distributed latent space. Moreover, we have designed a conditional version for target molecule generation. Numerical experiments show that our model could generate plausible 3-D molecules, which outperform all other quantum (or hybrid) methods, and achieve competitive performance with significantly reduced parameters compared with their classic counterparts. Though we provide detailed quantum circuits compatible with NISQ devices, due to hardware constraints, so far we have not implemented our proposed quantum eircuits on real quantum devices. We leave it for future work and further collaboration with hardware groups where tailored hardware error correction may be needed.
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Appendix
A Related Work
Classic methods for molecule generation. Classic deep learning for molecule generation can be roughly divided into two categories. The first category focuses on generating molecular graphs, which has been extensively studied in the research community. These methods include generating the node type and adjacency matrix of the graph all at once 53)5 generating molecular graphs using fragments or motifs [56]59), or sequentially adding nodes and edges to molecular graph [60-62]. Another category focuses on a relatively under-investigated domain, which involves generating the 3-D molecular structure. G-SchNet (50) and G-SphereNet [23] use auto-regressive models to generate molecules in step-by-step manner via progressively connecting atoms. E-NF [22) introduces an cquivariant normalizing flow that incorporates a differential equation. Recently, with the emergence and success of diffusion models (DM) 63H65), there has been notable shift towards utilizing DMs
for 3D molecule generation 44 66
Quantum generative learning. There are four types of quantum generative models: Quantum Circuit Born Machines (QCBMs) 67 68), Quantum Boltzmann Machines (QBMs) [9]. Quan- tum Generative Adversarial Networks (QGANs) 77 122, and Quantum Variational Autoencoders (QVAEs) 71 72). The complexity of QCBM-baser and QBM-based methods high, which makes them difficult to implement on NISQ devices. QGAN conceptualized by 770 díscusses the potential merits of QGANs when either the generator, the discriminator, or both are implemented on quantum computers. However, balancing the training rates of quantum generators and discriminators remains a critical challenge. The QVAE introduced by [71] adopts a classic VAE structure [2]] 73 and a quantum prior distribution in the latent space realized by QBM model. However, it remains an open question whether there is a quantum circuit realization of QBM. In contrast, we explicitly present a full quantum VAE circuit. As for quantum generative learning for molecule design, QGAN-HG [14] introduces a hybrid model upon MOLGAN, and SQ- VAE [3] proposes a circuit- QVAE, which more hardware efficient compared QVAE based on QBM model. In this paper, follow the paradigm of circuit-based quantum VAE and propose a fully quantum VAE for 3-D molecule genera- tion with the capacity of conditional generation. Although SQ-VAE is also designed for molecular generation, 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 nulti-conditional control as encoder/decoder while SQ- VQE uses a hybrid quantum-classical classical layer. In defining the latent variable space, we employ the vMF distribution harness the inherent properties of quantum states, while they simply imitate classical VAE using Gaussian distribution.
B Quantum Computing and Parameterized Quantum Gates
Quantum states are typically represented using bracket notation. It is common practice to create linear combinations of these states, known as a superposition, exemplified by [w) = a (0) + 1). In formal terms, a quantum system composed of n qubits is represented by an n-fold tensor product Hilbert space H. = (C²) Sd
which has a dimension of 2d. For any state (&b) within H. its conjugate transpose is denoted as { = |s)), The inner product (vb) = w1 calculates the square of the 2-norm of sb. The outer product (e)) (1b) forms rank 2 tensor. The computational basis states are defined as (0) = 0) and [1) = (0, 1), while composite basis states, such as (01) = 10) @ [1] (0, 1, 0,0,0, 0),
extend these definitions.
The single-qubit and double-qubit parameterized quantum gates used this paper contain:
cos( 0 sin
sin ( cos
e 2 0
Ry
R2
(8)
?)
2
0
0
1
CR
=
0
0
cos( ())
i sin( 9 cos(?
0
0
sini ()
15

