Quantum 3D graph structure learning with applications to molecule computing
Authors: Ge Yan
Huaijin Wu
Junchi Yan
Venue: Submitted to ICLR 2023 (2023)

Abstract: Graph representation learning has been extensively studied over the last decades, and recent models start to focus on an under-explored area of 3D graph learning with 3D spatial position as well as node attributes. Despite the progress, the ability to understand the physical meaning of the 3D topology data is still a bottleneck for existing models. On the other hand, quantum computing is known to be a promising direction for theoretically verified supremacy as well as increasing evidence for access to a physical quantum device in the near term. For the first time, we propose a quantum 3D embedding ansatz that learns the latent representation of 3D structures from the Hilbert space composed of the Bloch sphere of each qubit. We convert the 3D Cartesian coordinates of nodes into rotation and torsion angles and then encode them into the form of qubits. Moreover, Parameterized Quantum Circuit (PQC) is applied to serve as the trainable layers and we take the output of the PQC as the node embedding. Experimental results on two downstream tasks, molecular property prediction and 3D molecular geometries generation, demonstrate the effectiveness of our model. Though the results are still restricted by computational power, we have shown the capability of our model with very few parameters and the potential to execute on a real quantum device.

Under review as a conference paper at ICLR 2023
QUANTUM 3D GRAPH STRUCTURE LEARNING WITH APPLICATIONS TO MOLECULE COMPUTING
Anonymous authors Paper under double-blind review
ABSTRACT
Graph representation learning has been extensively studied over the last decade, and recent models start to pay attention to a relatively new area i.e. 3D graph learning with 3D spatial position well node attributes. Despite the progress, the ability to understand the physical meaning of the 3D topology information is still a bottleneck for existing models. On the other hand, quantum computing is known to be a promising direction for its theoretically verified supremacy for large-scale graph and combinatorial problem as well the increasing evidence for the availability to physical quantum devices in the near term. In this paper, for the first time to our best knowledge, we propose a quantum 3D embedding ansatz that learns the latent representation of 3D structures from the Hilbert space composed of the Bloch sphere of each qubit. Specifically, the 3D Cartesian coordinates nodes are converted into rotation and torsion angles and then encode them into the form of qubits. Moreover, Parameterized Quantum Circuit (PQC) is applied to serve the trainable layers and the output of the PQC is adopted the final node embedding. Experimental results on two downstream tasks, molecular property prediction and 3D molecular geometries generation, demonstrate the effectiveness of our model. Though the results are still restricted by the computational power on the classic machine, we have shown the capability of our model with very few parameters and the potential to execute on a real quantum device.
INTRODUCTION
Graph representation, or specifically 3D graph representation as considered in this paper, has re- ceived extensive attention over the last decade. Beyond tasks like node classification or link pre- diction, it further facilitates various downstream applications such as molecular property prediction (Liu et al., 2021) and drug design (Gaudelet al., 2021). Recently, machine learning approaches have been well developed for learning latent node embedding on molecules (Schütt et al., 2017; Unke & Meuwly, 2019; Gasteiger al., 2019; 2021). However, the mainstream of such researches is still facing the challenges of better processing the 3D Cartesian coordinates and learning the latent representation of the 3D graph structure.
On the other hand, there are also emerging lines of researches in the area of quantum computing. State-of-the-art quantum computing hardwares are now stepping into the Noisy Intermediate-Sctt Quantum (NISQ) era, which leads to the possibility to implement applications in specific scientific domains in the near term (Preskill, 2018; Arute et al., 2019; Zhong et al., 2020; Huang al., 2020). The overlap between quantum computing and machine learning has emerged as one of the most en- couraging areas for quantum computing, as termed by quantum machine learning (Biamonte al., 2017). Quantum paradigms or hybrid paradigms have been carefully designed to fulfill quantum supremacy quantum chemistry problems (Aspuru- -uzuk et al., 2005; O' Malley et al., 2016). Ex- isting approaches mainly focus the quantum simulation of molecular energies, which enables effective prediction of chemical reaction rates. However, these quantum approaches (Romero et al., 2018 8; Peruzzo et al., 2014; O' Malley et al., 2016; Yung al., 2014) are still simulating the energies of certain small molecules like H2, LiH, etc.
