QUANTUM 3D GRAPH STRUCTURE LEARNING WITH APPLICATIONS TO MOLECULE COMPUTING

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 as well as 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 as 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 of 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 as the trainable layers and the output of the PQC is adopted as 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.

1. INTRODUCTION

Graph representation, or specifically 3D graph representation as considered in this paper, has received extensive attention over the last decade. Beyond tasks like node classification or link prediction, it further facilitates various downstream applications such as molecular property prediction (Liu et al., 2021) and drug design (Gaudelet et 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 et 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-Scale 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 et al., 2020) . The overlap between quantum computing and machine learning has emerged as one of the most encouraging areas for quantum computing, as termed by quantum machine learning (Biamonte et al., 2017) . Quantum paradigms or hybrid paradigms have been carefully designed to fulfill quantum supremacy in quantum chemistry problems (Aspuru-Guzik et al., 2005; O'Malley et al., 2016) . Existing approaches mainly focus on the quantum simulation of molecular energies, which enables effective prediction of chemical reaction rates. However, these quantum approaches (Romero et al., 2018; Peruzzo et al., 2014; O'Malley et al., 2016; Yung et al., 2014) are still simulating the energies of certain small molecules like H 2 , LiH, etc. In this paper, we aim to develop quantum machine learning approaches to learn the latent representation of the 3D graph structure of molecules instead of directly simulating the molecular energies with Hamiltonians. Graph learning may not be as precise as molecular simulation approaches for property prediction, but they have the ability to learn hundreds or thousands of molecules and predict 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 so that each time a focal atom is picked to 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 to ten qubits to 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 et al., 2017; Huang et al., 2021) , we apply a Parameterized Quantum Circuit (PQC) after the encoding stage. The trainable parameters are the θs of the rotation gates R x and R y in the PQC. The gradient of each parameter θ 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 it 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-art baseline models, our quantum 3D embedding 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 a qubit on a Bloch sphere to encode the rotation and torsion angle of two atoms is 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 a quantum circuit full-amplitude simulator with transition unitary for the PQC on a classical processor. It replicates the results yet over 20 times faster than the QASM simulator from IBM Qiskit's simulator, which enables us to conduct experiments on more tasks. 4) The numerical experiments on two different well-studied molecular tasks show that our embedding 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 a binary state. The two possible states for a qubit are the state |0 and |1 , which correspond to the state 0 and 1 for a classical bit respectively. We refer the readers to the textbook (Nielsen & Chuang, 2002) for comprehension of quantum information and quantum computing. 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: |ψ = α|0 + β|1 . Formally, a quantum system on n qubits is an n-fold tensor product Hilbert space H = (C 2 ) ⊗d with dimension 2 d . For any |ψ ∈ H, the conjugate transpose ψ| = |ψ † . The inner product ψ|ψ = ||ψ|| 2 2 denotes the square of the 2-norm of ψ. The outer product |ψ ψ| is a rank 2 tensor. Computational basis states are given by |0 = (1, 0), and |1 = (0, 1). The composite basis states are defined by e.g. |01 = |0 ⊗ |1 = (0, 1, 0, 0). Analog to a classical computer, a quantum computer is built from a quantum circuit containing wires and elementary quantum gates to carry around and manipulate the quantum information. A