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Similar to classical computer, quantum computer is constructed using quantum circuit composed of wires and basic quantum gates, which facilitate the transport and manipulation of quantum information. Fach quantum gate represents unitary operation, denoted as U, on a Hilbert space H. When simulating a quantum cireuit on a classical computer, the complete unitary transformation achieved by tensoring and multiplying these unitary gate operators together.
projective measurement involves an observable, denoted as M, which Hermitian operator acting on the state space of the observed system. This observable can be broken down into a spectral decomposition, represented as MM = Em mP m. where each Pmm. m is the projector onto the eigenspace of M corresponding to the eigenvalue m. When the state (bb) is measured, the probability of obtaining the result is determined by p(m) (v) P ml 1)).
C Encoding Details
The dataset is set of M molecules, which we represent as M 3-D point clouds, each with n, nodes, To ensure that molecules can be uniquely encoded to a quantum state vector, we need to fix the order of atoms and the molecular conformation during input. Furthermore, the probability vector containing the amplitudes restricts the generated vector to the positive octant, which requires us to keep the input data of molecule within the positive octant as well.
To fix the order of atoms, we first convert the 3-D molecules to canonical SMILES strings using rdkit. We then encode the coordinates and atom type of each atom onto the quantum state vector according to the canonical SMILES strings. As for the molecular conformation, we first align the center of mass of the entire molecule to zero and then rotate the position of the first atom in the SMILES string onto the z-axis. Now molecules in dataset with arbitrary conformations and atom orders can be uniquely represented by [(v). x })) - 1==0
In addition, to keep the molecule within the octant, we define an axis-aligned bounding box (Vmin, Vmaz) across the entire dataset. This is done by defining the minimum and maximum values along each axis.
Umin .a
min M
8,0
Umax ,a
max M
1.00
where v/, E R is the coordinate of vertex of along axis E {x, y, 2). To further achieve isotropic re-scaling, we turn the bounding box into a cube with side length s = m&X Umax,a Umin.e Then the coordinates of each atom can be shifted and re-scaled by
(o ", = H{
Vmin ny 1
1=0 1==
Now we get the final normalized dataset.
Initial state preparation. There are several techniques for encoding classical data into quantum states, including basis encoding, angle encoding, and amplitude encoding (74). We adopt the amplitude encoding scheme encode the input 3-D molecule or sampled classical latent variable as a quantum state. The reason we adopt amplitude encoding is that it can encode 2" classical values using only qubits, offering exponential encoding advantage. In the existing literature quantum algorithms, amplitude encoding is commonly employed to harness the power of quantum computation (75]. The implementation of amplitude encoding is complex, but there are many works dedicated to investigating how to efficiently encode information into amplitudes (38), such as top-down encoding [39] bottom-top encoding [39], MPS encoding (76), or quantum architecture
search (77).
The qubits complexity analysis of QGAN-HG and SQ-VAE. QGAN-HG [4] is hybrid model with a classic discriminator and a hybrid generator, which utilize quantum circuits to generate feature vectors. QGAN-HG obtains the output of the circuit through measurements and yields q-bit string (here q is the number of qubits). Thus the qubits complexity comes to 4 = d = O(d, where d is the needed feature dimension.
SQ-VAE [5] uses amplitude encoding to encode molecular graphs. First, the molecular graph represented by node matrix (n x k) and bond matrix (n x x 6), is the number of atoms,
16

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is the number of atom types and b is the bond types. Then, the node matrix and bond matrix are flattened get a vector with the dimension of n l + n n k. Thus the qubits complexity comes to [log2 (n k + k)] =
log(n))
D Loss Function Design
Classic loss. The classic reconstruction loss consists of four parts: 3-D coordinate loss Li, atomic classification loss L2, constraint loss L3. and auxiliary loss C1. We use geometric distance error for 3-D coordinate loss, weighted cross entropy for atomic classification loss with weight Wy and one-hot label for each atom type t. Note that the sum of the probability of each atom type 8; +3 ................................ + a +++3+(k-1) should be as equal as possible for each i, with the expectation of R[s:] = 2 )2= = I. To this end, we design a constraint loss via mean square error. Furthermore, the auxiliary loss is designed for the auxiliary value and the padding entries. In summary:
än
(
2
2
1/2
L1 Σ
04
Di
0rr
2j
(18)
2
2/m
2/T1
L2 =
-2=-E,, Weya log(ast):
E,E,
Lz
(si
1
(19)
4n
64= Σ 0r44+3
3 x
y
2
Σ
layl-
(20)
2/Ti
(=(n=(4++))
Specifically, L1 supervises the reconstruction of the molecule 3-D position by geometric distance error, and L2 supervises the reconstruction of atom types by weighted cross entropy. Lg 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 L4 is designed to supervise the reconstruction of these entries of zero. With M data samples in total, the final loss comes to:
1 E(s + aCz + BCs + Caa), M
C=
(21)
where ,3 y are the hyperparameters to balance the losses. The best hyperparameter configuration can be determined through grid search.
Fidelity loss. It quantifies the "closeness" between two quantum states, and the formal definition between two quantum states p. F(p, a)) = (tr vp)² For pure states, p = 1b)p) ( and -/>>) (vbr,, the fidelity equals to (wp We construct the fidelity loss:
f
F( (hr) = ((to 1
(22)
The design of fidelity loss does not consider the real physical meaning 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.
E Discussion of Circuit Design
We choose R2 (0) as the single-qubit trainable layer, and next we will introduce the reason. An arbitrary single-qubit operator with the unitary matrix as U can be decomposed into a sequence of R2, Ry, and R2 gates, and global phase (78).
= e R2 BB Ry (0, (2))
(23)
Proof. A 2 x 2 unitary matrix has the following general expression:
U= aa b
b"
cia
Vva,
(24)
where a. b are complex numbers, and a is a real number, in is the phase part, and V is also a unitary
matrix. Notice that the determinant of V satisfies det. V aa" + bb*" |a|2 |b/2 1 1( denotes the conjugate operator). Consequently, we have
det U e 2ic det V E Zia
(25)
17