In this paper, we aim develop quantum machine learning approaches learn the latent represen- tation of the 3D graph structure of molecules instead directly simulating the molecular energies with Hamiltonians. Graph learning may not be precise as molecular simulation approaches for

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property prediction, but they have the ability learn hundreds or thousands of molecules and pre- dict the properties for more complex molecules. Specifically, we first convert the 3D Cartesian coordinates of the atoms into three geometries: distance, rotation angle, and torsion angle. Then we encode the angles and distance as well as the atom type (a discrete variable), into qubits. A distance threshold is used that each time focal atom is picked learn the embedding, one only need to consider the neighboring atoms within the threshold. Considering the size of the molecules and the size of the neighborhood, we only require up ten qubits learn the representation, which makes our proposed model easy to simulate on a classical processor and capable of running on a NISQ device. Analog to the hardware efficient ansatz (Kandala al., 2017: Huang et al., 2021), we apply a Parameterized Quantum Circuit (PQC) after the encoding stage. The trainable parameters are the Os of the rotation gates R₂ and Ry in the PQC. The gradient of each parameter O is calculated by the shifting technique (Mitarai et al., 2018), and those parameters are updated by the backpropagation and gradient descent approach analog to classical neural networks. We apply a tomography at the end of the circuit and concatenate the real part and imaginary part of the output vector and then take as the node embedding. We conducted numerical experiments on the filtered QM9 dataset for both molecular property prediction task and molecular geometries generation task. Experimental results show that compared with classical state-of-the-ar baseline models, our quantum 3D embed- ding model achieves comparable results on small datasets with much fewer network parameters. We summarize our contributions as follows:
1) To the best of our knowledge, we are the first to use qubits to encode 3D relative positional information, which aims to effectively preserve the property of equivariance and invariance. In fact, using qubit on Bloch sphere to encode the rotation and torsion angle of two atoms more intuitive than using 3D Cartesian coordinates, which is also supported by the success of spherical representation on not only in molecules but also point clouds in recent studies.
2) We use two qubits to represent each atom, and we only consider the focal atom and its neighbors at each iteration. Therefore, the maximum number of qubits is 10 in our model. So we are able to test our model on Qiskit (http://qiskit.org with quantum cloud service from IBM-Q with simulator yet it guarantees that the code can also be seamlessly deployed and runnable on IBM's NISQ device.
3) We manage to implement quantum circuit full-amplitude simulator with transition unitary for the PQC on a classical processor. replicates the results yet over 20 times faster than the QASM simulator from IBM Qiskit's simulator, which enables to conduet experiments on more tasks.
4) The numerical experiments on two different well-studied molecular tasks show that our embed- ding approach is able to extract geometry and neighborhood information with very few parameters (only 64 parameters in the PQC) and achieve relatively good results.
2 PRELIMINARIES AND RELATED WORKS
In this section, we first briefly review basic concepts of quantum computing as well as quantum machine learning. We further present some previous works on quantum graph learning approaches.
2.1 QUANTUM COMPUTING
In quantum computing, qubit (abbreviation of quantum bit) is a key concept which is similar to a classical bit with binary state. The two possible states for a qubit are the state [0) and [1), which correspond to the state 0 and for a classical bit respectively. We refer the readers to the textbook (Nielsen & Chuang, 2002) for comprehension of quantum information and quantum com- puting. In this paper, we give a compact description of background for self-containess.
A quantum state is commonly denoted in bracket notation. It is also common to form a linear combinations of states, which we call a superposition: | = a (0)) + B11). Formally, a quantum system on qubits is n-fold tensor product Hilbert space H. (C2)@d with dimension 2d. For any ) E H. the conjugate transpose (e*] = | The inner product (vlw) = | denotes the square of the 2-norm of vb. The outer product [n) ( is rank 2 tensor. Computational basis states are given by 10) = (1.0). and [1) = (0, 1). The composite basis states are defined by e.g.
(01) = (0) @ [1) O.)
Analog to a classical computer, a quantum computer is built from a quantum circuit containing wires and elementary quantum gates carry around and manipulate the quantum information. A
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quantum gate is unitary operation U on Hilbert space H. When simulate the quantum circuit on a classical computer, we can obtain the overall transition unitary by tensoring and multiplying those unitary gate operators together.
A projective measurement is described by an observable, M, Hermitian operator on the state space of the system being observed. The observable has a spectral decomposition, . -_n. m.m. m. where Pm is the projector onto the eigenspace of M with eigenvalue m. When measuring the state (&)), the probability of getting results m is given by p(m) = (vb (m)pm/))
2.2 QUANTUM MACIIIEE LEARNING
(Cerezo et al., 2021) proposed the concept of Variational Quantum Algorithms (VQA), which lever- ages quantum advantages to solve machine learning problems on near-term quantum device. Then, Parameterized Quantum Circuits (PQC) are the concrete implementation of certain VQA. For each qubit we have rotation operator (Ø) which rotate through angle 0 (radias) around the r-axis. A PQC is mainly composed of ()), (0) and (0) with 0 as the parameters. The parameters O are updated by a classical optimizer to minimize the loss function L(0) which evaluates the dissim- ilarity between the output of PQC and the target result. The derivative of the i-th parameter ( can be computed by using the shifting technique proposed by (Mitarai et al., 2018). It requires running the whole circuit twice but with shifting ((i) (i) + //2 and O(i) -7/2
aC(O) L(1(1), (i) + 1/2 ...) L()(1), O(i) ........) (0())
2
Also using gradient backpropagation, classical learning models are adapted into their quantum ver- sion, e.g. QCNN (Cong et al., 2019), QRNN (Bausch, 2020), QGAN (Huang et al., 2021), QL- STM (Chen et al., 2022), and etc. which yet show that the quantum counterparts on NISQ device may not be as powerful as the SOTA classical ones (usually with millions of parameters). Involving quantum computing is an interesting experiment to seek potential supremacy and the connection between latent space and the mystery quantum entanglement.
2.3 UNITARY COUPLED-CLLUSEER
One of the most promising area to demonstrate the quantum computing supremacy is quantum chem- istry. There have been continuous work in this research area and the mainstream of these work is Unitary Coupled-Cluster (UCC) (Romero et al., 2018; Peruzzo et al., 2014; O' MaMlley et al., 2016; Yung et al., 4). UCC focused on solving the time-independent Schrödinger equation for molec- ular system to predict the chemical properties. The coupled-cluster theory is used to obtain the Hamiltonian of a certain molecule and then use Trotter-Suzule decomposition to approximate the Hamiltonian on a quantum circuit. The parameters in the rotation gates allow us to train for the min- imal ground-state energy. This method provides a hierarchy of wave functions that can be prepared on quantum computer using a polynomial number of gates. It is believed that UCC can provide better accuracy than classical coupled cluster (Wierschke, 1994; Hoffmann & Simons, 1988; Bartlett et al., 1989), which is also regarded as the "gold standard" of quantum chemistry (Bartlett & Musial, 2007). However, UCC is an unsupervised learning method with no ground truth and can only evolve one molecule at time since the circuit is uniquely designed for a certain molecule. There also evidence showing that the number of parameters in UCC might be still too large to allow practical calculations for large molecules.
2.4 QUANTUM GRAPH LEARNING
Different from evolving Hamiltonian and solving the Schrödinger equation with the quantum circuit, we also have quantum graph learning approaches trying to learn the latent representation of the vertex and the graph. A hierarchical architecture based on quantum random walks is employed to extract multi-scale properties of the graph (Dernbach et al., 2018). However, it is vague that how to efficiently construct the diffusion matrix from the quantum states generated by the quantum walkers. The information aggregation is performed by the classical system, which further incurs additional expenses consequence of the interaction between quantum and classical environment. (Zhang et al., 2019) and (Ai et al., 2022) suggest to exploit the quantum Hilbert space to rebuild the quantum representation of the graph in the quantum state. But the number of qubits to represent a graph with
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(b)
Q1
Estunglement Layer
Entanglement Lager
Q2
Trainable Layer
Than Lager
Inpus Laper
x
Qu-1
QN
(c)
(d) Parameterized Quantum Circuit
(a)
(c) Molecular Praperly Predicition
(f) Molecular Geometries Generation
stor type
position tuplo
embedding
multiayer pertapiron
prediction
The quantum 3D embedding scheme. (a) The 3D molecular graph with the gray node (in the black circle) is picked as the focal atom and three white nodes within the distance threshold the neighbors. (b) We convert the 3D Cartesian coordinates of the atoms into the relative position tuple (d, 0, sp). (c) We encode the position tuple as well as the atom type into two qubits for each atom. (d) The PQC for our model, the input layer includes Rx and Ry on each qubit, which encodes the up mentioned data. Trainable layers with parameters Os and entanglement layers are applied alternately to analog the classical machine learning layers. (e) The task property prediction. We use the embeddings from the PQC to predict chemical properties and compare them with the labels. (f) The task of 3D molecular geometries generation. We generate a molecule from scratch based on autogressive flow model with picking one focal atom and then deciding the relative position.
Figure 1:
its node attributes scales linearly with the number of nodes, and the encoding strategy not carefully designed. (Bai et al., 2021) and (Chen et al., 2021) develop hybrid graph learning model which consists of quantum layer and classical layer aimed reflecting richer graph characteristics. But they both lack formal justifications for the quantum model selections, which lead us to question whether the quantum layer is necessary. Thus, we propose a full quantum paradigm with quantum friendly encoding specially designed for molecular problems.
METHODOLOGY
3.1 PROBLEM SETTING AND METHOD OVERVIEW
Problem Setting. In this paper, we aim develop quantum machine learning approach for learn- ing node embedding with node-wise 3D coordinates. We take molecules with 3D graph structures as an example. Let G denotes the graph of certain molecule and V denotes the node set of graph G. The number of nodes (in other words atoms) is n = [V]. Each node V; E V has an attribute which is the atom type in our setting. Our target to learn the embedding for each atom and then obtain the final embedding for the molecule. The embeddings are then tested on different molecular tasks (e.g. molecular property prediction, 3D molecular geometries generation, etc.).