---

Algorithm vMF sampling
Input: dimension m, direction jr, constant
I: sample U (sm==2:
2: sample w ~ g(w rs, m) & exp(wx) (1 - w²) 2
3: z' <- [w: (vT w²) T)=
4: U < Houschollerr (e1, μ); [Houscholder transform)
5: Return: Uz'
Algorithm 2 Householder transform
Input: mean μ4, modal vector e₁.
I: u' +- 6 JG
2: u T
3: U t 2
4: Return: U
Let e
-0 and b = -
cos 0 2
sin
01
then the unitary matrix V becomes:
2
to
sin 01 cos
V
(26)
sin
2
Decomposing the matrix V, we can obtain:
cos 01
sin 01
c 80
V
+
sin gf
cos
9
2
0
e d0
(27)
Due to
cos( sin
sin ( cos( 5
Ry (0)=
R2
(28)
el
the matrix V can decomposed as the sequence of parameterized rotation gates R2 R,, i.e.
V (02) (z)))((()))
(29)
Therefore, arbitrary unitary matrix be represented by sequence of Rz, Ry and R2 gates, and phase eia Because it does not affect the outcome of experiment, we discard the global phase as consensus. Thus, we use the combination of Rg, Ry and R, in the trainable layer.
F Discussions of Latent Space
Trace out and quantum state tomography The tracing out operation similar to marginalizing out specific qB binary dimensions of g-dimensional probability distribution when treating the squared amplitudes of (v)) AB as a distribution. After tracing out qB qubits, the dimension of the quantum state comes to 294.
Readout of the quantum state may require an exponentially large number of measurements. However, there are many methods proposed to tackle this problem. For instance, Schmale et al. [40] propose an efficient quantum state tomography with convolutional neural networks. Moreover, Rambach et al. [42 provide practical, efficient, and robust self-guided tomography for measuring high dimensional quantum states.
Rejection Sampling Here we provide the sampling procedure in Algorithm
G Supplementary Experiments
Metrics. Validity is defined as the percentage of molecular graphs that do not violate chemi- cal valency rules. Uniqueness measures the proportion of molecules that have different SMILES
18