Method Overview. We develop quantum machine learning approach to learn the embedding on 3D graph. The trainable parameter refers to the f in those rotation gates in the PQC.
Specifically, we first encode the 3D coordinates and the atom types into qubits. We use relative coordinates instead of the 3D Cartesian coordinates to ensure both equivariance and invariance. The relative coordinates can be written in the form of position tuple (d, 0, P)), where d. 0 and 4 denote the radial distance, polar angle, and the azimuthal angle, respectively. We set up a distance threshold to pick the neighbors which variables and entangle the qubits together. We further apply a tomography at the end of the PQC and then concatenate the real part and the imaginary part. The overall pipeline is shown in Fig.
interact with the focal atom. A PQC is then used to learn the latent
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3.2 THE PROPOSED ATOM2QUBIT
Considering molecule with atoms, we take it mathematically graph G with nodes. For each node Ui we have a corresponding attribute ag which denotes the atom type and a 3D Cartesian coordinate set {I; Vr, 2i). Without loss of generality,
first pick 8; as the focal
atom and learn the embedding of node vi. The distance between v; and other nodes vi E V is
d;j = V (zg xz)² + (vj y:)² + (2j 21)². Note that not all of the node pairs in the graph have interaction in the pairs, we set a maximum distance threshold dmaz as a hyperparameter. So that "/ (e;)), if + j and dis S dmaz. which means only the nodes v, with dij S dmaa are consid- ered as the neighbors of vs. We then need to convert the 3D Cartesian coordinates of vn E (Nsss) into the position tuple (dij. Ozg, Piy)) The definition of rotation angle O and torsion angle sp are shown in Fig. (b). Now each node Vy E (N())) can be uniquely defined by {az, daj. O.j. Piy))
When we encode classical information into the quantum form, we have two different ways. The first one is amplitude encoding and the second one is angle encoding. The amplitude encoding can encode classical one-hot vector of dimension n with only logz(n) qubits, but it is quite hard to encode continuous variables while requires (n) times to encode the information. On the contrary, the angle encoding requires minimum of n/3 qubits to encode n classical information, but it is capable of encoding both discrete and continuous variables. Furthermore, the angle encoding is better fit for the rotation parameters in the circuit. In this paper, we pick angle encoding as our way to encode the information set {a,, daj. ozg. Pis)) into qubíts.
For each qubit, we have three rotation operators Rx, Ry and Rg We can theoretically encode three different pieces of information on one qubit. However, if we consider the qubit on a Bloch sphere, we can uniquely define the rotation track on the Bloch sphere using only two rotation operators. To avoid the decomposition of the third input, we only use two of the rotation operators R, and R, in this paper (R, does not change the outputs our measurement method), Therefore, we need two qubits [V i) and (V2) to encode each node Vj.
1 } U2(Og)) Uy(P()) 10)
U2)2 = Uz( Ux(= dis
x 2 Uy( aj 0uum
2 10)
dmax
where anam denotes the number of atoms occurred in the dataset and aj is an integer [1, anum] ]-
3 8 (22) is the quantum encoding state of one node generated from initial state J00). If n =
jo U W (we). the initial state (v0) for the PQC in Sec. 3.3 after the Atom2Qubit encoding stage is
bT) = 1 X 422) ② ... -& (12m-1) & (22)
(4)
3.3 QUANTUM 3D EMBEDDING ANSATZ
We first discuss the number of qubits we need for our approach on molecule problems. Each time learn the emhedding of node ú. we need to encode the information of ;; U W(x)) into qubits. Therefore the qubit number is linear with the size of (wx).. The interaction between atoms in a molecule is bounded by the bond length between atoms. As the bond length increases, the interaction becomes much weaker, which means we barely have multi-hop message passing in our graph. This gives us the possibility to run the test on an existing near-term quantum device. Therefore, we choose hardware-efficien ansatz that has been proved on a superconducting quantum processor with six fxed-frequent transmon qubits by (Kandala et al., 2017) and 56-bit superconducting quantum processor Zuchongzhi by (Huang et al., 2021).