---

Table 3: Hyperparameter choices of and the training phase settings.
Hyperparameters
Values
Quantum parametric layer (L)
8
Ansatz
Qubits in subsystem B (qn)
2
Concentration parameter in vMF (k)
10
Generate threshold (I'
0.2
Batch Size
512
Training
# Epochs
50
Optimizer
Adam
Learning Rate
0.01
Weight Decay
le-4
Table 4: Comparison of MMD distances for bond length distributions in different methods.
MMD distances
Method
C-C
C-N
C-O
C-F
C=C
N-N
N-O
C=O
Average
MLP-VAE 21
2.85
7.39
2.65
4.12
1.97
7.26
2.14
1.49
3.73
E-NFs 22
0.56
0.18
1.02
3.31
2.99
1.05
0.69
1.76
1,45
G-SchNel 50
0.18
0.08
0.32
0.184
5.34
0.25
0.25
1.58
1.02
G-SphereNet (23
1.02
0.17
0.78
0.23
2.47
1.65
0.91
0.22
0.93
EDM [49
0.04
0.03
0.06
0.34
5.14
0.17
0.07
1.32
0.90
QVAE-Mole
2.26
1.77
2.88
1.70
2.35
1.85
2.26
3.63
2.34
QVAEMoole (fidelity loss)
2.69
2.38
1.79
3.42
3.05
3.41
3.34
4.46
3.07
strings (which implies that they are non-isomorphie). Novelty measures the proportion of generated molecules that are the training set. To evaluate the generated molecular geometry, we evaluate by the Maximum Mean Discrepancy (MMD) [48] distances of bond length distributions. Specifically, for specific type bond, extract its length distribution from both the generated geometries and the dataset's ' geometries individually. We then calculate the statistical discrepancy between these two length distributions using the MMD distance. Avg.MMD is the average MMD of all bond types.
Target properties. SA represents the difficulty of drug synthesis, with higher values indicating easier synthesis as it is normalized between O and 1. QED quantifies the likelihood of a molecule being a potential drug candidate. LogP indicates the molecule's 's partition coefficient between octanol and water, where log? values between -0.4 and 5.6 are considered favorable for drug candidates [79]. HUMO-LUMO gap sigmifies the energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO).
Training Details. In the TorchQuantum simulator, it is known that the running of quantum circuits is simulated with unitary operations, and the unitary operations can be implemented through Py PyTorch's ' underlying tensor operations. Thus obtaining the quantum state vector in the simulator only requires one forward unitary computation, which makes the entire process differentiable TorchQuantum. In this way, we can directly get the gradient of quantum parameters, perform backpropagation, and train the model. We use stochastic gradient descent with Adam optimizer [80 to train our model for a maximum of 50 epochs with batch size of 512 and a learning rate of 0.01. The choice of hyperparameters is shown in Table 3 The parameters in both the training and inference stages of QVAE-Mole and QCVAE-Mole remain the same.
Evaluate 3-D molecular geometry. Following 23 we compute the MMD distances on eight most frequently appeared types of chemical bonds, including carbon-carbon single bonds (C-C), carbon-nitrogen single bonds (C-N), carbon-oxyee single bonds (C-O), carbon-carbon double bonds (C=C), nitrogen-nitrogen single bond (N-N), nitrogen-oxygen single bond (N-O), carbon-oxygen double bond (C=O). All metrics are computed from 10k generated molecular geometries and the results are shown in Table Since SQ-VAF and QGAN-HG cannot generate 3-D molecules, here we only compare QCVAE-Mole with classic SOTA methods.
19

---

H Impact Statements
Quantum computing can play a significant role beyond molecule generation. Thus we shall be cautious about this technology especially when quantum generative Al could have a broad impact on society.
20