Analog to classical neural network models, the PQC constructed by layers and each layer has an identical arrangement of quantum gates. Fig. 2 illustrates the general framework of the quantum 3D embedding ansatz. The overall unitary U()) = II (UentU.()) where Uent is the entanglement layer and (i) the /-th trainable layer. particular, we have the I-th trainable layer (i) =
N
N
(U, (K.0)
(U, where U² is the unitary of gate R2 and 0 (k.I) is the parameter
k=1
k=1
for R2 at the 1-th layer the k-th qubit. The entanglement layer Uent consists of CNOT gates and
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Rx(O,1)
Ry(B,
R, 21)
Rs(0,22
10 M 25
(0, 1-1)
R2 m
Ry Mi
Uent
Uent
UL+1
Uent
layen-1
layen:
layer
Figure 2: The circuit for our quantum 3D embedding ansatz. Each layer includes trainable param- eters block Uj and entanglement block Vent. We have N qubits in the circuit so there are N parameters in each layer. The entanglement layer is composed of CNOT gates to pairwisely entangle all the N qubits.
it entangles all the qubits together shown in Fig. 2. The quantum state (M/') after layers is
( = Uent U1 x
(Wene x Ul-1 x (Uens x U1(D))))
(5)
N
= Uent
k.1 ((((t()))) (... (Unnt x (e,
(U, ( () (6)
kml
kml
The quantum state (vr)) is the initial state, which is also the output of the Atom2Qubit stage. With the parameters (e.11) and t(*,.), we can learn the latent representation of each node. Note that the model we proposed is graph representation learning model, thus we need further attach downstream tasks to test the efficiency of our model, and the loss function is also obtained from the downstream model. The loss function L which is employed to optimize the trainable parameters (k.)) (k.1) where @ is concatenation, for our model M it yields:
min ((Mo(
The parameters A are then updated at each iteration by gradient decent from Eq. 1.
3.4 ATOM EMBEDDING
The quantum circuit mentioned above is N qubit eircuit and works in 2 dimensional Hilbert space. We apply a tomography at the end of the circuit thus we can get a 2 dimensional vector with a complex number a, for each dimension. quantum state (e) can be written in the form of combination of the computational basis states,
(+) = a [0 10..-00) 00) +02
10...07 -1) +03 10... 10
a2N
(8)
2N
2N
2N
2N
s N are the complex coefficients and the vector (a1, .2..... Q22N) is the result of the tomography. Each a; can be written in the form of a; = Re(a,) + Im(a;) where Re denotes the real part, Im denotes the imaginary part and the imaginary unit. We then concate-
where a; 15
nate the real part and the imaginary part of the tomography and get the node embedding vector (Re ai, Re ag2 ... Rga, Im aj Im a2, ... IIm agv) with the dimension N-11
NUMERICAL EXPERIMENTS
All the experiments are performed single machine with 1TB memory, physical CPU with 28 cores Intel(R) Xeon(R) W-3175X CPU 3. 3.10GHz), and two GPUs (Nvidia Quadro RTX 8000). The source code is written by Poroc,, where simulate the whole quantum circuit process using

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transition unitary. We have also implemented Qiskit version of our modelOn 2MM9-pred the erage training time for each epoch is over 2 hours using QASM simulator on Qiskit and it takes an simulator, which is about 23 times faster. Note that all our models are not
average of 310s implemented on quantum hardware yet, but the model and the circuit we proposed are easy to adapt to NISQ devices. To test the performance of our embedding model, we perform numerical exper- iments on two different tasks and compare the results with state-of-the-art classical 3D molecular representation learning models.
Dataset. The benchmark dataset we used is QM9 (Ra- makrishnan et al., 2014), which is widely used for pre- dicting various properties of molecules and 3D molecules generating tasks. It includes quantum chemistry struc- tures and properties of up to 134k stable small organic molecules. These molecules consists of up to heavy atoms CONF, not counting hydrogen, and their corre- sponding 3D molecular geometries are computed by den- sity functional theory (DFT).
Table I: Statistics of datasets.
Datasets
QM9-pred
QM9-gen
#Max Nodes #Nodes/Gruph
9.73
9.39
#Edges/Gruph
9.37
9.16
4.1 IMPLEMENTATIO DETAILS
In Sec. 3.2 we have shown how to convert the information of each atom into rotation angles on qubits. Now we discuss more precisely how to calculate the position tuple (d, 0, sp). The 3D coordinates for atom U; QM9 three real numbers I4, Y/ and ze. If we pick es as the focal atom, Vo, E (u:), we need to calculate the position tuple for atom vy against U/.
di = (xg x2)2 + (U) y1) + (2g 22
(9)
0g arctan v (xz x2) (zj )
(10)
Yj Vi
zj - Jz
Pi = arctan
(11)
2j 2
In order to fit the definition of rotation angle and torsion angle, those angles should fit into the
domain 0,5 E [0, and Pig E [0, 2T). We need to adjust the results in Eq. 10 and Eq. 11.
O.j - Oi + t
if yj - y: < 0
(12)
0, +- 0,
= 0 if y3
(13)
Pij -- Pij + n,
if 2, 2, < 0 (Ir - < - xz =,
(14)
Pij - Pij + 2 if 2j
0 OAIz -x, <0
(15)
Specifically, we set dmax 1.77, which is the maximum bond length in the dataset, and we set anum = 6 as there are five different types of atom in the dataset addition with a null type.