---

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Reviews:
[{"review_id": "F7Zos3tCK9", "reviewer": "Reviewer_VsUh", "summary": "The paper aims to realize 3D molecule generation on quantum hardware and proposes quantum parameter circuits for 3D molecule generation. The 3D coordinates and atomic types are explicitly encoded as the initial quantum state and input into the network. The paper selects the classic generative model Variational Autoencoder (VAE) to encode the quantum state into the latent space and decode the samples to generate novel molecules, and names the proposed architecture QVAE-mole and QCVAE-mole for conditional generation. In order to inherently meet the limitations of the quantum system, in the proposed architecture, the Von Mises-Fisher (vMF) distribution replaces the normal distribution used by VAE.  The paper conducts experiments on the QM9 dataset and compares the results with classical and quantum methods. The results show that the proposed method achieves a good balance between performance and speed.", "questions": "* Why do the methods proposed in Table 1 have zero classical parameters? Would moving some quantum parameters to classical parameters speed up the model?\n* 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]?\n* In line 348: \" ..., which proves classical data tends to follow a normal distribution.\". Do VAE and QVAE use different datasets? Please clarify this.\n* 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?\n* 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?\n\n\n\n\n[1]. Quantum autoencoders for efficient compression of quantum data.", "limitations": "The paper addresses the limitations of the hardware in Section 5. However, some limitations should be explicitly pointed out:\n* How to improve the performance to obtain similar performance to classical methods.\n* How to accelerate the method to make it as fast as other quantum-based methods.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "* The paper proposes a method to encode atom type, geometric information, and constrained unit form information into the initial quantum state through amplitude encoding. The paper briefly explains the reasons of choosing amplitude encoding instead of angle encoding.\n* The paper proposes to use vMF in a hyperspherical space, which can inherently satisfy the constraints of quantum systems. Ablation studies show that architectures with vMF distribution perform better than normal distribution except the metric Novel x Valid.\n* The paper verifies the effectiveness of single condition generation from four properties: SA, QED, logP and gap, and multiple conditions generation. The trained conditional generation architecture performs better than the random generation of QVAE-Mole, which shows the effectiveness of conditional generation.", "weaknesses": "* Compared with classical molecule generation methods, the performance of the proposed method is still lag behind. For the metric Unique x Valid, the proposed method only outperforms the MLP-VAE proposed in 2013. For the metric Novel x Valid, the performance of proposed method is still lower than most classical methods.\n* Compared with quantum methods QGAN-HG and P2-QGAN-HG, the speed of the proposed method is still lag behind.\n* It is unclear what **Valid, Unique** and **Novel** mean and how to define these metrics. The ambiguity of the evaluation metrics will lead to unfair comparisons.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n> **W1: Compared with classical molecule generation methods, the performance of the proposed method is still lag behind.**\n\nThanks. 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.\n\n> **W2: Compared with quantum methods QGAN-HG and P2-QGAN-HG, the speed of the proposed method is still lag behind.**\n\n\nTable 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.\n\nQGAN-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. \n\n\n> **W3: It is unclear what Valid, Unique and Novel mean and how to define these metrics.**\n\nThanks for the suggestion, we will add the detailed definition of each metric to our paper.\n\n+ 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.\n\n+ 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.\n\n+ 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.\n\n> **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?**\n\nAs 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.\n\n> **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]?**\n\nThanks, [1] aims to propose a quantum AE， and traditionally, AEs are used for dimension reduction or feature learning [2,3,4].\nIn 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.\n \n> **Q3: In line 348: \" ..., which proves classical data tends to follow a normal distribution.\". Do VAE and QVAE use different datasets? Please clarify this.**\n\nThanks 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.\n\n> **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?**\n\nThank 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "v9gyl40t0z", "reviewer": "Reviewer_CAJp", "summary": "The authors introduce the first Variational Autoencoder (VAE) and Conditional Variational Autoencoder (CVAE) formulated entirely as parameterized quantum circuits (PQC), as opposed to hybrid methods that combine learnable parameters in quantum circuits with classical learnable parameters. In previous hybrid models, classical parameters were necessary to translate between the normalization constraint of quantum states, where the norm evaluates to unity, and latent normal distributions, which are not constrained by norm. The authors suggest using the von Mises-Fisher distribution on a hypersphere as prior latent space distribution, which automatically normalized the samples to unity. This approach allows them to formulate the entire VAE as a PQC.\n\nTo construct a CVAE, the authors encode conditions into the initial quantum states. Experiments on the QM9 dataset demonstrate that using the von Mises-Fisher distribution as a latent prior improves the performance of the proposed VAE. Furthermore, the use of the CVAE helps to align sample properties with the conditions, compared to unconditional sampling.\n\nThe authors compare the share of valid, unique, and novel VAE samples with classical and hybrid quantum methods. The proposed VAE outperforms the hybrid and two classical models but performs worse than three other classical models.", "questions": "1. How were the performance numbers for the baselines reported in Table 1 obtained? Were the baselines retrained and how were the hyperparameters chosen? Please also compare with the weaknesses section.\n2. Line 121: Why does the amplitude encoding allow to use the exponentially large Hilbert space in contrast to the angle encoding?\n\nPost-rebuttal update: I raised my score to 4", "limitations": "The biggest limitation of the presented method is that it is outperformed by classical approaches, e.g. EDM as reported in Table 1. The paper currently lacks a sufficient discussion of this limitation and possible further disadvantages.", "rating": 4, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "1. The von Mises Fisher distribution as prior for the latent space is introduced. This distribution is preferable for quantum states since it satisfies the constraint that the norm of sampled vectors is unity.\n\n2. On the quantum simulator, the proposed method can generate samples faster than all methods that have better metrics.\n\n3. The authors compare the proposed method to many baselines, however, the description of the baselines lacks details.", "weaknesses": "1. Lack of novelty: The paper combines the 3D representation, PQC (parametric quantum circuit) and dimension reduction approach from 3D-QAE (https://arxiv.org/abs/2311.05604) and the variational Autoencoder from SQ-VAE (https://arxiv.org/abs/2205.07547) with the only major modification that the latent prior is chosen as the von Mises-Fisher distribution and that the 3D points can have a type (the element), which is one-hot encoded.\n\n2. The data reported for the outperformed baseline SQ-VAE cannot be found in the respective paper, for E-NFs and QGAN (https://arxiv.org/pdf/2101.03438), the metrics are different than reported. It is not stated whether these models were retrained or if so, which hyperparameters were chosen or how this retraining could be reproduced for validation. The experiment section of the MLP-VAE paper does not contain any application to molecular or 3D data at all, and it is unclear how hyperparameters were chosen for the encoder and decoder networks for this specific task.\n\nSince the method lacks major novelty, detailed comparison to the baselines and ablations are essential.\n\n3. For the conditional generation, it is not stated how\na) the properties like logP values of generated samples are obtained (this typically requires MD simulation or quantum chemical calculations) and\nb) How the equality of continuous properties is defined for Table 2: The evaluation here seems to be problematic. For example, the logP values in the violin plot in Figure 5 are negative for a significant portion, rounding all these values up, as described in the caption of Table 2, seems to be an arbitrary decision. This would mean that values are rounded to either 0, if negative, or 1, if positive, instead of rounding them to the nearest integer which would result in worse performance than the values reported in Table 2.\nc) The data does not support the results presented for logP in Table 2: An improvement for QCVAE-Mole from 2.6 to 45.6% for logP=1 is reported. We find that this is not supported by the data. The violin plot in Figure 5 shows very low support of the distribution for logP=1, and thus the data does not support the findings reported in Table 2. \n\n4. It is not reported explicitly how the metrics validity, novelty and uniqueness are calculated.\n\n5. Line 219: It is stated without proof or explanation that the KL term in the ELBO loss is constant. This does not seem to be the case since the KL divergence should also depend on the location of the mean and not only on the variance.\n\n6. Line 180: “In common VAEs, both the prior and posterior are defined as normal distributions.”: The priors can be defined at wish, often one chooses normal distributions, the VAE does not fundamentally depend on this choice and alternative priors have been suggested in the literature.\n\n7. Line 190: “where we take the absolute value of $|\\psi_E ⟩| \\in R^{2^q A}$ to transform it from the complex domain to the real domain.” How is this operation defined? If it is defined as taking the absolute value of each amplitude (each vector component in the qubit basis), the norm of the amplitude vector (not of the quantum state) might not be unity anymore.\n\n8. Line 348: “We also observe that utilizing normal distribution leads to better performance in classic VAE, which proves classic data tends to follow a normal distribution.”: Here the use of the word “proves” is perhaps not the best choice, as the experiments do not necessarily “prove” this statement. How is “classic data” defined? The input data follows the data distribution, only the latent space follows a normal distribution, and which distribution is more suitable in latent space highly depends on the complexity of the encoder.\nMaybe rather: “which indicated that in the setting at hand, imposing a normal distribution as latent prior in comparison to the von Mises-Fisher distribution is beneficial for classical variational autoencoders.”\n\nTechnical remarks:\n9. Line 289: “To the best of our knowledge, we are the first full quantum model” -> …, we propose …", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n> **W1: About novelty (SQ-VAE (https://arxiv.org/abs/2205.07547) and 3D-QAE).**\n\nThanks, and we would like to humbly point out that you may have some misunderstandings that may affect the interpretation of our work.\n\n+ 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.\n\n+ 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. \n\nFor the position and contribution of our work, please see **general response** above.\n\n> **W2 and Q1: Questions of baselines.**\n\nAs 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.\n\n> **W3-a: How to calculate properties like logP.**\n\nIn 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. \n\n> **W3-b: How equality of continuous properties is defined.**\n\nWe 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.\n\n> **W3-c: The violin plot does not support the results presented for logP in Table 2.**\n\nWe 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.\n\n> **W4: It is not reported explicitly how the metrics validity, novelty and uniqueness are calculated.**\n\nThanks for the suggestion, we will add this part, see **Supplement to W2 & Q1** in official comment below.\n\n> **W5:: It is stated without proof or explanation that the KL term in the ELBO loss is constant.**\n\nThanks, here we provide the detailed proof, see **Supplement to W5** in official comment below.\n\n> **W6: Writing suggestion for Line 180.**\n\nThanks, we will modify this according to your suggestion.\n\n> **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?**\n \nWe 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.\n\n> **W8: Question with Line 348:**\n\nThanks for the the suggestions. We admit that using \"prove\" here is not appropriate, and we will modify this.\n\n> **Q2: Amplitude encoding vs. angle encoding.**\n\n- **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 $|\\psi\\rangle = \\bigotimes_{i=0}^{n-1} R_y(x_i)|0\\rangle$. \n- **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 $|i\\rangle$.