Now discuss more details of the transition unitary based full-amplitude circuit simulation classical processor. The quantum gates we used in our PQC are only (0). R, (0) and CNOT. The matrix representations of the single-qubit gates are as follows:
(0) cos - sin ?
i sin ( cas( @
)
cos( sin
sin 1 Ø
(0)
(16)
cos 2
2
The two-qubit gate unitary matrix is as follows:
0
0
0
1 0
0
CNOT
(17)
0
0
1
1
0
With these matrices of the basic gates, we obtain the unitary of circuit block. We first divide the block by layers, and each layer has at most one gate for each qubit. We tensor the gate matrix within each layer and thus we get the unitary for the layer. We then use matrix multiply between different layers, so the final unitary is calculated through torch.tensor() and then torch.matmul(. For N qubit circuit, the overall unitary U C2* x2 The maximum number of qubits we need is 10, so the largest unitary we need is in C1024x1024 which is still affordable on classical processor.
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Table 2: Performance comparison between the baselines and our proposed method on QM9-pred in terms of MAE for three properties (CHOMO. CLUMO and Ac) and the std. MAE for all three properties. We use the unit eV for these three energy properties.
SchNet
DimeNet++
SphereNet
ComENet
EGNN
Ours
Property eV
std.%
cV
std.%
eV
std.%
eV
std. %
std.
std.
FHооо
0.683
90.8
0.427
56.8
0.349
46.4
0.504
67.0
0.409
54.4
0.419
55.7
CLUMO
0.605
45.1
0.451
33.6
0.287
21.4
0.538
40.0
0.580
43.2
0.451
33.6
Åc
0.704
60.2
0.576
49.2
0.403
34.4
0.665
56.8
0.704
60.2
0.486
41.5
overall
0.664
65.2
0.485
46.4
0.347
34.0
0.569
54.5
0,564
52.6
0.452
43.5
4.2
MOLECULAR PROPERTY PREDICTION
We first conduct experiments on the task of molecular property prediction to evaluate our embedding model. The downstream model we used is a simple multilayer perceptron predictor, which can perform linear regression on the embeddings from the embedding model.
Setting. We filter the QM9 to generate the dataset for our prediction task. Our quantum model suffers from the extremely high time cost of simulating the quantum circuit on classic computers, and it impossible for us to run on the whole 134k molecules with molecules prediction task. We denote the dataset as QM-pred, and statistics of it are listed in Table 1. We split the dataset into vaaining/vllidaaioontesst sets with ratio of 8:1:1. Training molecules used to optimize the model parameters. The validation molecules are used to fine-tune the hyper-parameters well conduct the early stopping, and then report the results test molecules. Among all sixteen properties listed QM9, we selected three important energy-related properties, namely ehoMo, ELUMO and Ae. Ae, also known the HOMO-LUMO gap, is one of the most practically- relevant quantum chemical properties of molecules (Bredas, 2014). In line with (Liu et al., 2021), we report the mean absolute error (MAE) for each property as well as the overall mean standardized MAE (std. MAE) for all these three properties.
QM9. We sieve the dataset more than 10 atoms, and randomly pick 500 of them to form our dataset for the
Baselines. To the best of our knowledge, there are no other quantum models considering represen- tation learning for 3D graphs, thus we compare our method with four baselines in the classical do- main: the seminal work in this area SchNet (Schütt al., 2017), DimeNet++ (Klicpera al., 2020), SphereNet (Liu et al., 2021), ComENet (Wang et al., 2022) and EGNN (Satorras et al., 2021).
Prediction model. The embeddings obtained from our model are fed to a simple predictor, which is multilayer perceptron reducing the size of the embedding from 2 to 1. We use stochastic gradient descent (SGD) with Adam optimizer (Kingma
Ba, 2014) 4) to train our model for a maximum of 100 epochs with batch size of 32 and a learning rate of 0.01. Mcanwhile, as the rumning time will increase dramatically when the number of trainable layers increases, we set the number of layers in the PQC four to balance the training time and accuracy.
Results. The results of the property predic- tion task are presented in Table 2. Our model achieved the second best on all three prop- erties. SphereNet is 21.8% better than ours on the overall mean standardized MAE and is 17.1% better than us on the HOMO-LUMO gap. However, SphereNet used 1, 898, 566 pa- rameters and we only used 102, 881 parameters (64 of them are from the PQC). Compared to the fundamental baseline, we are 33.3% bet- ter than SchNet on overall mean standardized MAE and 31% on HOMO-LUMO gap. Notice that SchNet still used 455, 809 parameters. As shown in Fig. 3, our model converges very fast in the first few epochs, which also demonstrates the efficiency of model.
SchNet
DimeNett+
SphereNet
Log MAE
ComtENet
Our
10
Epoch
Figure 3: Training loss the prediction task.
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Table 3: Performance of the G-SphereNet and our proposed method on 500 randomly generated molecules for chemical validity percentage and MMD distances of bond length distributions.
MMD distances +
Method
Validity 1
H-C
H-N
H-O
C-C
C-N
C-O
Average
G-SphereNet
68.55% %
0.161
0.280
1.104
0.399
0.438
0.277
0.443
Ours
67.00%
0.237
0.409
0.770
0.326
0.407
0.378
0.421
4.3 3D MOLECULAR GEOMETRIES GENERATION
This study evaluates the performance of our proposed embedding model when adapted to the exist- ing random molecular geometry generation method. To be more specific, the embeddings from our model are used to extract 3D conditional information in the generation process.
Setting. We also use filtered QM9 for evaluation. Different from QM-pred, we select 806 molecules that contain no more than 10 atoms to form our dataset, 50 of them are used for validation and the remaining are used for training. We entitled this filtered QM9 as QM9-gen and the statistics of QM9-gen is presented in Table 1. The generated molecular geometries can be converted to molecu- lar graphs according to the approach proposed in (Gebauer al., 2019). As for metrics, we use the chemical validity percentage (Validity) which is defined as the percentage of molecular graphs that obey the chemical valency rules to evaluate the generation accuracy. addition, we adopt Max- imum Mean Discrepancy (MMD) (Gretton et al., 2012) distances of bond length distributions to evaluate the 3D structural accuracy of the generated molecular geometries. We calculate the length distribution in the generated geometries and in the dataset geometries separately for each type of bond, then we can obtain the statistical discrepancy between them with the MMD distance. In line with (Luo & Ji, 2022), we compute the MMD on hydrogen-carbon single bonds (H-C), hydrogen- nitrogen single bonds (H-N), hydrogen-oxyge
single bonds (H-O), carbon-carbon single bonds (C-C), narbon-nitrogen single bonds (C-N), carbon-oxygen single bonds (C-O) these six types of chemical bonds respectively as they are most frequently appeared.
Baseline. We use G-SphereNet (Luo & Ji, 2022) as the baseline in this molecular geometries gener- ation task. We select G-SphereNet (also from ICLR 22) produced by the same group as SphereNet, which uses SphereNet as the embedding model to extract 3D conditional information.
Generation Model. As for generation model, we employ the same generation pipeline as G- SphereNet, which adopts a flexible sequential generation strategy by adding atoms 3D space one by one based on autoregressive flow models. We use Adam optimizer to train the our model for 100 epochs, with a batch size of 64 and learning rate of 0.001. Also, we set the maximum number of atoms that can be generated for each molecule as 13.
We present the performance of our model against G-SphereNet in Table 3. We reach com- parable results with baseline model on QM-gen. More specifically, our model slightly outperforms the baseline model on MMD distances for 3 types of bond length, which shows that our method bears strong capability of extracting the 3D conditional information of molecular geometries.
Results.
CONCLUSION
3D information is important for graphs such as molecules in quantum chemistry and learning the 3D representation for those graphs has attracted increasing attention. Existing classical models face the inherent challenge of understanding the physical meaning of the 3D Cartesian coordinates. To our best knowledge, we are the first use qubits encode 3D spatial information and use a Parameterized Quantum Circuit (PQC) to learn the representation of each node as the embedding. The experiments on two well-studied downstream tasks demonstrate the efficiency and capability of model, and the potential to execute on real quantum devices.
Limitation & future works. Our method limited by the time consumption when simulating quantum circuits, while superconducting NISQ device is entering the 50+ qubit era (Gong et al., 2021), which gives us the confidence to test our model on one of them. Meanwhile, the noise on the gates not fatal with such shallow circuits. But will need to adjust the readout procedure of our embedding when testing on NISQ device. It is aimed to extending our experiments to 10 thousand molecules and reaching the chemical accuracy of 1.6 10-3 Hartree.