\n- 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. \n \n> **L1: The performance does not match that of classical methods.**\n\nPlease refer **Supplement to L1** for detailed discussion. \n\n---\n\nWe 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hvHUoTxc1U", "reviewer": "Reviewer_5Ebq", "summary": "This paper introduces a Variational Autoencoder (VAE) with a von Mises-Fisher (vMF) latent space for 3D molecular (conditional) generation. This approach leverages the capabilities of quantum computing, particularly within the Noisy Intermediate-Scale Quantum (NISQ) era, to achieve efficient and effective molecular generation.", "questions": "1. Please clarify the term \"fully\" and, if necessary, explain how it compares to its counterparts.\n2. Is the latent space defined as discrete [1]? If so, please provide a rationale for this choice. \n\n[1] Van Den Oord, Aaron, and Oriol Vinyals. \"Neural discrete representation learning.\" Advances in neural information processing systems 30 (2017).", "limitations": "The emphasis on the QM9 dataset may constrain the generalizability of the results to other molecular datasets or different types of data.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper is well-structured, featuring efficient schematic diagrams and theoretical explanations.", "weaknesses": "Formulas 8 and 9 can be represented using more rigorous mathematical notation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for acknowledging our work and your suggestions have been immensely helpful. Below is our detailed response.\n\n> **W1: Formulas 8 and 9 can be represented using more rigorous mathematical notation.**\n\nThanks for your suggestion and we will revise this.\n\nThe von Mises-Fisher (vMF) distribution is a probability distribution on the unit sphere in $\\mathbb{R}^d$.\n\n- 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 $\\{x|x\\in\\mathbb{R}^d, \\Vert x\\Vert=1\\}$.\n\n- 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.\n\n> **Q1: Please clarify the term \"fully\" and, if necessary, explain how it compares to its counterparts.**\n\nThank 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.\n\n> **Q2: Is the latent space defined as discrete? If so, please provide a rationale for this choice.**\n\nThanks, 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.\n\n\n> **L1: The emphasis on the QM9 dataset may constrain the generalizability of the results to other molecular datasets or different types of data.**\n\nThanks, 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). \n\nHere 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.\n\nIn 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.\n\n| Method               |  Atom stability ($\\uparrow$) | Wasserstein distance ($\\downarrow$) |\n| -------------------- | -------------------- | ------------------ |\n| Dataset               |    86.5           |           0       |\n|  MLP-VAE         |          41.2    |      5.21          |\n| EDM (classical SOTA)              |   81.3           | 1.41    |        \n| **QVAE-Mole (ours)** |       **69.1**    |  **3.12** |\n\n\nIt 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.)\n\n---\nWe hope this response could address your concerns, looking forward to receiving your feedback soon.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "eLkXjmn7S4", "reviewer": "Reviewer_hKnP", "summary": "This paper proposes a fully quantum VAE framework, QVAE-Mole, for 3D molecule generation. It introduces a quantum encoding scheme and adopts a von Mises-Fisher distributed latent space. A conditional version, QCVAE-Mole, is also presented for property-specified generation. Experiments show that the model outperforms other quantum or hybrid methods, and achieves competitive performance with fewer parameters compared to classic methods.", "questions": "See **Weaknesses** Section.", "limitations": "See **Weaknesses** Section.", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I'm not an expert in this direction, but the approach looks novel. This paper proposes the first fully quantum VAE for 3D molecule generation, which has the potential quantum advantage, especially in the NISQ era. Adopting a von Mises-Fisher (vMF) distributed latent space to meet the inherent coherence of the quantum system, which is more suitable than the normal distribution used in previous methods. The model outperforms all other quantum (or hybrid) methods and achieves comparable results to several classic methods with significantly reduced parameters.", "weaknesses": "I have no major concerns about this article, just the following two minor shortcomings:\n\n1. 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).\n2. The **Atom stability** and **Molecule stability** metric are compared in most 3D molecular generation paper, while this paper does not show this result.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for acknowledging our work and your suggestions have been immensely helpful. Below is our detailed response.\n\n\n> **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).**\n\nThank 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..\n\n**Single-qubit quantum state.**\nIn 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 $|\\alpha_2|^2$. \n\nMathematically, 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. \n\n**Multi-qubit quantum state.** \nMulti-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 $|0\\rangle \\otimes |0\\rangle$. \n\n**Quantum circuits.** \nQuantum 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.\n\n\n> **W2: The Atom stability and Molecule stability metric are compared in most 3D molecular generation paper, while this paper does not show this result.**\n\nIt 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. \n\nMoreover, 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.\n\n| Method                  | Atom stability ($\\uparrow$) |\n| ----------------------- | --------------------------- |\n| Dataset                 | 99.0                        |\n| MLP-VAE        (classical)             | 88.6                        |\n| E-NFs         (classical)              | 85.0                        |\n| G-SchNet        (classical)            | 95.7                        |\n| G-SphereNet   (classical)         | 94.7                        |\n| EDM (classical SOTA)    | 98.5                        |\n| SQ-VAE        (quantum) | 86.2                        |\n| QGAN-HG (hybrid)        | 90.2                        |\n| P2-QGAN-HG   (hybrid)   | 69.1                        |\n| **QVAE-Mole (ours)**        | **94.3**                      |\n\n\n---\nWe hope this response could facilitate your understanding of our work and ease your concern, looking forward to receiving your feedback soon.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
# General Response by Authors

We express our gratitude to all the reviewers for dedicating their time and providing valuable comments. They acknowledged that our work is well-structured (VsUh, 5Ebq, hKnP), contributive (VsUh, hKnP), effective (VsUh, 5Ebq), and presents a novel approach (hKnP). While the overall feedback from the reviewers is positive, reviewer CAJp has some reservations regarding the novelty and performance of this paper, as well as the lack of detailed descriptions of baselines. To clarify potential misunderstandings that might affect the evaluation, we first restate the position and the contribution of our work within the field of quantum machine learning.

The position of this work is to explore a quantum version of VAE for 3-D data generation (which is the first time in literature to our best knowledge), especially for molecule generation, with potential supremacy on future quantum computers. Like many works in the field of quantum ML, e.g. QCNN [1], QGAN [2], and QLSTM [3], we follow the architecture of its classic design, the VAE in our case. Though our paper gets some inspiration from other works and incorporates common techniques in quantum machine learning, proposing a quantum counterpart as well as its detailed quantum circuits compatible with NISQ devices is still highly nontrivial. Here we list the following efforts as **contributions**:

1. We propose the first fully (to our best knowledge) quantum VAE for 3-D data generation and its detailed quantum circuits compatible with NISQ devices. For the generated quantum vector, we fulfill its inherent and strict normalization requirement, via the von MisesFisher (vMF) distribution in a spherical latent space. In addition, we provide the theoretical analysis of the expressive power of our designed quantum circuit.

2. To our best knowledge, our method presents the first quantum **conditional** VAE framework and is capable of conditional **3-D** molecule generation. This is attributed to two main factors: 1) We designed a quantum state encoding scheme specifically for 3-D molecular data, ensuring maximum preservation of the original information. 2) By employing angle encoding, we integrated conditional vectors into our proposed QVAE framework for training and generation. This approach endowed our model with conditional generative capabilities, enabling it to learn more specific 3-D molecular representations under given conditional information.

3. We carefully conducted all the experiments in a TorchQuantum-based simulation environment in line with many QML works [4~6]. Extensive experimental results demonstrate that our model **outperforms all other quantum (or hybrid) methods** and delivers **comparable results** when compared to **several classical methods**.

In the following response, we provide detailed answers to all the questions and comments point-by-point. In particular, we have provided the details of the baselines in the official comment to reviewer CAJp. We deeply appreciate the suggestions for improving this paper. If you have any further questions, please let us know so that we can provide a timely follow-up response.

**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] Recurrent quantum neural networks. NeurIPS 2020.

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

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

Meta-review: This work proposes a quantum VAE method to generate molecular configurations in 3D. The majority of the concerns reviewers initially had were resolved during the rebuttal process, and my impression is overall the strengths overweight the weaknesses, hence the recommendation of acceptance. The authors should account for these discussions in a revision.
Decision: Accept (poster)