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Under review as a conference paper at ICLR 2023
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12


Reviews:
[{"review_id": "mfWPAUexl8", "reviewer": "Reviewer_YuG5", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "The authors present a quantum neural network architecture for 3D graph learning. the main idea of the paper is to encode the spatial information into a qubit before training the neural network. This idea is quite straightforward and the evidence that this outperforms classical or other quantum architectures is really minimal.  ", "summary_of_the_review": "A simple idea for a quantum neural network for 3D structure that misses both theoretical analysis of convergence (barren plateau, etc) and also no siginificant experimental evidence that it is better than other techniques.", "strength_and_weaknesses": "+ interesting idea of encoding directly the spatial information in the degrees of freedom of a qubit\n- very little evidence that this outperforms many other quantum neural network ideas out there\n- no theoretical understanding of the architecture. For example, barren plateau will appear in this architecture", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Q0ah-ZNwUA", "reviewer": "Reviewer_2YoU", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "This manuscript tackles the challenging problem of graph learning in 3D. The authors proposed the first use of quantum computing to learn quantum 3D latent representation of 3D structures from the Hilbert space composed of the Bloch sphere of each qubit. The proposed method preserves equivariance and invariance of 3D graphs. The authors implemented the quantum circuit full-amplitude simulator on a classical processor. The model has a smaller number of parameters compared with existing models but achieved comparable results on small graphs. ", "summary_of_the_review": "This paper present a novel application of quantum computing to learn 3D graphs, which lead to a model with substantially small number of parameters. The lackluster experiment results show that the proposed model works on small graphs. Several weaknesses should be addressed. The proposed method can potentially take advantage of large-scale quantum devices. ", "strength_and_weaknesses": "Strengths:\n(1) The paper proposed an interesting methodology and is the first to use qubits to encode 3D relative positional information. It translates the 3D Cartesian coordinates to the relative position tuple (d,θ,ϕ) between the two atoms. Relative coordinates are used to ensure equivariance and invariance.  \n(2) The paper is clearly written with a coherent structure. \n\nWeaknesses:\n(1) In Section 3.3, the authors justify their usage of a distance threshold by saying “As the bond length increases, the interaction becomes much weaker, which means we barely have multi-hop message passing in our graph”. It is known that multi-hop interactions can have non-trivially effects the quantum properties of molecules. It may be interesting to compare the performance of 1-hop vs 2-hop. Section 4, the distance threshold is set to 1.77. Some explanations about this choice of threshold will help. How would a higher distance threshold affect the computing speed and results? \n(2) Table 2, why not report the std of each property prediction task? SphereNet performs significantly better than the proposed approach. Is it possible to give some (theoretical) estimations on improvement of performance if more parameters are used in the proposed model?\n(3) The idea of using relative coordinates is not novel.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "3EfdA5byWr", "reviewer": "Reviewer_hAux", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper provides a novel aspect of using quantum ML for geometric data modeling. This is the first work in this research line, especially for quantum chemistry.", "summary_of_the_review": "I appreciate the authors for introducing quantum computing for geometric modeling. I think it is a promising direction. I’m familiar with the existing SE(3)-equivariant GNN literature, and from this aspect, the current story is not clear to me because:\n- There have been a lot of existing SE(3)- / E(3)-invariant or equivariant GNN models. They are mainly using the spherical harmonic basis (not 3D coordinates), and they have done quite well on the QM9 datasets.\n- Empirically, this work is not doing overwhelmingly well on QM9 prediction; and this work is not giving sufficient insights on how this new framework is better than previous work. I would like to raise my score if either of the questions could be solved.\n", "strength_and_weaknesses": "Strengths:\n- This paper introduces using the quantum computing for geometric data modeling. This is a promising direction.\n- The paper is self-contained, covering the basic information for quantum ML.\n\nWeaknesses:\n- Turn the atom type, distance, and angle into qubits. Then what is the advantage of this operation? The authors claim that this is more “intuitive”, which can be subjective. Can authors provide more objective reason for this?\n\n- Since this work is problem driven, i.e., the main focus is on modeling the quantum chemistry data, then authors may give an overview of this research line. But it is not present in Sec 2. Now the authors give a concrete discussion on quantum ML (sec 2), which is good. But none of them can be used as the baseline (or to be directly related to this problem).\n\n- The insights or motivation of the method is missing. Existing SE(3)-invariant GNNs (DimeNet, SphereNet, GemNet) are using the exact same information for modeling (atom type, distance, and angle), not directly on the 3D Euclidean distance. They can be formulated with the Spherical Fourirer-Bessel basis framework, and this work seems to be using an alternative solution. \n\n- The experimental results are not strong enough to verify the effectiveness of using quantum computing for geometric data modeling. Besides, there are more tasks in QM9 and some key baselines are missing. The baseline compared in this work only covers the SE(3)-invariant NN. There exists another large track of method on SE(3)-equivariant GNNs, such as EGNN[1], Allegro[2].\n\n\n\n- A minor point. “However, these approaches are still simulating the energies of certain small molecules like H2…” I think the authors should give reference for this sentence. Actually a lot of work, like NequIP[3], Allegro[2] are considering the MD17 dataset, where the drug molecules are much larger.\n\n[1] Satorras, Vıctor Garcia, Emiel Hoogeboom, and Max Welling. \"E (n) equivariant graph neural networks.\" International conference on machine learning. PMLR, 2021.\n\n[2] Musaelian, Albert, et al. \"Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics.\" arXiv preprint arXiv:2204.05249 (2022).\n\n[3] Batzner, Simon, et al. \"E (3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials.\" Nature communications 13.1 (2022): 1-11.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject