3D graph data processing method of quantum graph neural network based on rotation arrangement equivariant

Abstract

The invention provides a 3D graph data processing method of a quantum graph neural network based on rotation arrangement equivariant, and the method comprises the steps: 1, building a rotation arrangement equivariant quantum graph neural network which comprises a rotation and arrangement equivariant module, a convolution and entanglement module, and a geometric deentanglement module; the rotation and arrangement isovariant module is used for processing geometric information and keeping rotation and arrangement isovariant of the quantum graph neural network; the geometric information entanglement module is used for distance and angle entanglement; the convolution and entanglement module is used for processing non-geometric information; and step 2, applying the quantum graph neural network to 3D graph data processing. According to the method, the limitation of an existing method in processing the three-dimensional diagram data can be solved, the model can cope with the change of the diagram data under spatial rotation due to the rotational symmetry, the consistency of the structure information of the diagram under different node arrangements is ensured due to the arrangement symmetry, and the method is suitable for the three-dimensional diagram data processing. According to the model, geometric information in three-dimensional diagram data can be more comprehensively understood and utilized.

Description

Technical Field

[0001] The present invention relates to the field of quantum neural network technology, and in particular to a 3D graph data processing method based on a rotation-permutation-equivariant quantum graph neural network. Background Art

[0002] Graph data plays a central role in a variety of fields, including social network analysis, bioinformatics, and recommender systems. As data complexity increases, graph data is no longer simply a collection of node features; it now contains rich geometric information, such as node coordinates. The introduction of geometric information increases the complexity of graph structures, posing greater challenges to model design, especially in capturing complex geometric relationships. To effectively process complex graph data, graph neural networks (GNNs) must be able to capture and utilize geometric information. By embedding symmetry-related inductive biases in GNNs, their ability to understand and model geometric relationships between nodes can be enhanced. This demand has driven the development of equivariant graph neural networks (EGNNs). By incorporating equivariant designs with respect to certain symmetries, EGNNs can impose constraints on the network structure, accurately mapping and preserving the symmetries of the input data. EGNNs' ability to preserve and effectively exploit symmetries in data enables them to process complex data and demonstrate significant potential and value in a variety of practical applications, including physics, chemistry, and finance.

[0003] Quantum graph neural networks (QGNNs) face similar challenges to GNNs when processing 3D graph data. Therefore, to enable QGNNs to effectively learn the structural information of 3D graph data and improve their performance, researchers have been working to incorporate equivariance into QGNNs. The concept of equivariance quantum graph neural networks (EQGNNs) was first proposed in 2022, aiming to apply equivariance theory to the design of QGNN models as a unified framework for learning quantum graph representations. Existing EQGNN models sometimes only consider coordinate information, while others focus solely on angular information, which limits the full utilization of 3D graph information. Furthermore, existing EQGNN models only consider a certain type of symmetry in graph data, resulting in poor generalization performance.

[0004] Equivariance provides a powerful framework for modeling complex geometric relationships, enabling the model to reliably capture and represent detailed geometric features, such as molecular configuration changes and atomic interactions. However, existing quantum equivariant graph neural network models still do not fully utilize the information of nodes and edges when processing graph data, and are mainly limited to satisfying the inherent arrangement symmetry of graph data.

[0005] Current research is focused on understanding how to construct equivariant subgraph circuits and applying them to solve various practical problems. However, the lack of geometric information processing further exacerbates the challenges of equivariant subgraph models, as equivariant subgraph neural networks (EQGNNs) struggle to process global features. Current equivariant subgraph neural networks, such as the quantum 3D graph learning model (Q3DGL), only consider coordinate information, while the scalable quantum convolutional neural network (sQCNN-3D) focuses primarily on angular information, limiting the completeness of information extracted from 3D graphs. Furthermore, existing EQGNN models only consider one type of symmetry in graph data, which limits their generalization performance. Recently, the equivariant subgraph circuit (EQC) model has introduced permutation symmetry into quantum variational circuits to enhance the model's expressive power. Similarly, the quantum support vector machine with permutation-invariant coding (PI-QSVM) also employed permutation symmetry. These improvements introduced by equivariance have effectively improved the model's generalization and enhanced its robustness in extracting key features under various transformations. Summary of the Invention

[0006] Purpose of the Invention: This invention addresses the inadequate extraction of geometric and non-geometric information by existing quantum graph neural networks (EQGNNs) when processing three-dimensional graph data by introducing rotational and permutation symmetries and designing rotational and permutation equivariant modules. By constructing specialized encoding and entanglement circuits to learn geometric features from graph data and designing edge entanglement strategies to fully utilize heterogeneous edge information, the model's classification and prediction accuracy are significantly improved. Furthermore, this invention aims to enhance the model's generalization capabilities, strengthen its performance across a variety of complex tasks, and promote the development and potential of quantum graph neural networks in practical applications.

[0007] The present invention specifically provides a 3D graph data processing method based on a rotation-permutation-equivariant quantum graph neural network, comprising the following steps:

[0008] Step 1: Establish a rotation and permutation equivariant quantum graph neural network, which includes rotation and permutation equivariant modules. , convolution and entanglement modules , and the geometric disentanglement module ; is the general symbol for Hermitian conjugation.

[0009] Among them, the rotation and arrangement equivariant module Used to process geometric information and maintain the rotation and permutation equivariance of quantum graph neural networks, wherein the geometric information includes rotation and permutation equivariance.

[0010] The geometric information entanglement module Used for distance and angle wrangling.

[0011] The convolution and entanglement modules It is used to process non-geometric information and retain the relationship between edge features and nodes; the non-geometric information includes edge and node information, that is, feature information.

[0012] The geometric disentanglement module It is a rotation and permutation equivariant module The conjugate transpose of is used to ensure that quantum graph neural networks remain equivariant under rotation and permutation operations.

[0013] Step 2: Use the quantum graph neural network for 3D graph data processing.

[0014] In step 1, the rotation and arrangement of the equivariant module Contains the geometric information encoding module and geometric information entanglement module .

[0015] The geometric information encoding module Used to encode node coordinates, including:

[0016] Given a 3D graph data containing n nodes ,in and are the set of nodes and the set of edges in the graph data, respectively. is the 3D coordinate set of the node, each node All have corresponding attributes, including node characteristics and 3D coordinates , the node feature is non-geometric information, the 3D coordinate is geometric information, and the value of i ranges from 1 to n.

[0017] The geometric information encoding module Taking the geometric information of the node as input, set For the current node, when learning the current node When the geometric information is expressed in the quantum representation Convert to Euler angle configuration , current node With other nodes The distance between Expressed as: ,

[0018] in Represents the 3D coordinates of the j-th node.

[0019] The relationship between the unit representation of a symmetry group and Hilbert space: symmetry operations and quantum state transformations in quantum mechanics. When elements of a symmetry group act on a quantum system, they affect the representation of the quantum state in Hilbert space. These operations are typically implemented through matrix representations of the group elements, which in turn affect the properties of the quantum state.

[0020] The unit representation of a symmetry group is a mathematical tool for describing the symmetries of a physical system. It transforms the state of the system through the matrix representation of the group elements.

[0021] Hilbert space is the space of quantum states in which all states of a quantum system are represented.

[0022] Elements of the symmetry group (such as rotation matrices) act on quantum states in Hilbert space, changing their representation without altering their physical properties. These operations can be performed via matrix multiplication, ultimately affecting the properties of the quantum system.

[0023] Establish the unit representation of the symmetry group, the rotation matrix Determined by the selected data encoding form; at the same time, a rotation gate is used to rotate the graph data, and the characteristics of each graph node are encoded into a quantum bit. The formula is: (1), in Indicates right arrow state; middle, Represents the rotation operation around the Z axis, represents the coordinate of the i-th node on the Z axis, middle, Represents the rotation operation around the Y axis, represents the coordinate of the i-th node on the Y axis, middle, Represents the rotation operation around the X axis, Represents the coordinate of the i-th node on the X-axis.

[0024] The node coordinate information Input into formula (1), the quantum bit of each node will undergo rotation operations around the X axis, around the Y axis, and around the Z axis.

[0025] The revolving door is used to rotate the graph data so that the graph data can be mapped to the Hilbert space. It is represented as an arbitrary rotation on the Hilbert space, expressed as ; express Is a The matrix, The group is a unitary group.

[0026] The geometric information encoding module In encoding the node coordinates, each node represents a point in space and corresponds to a single quantum bit.

[0027] In step 1, the geometric information entanglement module It is a quantum entanglement module, through a double quantum control gate To construct, the information of adjacent nodes is applied as a group to the operation of the quantum gate, which is mathematically expressed as: , in yes The identity matrix, which represents the control quantum bit, is When , the target quantum bit does not change; is the operation applied to the target qubit, when the control qubit is When the target qubit is applied .

[0028] For 3D graph data containing n nodes , geometric information entanglement module Expressed as: , Geometric information entanglement module It consists of two or more quantum gates; where n represents the number of nodes, that is, the number of quantum bits. Indicates that the Hadamard gate acts on the i-th node, i and j are node indices, =( ...

[0029] In step 1, the convolution and entanglement modules It adopts the quantum graph convolution layer architecture and adopts the entanglement strategy based on edge features and edge types, including: known node features , the distance between nodes and relative angles The quantum graph convolution layer architecture includes a convolution layer based on node features and an entanglement layer based on edge features , the quantum state of the lth layer in the quantum graph convolution layer architecture Expressed as: , Among them, the convolution layer based on node features Used to process node features. 、 and is the trainable parameter of layer l; and It indicates that entanglement is increasing; It indicates the entanglement enhancement for distance information; Indicates the entanglement enhancement for angular information.

[0030] In the convolution and entanglement modules In the embodiment, nodes and edges are divided into different types so as to be processed differently, thereby realizing edge information enhancement in the present invention.

[0031] Edges are classified into different types: ,in , m is the total number of edge types based on edge characteristics; represents the kth edge type; e represents the set of types.

[0032] For each edge type , choose a two-qubit gate whose parameters are independent.

[0033] The entangled layer Including distance entanglement module and angle entanglement module , the following formula is the distance entanglement module and angle entanglement module Specific operational representation of the quantum state.

[0034] For a three-node Graph data, according to the edge and Different types of two-qubit gates are applied and ( and Represents the revolving gate around the x-axis and the revolving gate around the y-axis, respectively, which represent the geometric relationship between quantum bits in quantum computing), distance entanglement module Output Expressed as: , in It is the output quantum state, which represents the final state of the system after quantum computation. After the transformation of the quantum gate, the node information is updated and represented by the quantum state. Represents the distance between the i-th node and the j-th node; Represents the trainable parameters of the edge connecting the i-th node and the j-th node; is the input quantum state, which represents the initial quantum bit state at the beginning of the system. It is the starting state after the rotating gate operation.

[0035] Angle entanglement module Output Expressed as: , in, is the relative angle between the i-th node and the j-th node; i and j are 1, 2, and 3.

[0036] In step 1, the geometric disentanglement module The calculation formula is: , revolving door , , They are 、 ,and The Hermitian conjugate form of ; H is the Hadamard gate, which is self-adjoint, and the conjugate transpose is itself; It is a dual quantum control gate The Hermitian conjugate of , corresponding to the reverse control operation.

[0037] Subsequently, the module output is obtained through measurement and input into a classic linear layer to finally generate the complete graph prediction and classification results.

[0038] The present invention also provides an electronic device, comprising a processor and a memory, wherein the memory stores program code, and when the program code is executed by the processor, the processor executes the steps of the method.

[0039] The present invention also provides a storage medium storing a computer program or instruction, which executes the steps of the method when the computer program or instruction is run on a computer.

[0040] Glossary of relevant technical terms in this invention:

[0041] Rotational and permutation symmetry: In graph data processing, rotational and permutation symmetry means that the structural characteristics of a graph should not change when its nodes or edges are rotated or rearranged. For example, in 3D graph data, the model should be able to correctly identify the object regardless of the angle it is rotated. Permutation symmetry, on the other hand, means that the overall structure of the graph should remain unchanged when the order of the nodes in the graph is changed.

[0042] Equivariance: In machine learning, equivariance refers to a model's ability to adapt its output to changes in its input (such as rotations and permutations). If the input data undergoes some transformation, the model's output should also transform according to certain rules. For example, with rotational equivariance, rotating the graph data will not affect the model's output.

[0043] Encoding Circuit: In quantum computing, an encoding circuit is a quantum circuit that converts classical information into quantum states. For graph neural networks, the encoding circuit converts input graph data (such as the geometric coordinates of nodes) into quantum bit states, enabling processing in quantum computing.

[0044] Entanglement Circuit: Quantum entanglement is a special relationship between two or more qubits in quantum mechanics that makes their states inextricably linked. The Entanglement Circuit is used to introduce entanglement between qubits, allowing the model to capture the complex interactions between nodes.

[0045] Geometric Information: In graph data, geometric information generally refers to the spatial coordinates of nodes or other geometric data related to the graph. In 3D graph data, geometric information includes the three-dimensional positions of nodes and the distances between them.

[0046] Non-Geometric Information: In addition to geometric information, graph data also contains information related to structure, labels, or other attributes, such as edge types and node characteristics. This information helps in graph classification and prediction.

[0047] Convolution and Entanglement Module: This module combines convolution operations from convolutional neural networks and entanglement operations from quantum computing to extract important features from graph data. Convolution operations are used to extract local features, while entanglement operations are used to establish global relationships between nodes, thereby enhancing the model's expressive power.

[0048] Edge Entanglement Strategy: This technique uses the heterogeneity of graph edges (such as edge weight or type) to perform different types of quantum entanglement operations. This strategy helps capture the structure and characteristics of graph edges, thereby improving the model's classification and prediction capabilities.

[0049] In the present invention, constraints on rotation and permutation equivariance are derived, and a new quantum graph neural network model is proposed that combines rotation and permutation symmetry equivariance. The present invention constructs a rotation and permutation equivariance module for 3D graph data, which extracts geometric information of three-dimensional graph data by imposing equivariance constraints. Unlike previous studies, the model of the present invention does not rely on constructing a set of permutation equivariance quantum gates, but instead achieves permutation equivariance through carefully designed modules. To further extract non-geometric information, the present invention designs a convolution and entanglement module through which node features in graph data are learned. On this basis, in order to improve the prediction and classification accuracy of the model, the present invention proposes an edge entanglement strategy, in which the entanglement structure is designed based on the obtained scalar information (such as edge type, distance, and angle between nodes), thereby effectively improving the performance of the entanglement module.

[0050] The present invention has the following beneficial effects: The introduction of rotational and permutation symmetry: Current EQGNN models typically only process one type of symmetry in graph data (e.g., permutation symmetry), resulting in poor generalization performance and incomplete utilization of node and edge information. By introducing rotational and permutation symmetry, the present invention addresses the limitations of existing methods in processing three-dimensional graph data. Rotational symmetry enables the model to cope with changes in graph data under spatial rotation, while permutation symmetry ensures the consistency of the graph's structural information under different node arrangements. In this way, the model of the present invention can more comprehensively understand and utilize the geometric information in three-dimensional graph data.

[0051] Proposal of a rotation and permutation equivariant module: Although previous studies have proposed quantum graph neural networks and, to a certain extent, introduced equivariance, they still cannot simultaneously process multiple symmetries in the data and effectively extract geometric information from three-dimensional graph data. Based on the constraints of rotation and permutation symmetries, the present invention proposes a rotation and permutation equivariant module. By designing an encoding module and an entanglement module to learn the geometric information of three-dimensional graph data, it not only overcomes the shortcomings of existing methods in geometric information extraction but also significantly enhances the model's performance and generalization capabilities. As a result, the model of the present invention is able to handle complex three-dimensional geometric structures and demonstrates strong predictive accuracy.

[0052] Edge Entanglement Strategy: Current EQGNN models primarily focus on node information when processing graph data, while underutilizing edge information (especially heterogeneous edges). This limits the model's performance on complex tasks. The edge entangling strategy designed in this paper effectively extracts non-geometric information by performing different entangling operations based on heterogeneous edge information, enhancing the model's ability to process complex graph data. This innovation directly addresses the shortcomings of existing models in processing edge information, improving classification and prediction accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1It is a flow chart of the overall structure of the model of the present invention.

[0054] Figure 2 It is a diagram of the geometric information encoding module.

[0055] Figure 3 It is a diagram of geometric information entanglement modules.

[0056] Figure 4 It is a non-geometric information processing module. DETAILED DESCRIPTION

[0057] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, and the above and / or other advantages of the present invention will become more apparent.

[0058] An embodiment of the present invention provides a 3D graph data processing method based on a rotation-permutation-equivariant quantum graph neural network, comprising the following steps:

[0059] Step 1: Establish a rotation and permutation equivariant quantum graph neural network, which includes rotation and permutation equivariant modules. , convolution and entanglement modules , and the geometric disentanglement module ; is the universal symbol for Hermitian conjugation;

[0060] Among them, the rotation and arrangement equivariant module Used to process geometric information and maintain the rotation and permutation equivariance of quantum graph neural networks, wherein the geometric information includes rotation and permutation equivariance;

[0061] The geometric information entanglement module For distance and angle entanglement;

[0062] The convolution and entanglement modules Used to process non-geometric information and retain the relationship between edge features and nodes; the non-geometric information includes edge and node information, that is, feature information;

[0063] The geometric disentanglement module It is a rotation and permutation equivariant module The conjugate transpose of is used to ensure that quantum graph neural networks remain equivariant under rotation and permutation operations;

[0064] Step 2: Use the quantum graph neural network for 3D graph data processing.

[0065] The symbols used in the present invention are explained in Table 1.

[0066] Table 1 Symbol Explanation Table

[0067]

[0068] Step 1 includes the following: First, the rotation and permutation equivariance module. This module consists of two components: a geometry encoding module and a geometry entanglement module, which ensure that the model maintains rotation and permutation equivariance. The geometry encoding module encodes node coordinates, which are the core component of 3D graph data. The geometry entanglement module learns the interrelationships and structure of geometric information in the graph data.

[0069] Next is the Convolution and Entanglement module. This module consists of two parts: a non-geometric convolution module and a non-geometric entangling module. The convolution module extracts feature information from graph nodes, while the entangling module further processes and entangles node information to enhance the model's expressiveness. The Convolution and Entanglement module also introduces an edge entangling strategy to handle the heterogeneous edge types and features in graph data. This strategy aims to avoid over-smoothing information between nodes, thereby preserving richer edge features and relationships between nodes.

[0070] Finally, there's the geometric disentanglement module. This module is effectively the Hermitian conjugate of the rotation and permutation equivariance module, ensuring that the entire model remains equivariant under rotation and permutation operations. In other words, the geometric disentanglement module ensures the equivariance of the model, preventing it from losing its geometric properties due to improper operations. The model's final output undergoes classical operations and is ultimately used for graph prediction and classification tasks.

[0071] The present invention was applied to molecular property prediction and point cloud classification tasks. Given the limitations of current hardware, a compressed graph was used as the model input. The model output, obtained through quantum measurement operations, was then processed in downstream classical operations to produce the final prediction and classification output.

[0072] The rotation and permutation equivariant modules include: 1. Geometric information encoding module: The present invention first designs a node coordinate encoding module. Take a graph data containing n nodes as an example, each node All have corresponding attributes, including non-geometric information (node ​​features) and geometric information (3D coordinates) , these information satisfy different symmetry properties. The encoding module only processes the geometric information of the node as input. For the current node, when learning its quantum representation, its geometric information is converted into Euler angle configuration , and this node and other nodes The distance between them is expressed as: , Next, a unit representation of the symmetry group is established, since This is determined by the chosen data encoding format. Graph data is also encoded to ensure efficient representation in subsequent quantum computations. To reduce the number of required qubits, rotation encoding is used, where the characteristics of each graph node can be encoded into a qubit. A universal rotation gate is used to encode coordinates, enabling a natural mapping to Hilbert space. The universal selection gate formula is as follows: , The node coordinate information Input to the quantum circuit, and the quantum bit of each node will undergo 、 ,and Rotation operations. These revolving gates represent the rotation of a node in three directions.

[0073] The present invention uses a universal revolving door to encode the coordinates so that they can be naturally mapped to the Hilbert space. can be represented as any rotation on Hilbert space, express Is a The matrix of . The group is a unitary group, which consists of all The unitary matrix is ​​composed of unitary matrices, the determinant of which is 1. The definition of a unitary matrix is ​​that the matrix satisfy ,in is the conjugate transpose of the matrix, is the identity matrix.

[0074] In this process, each node represents a point in space and corresponds to a single quantum bit.

[0075] 2. Geometric Information Entanglement Module: A quantum entanglement module is built for geometric information. The core idea of ​​this module is not to directly embed the original coordinate information into the quantum circuit, but to extract the distance and torsion angle information between nodes from these coordinates. In order to achieve rotation and permutation equivariance, a controlled two-qubit gate is used in this invention. To build quantum circuits ( A quantum gate is a control gate that usually consists of two quantum bits: a control quantum bit and a target quantum bit), and applies the information of adjacent nodes as a group to the operation of the quantum gate. Its mathematical representation is: yes The identity matrix, which represents the control quantum bit, is , the target qubit does not change. is the operation applied to the target qubit, when the control qubit is When the target qubit is applied .

[0076] In this way, we can effectively use the geometric information of the graph to construct quantum circuits, and ensure that the designed quantum circuits have the necessary equivariance. , the entanglement module of geometric information Expressed as: , in, represents the Hadamard gate acting on the i-th qubit. This structure ensures that the quantum circuit maintains rotational equivariance and permutation equivariance, symmetries inherent in the geometric information in the graph data. (The data characteristics are symmetries, which correspond to equivariance in the model.)

[0077] Convolution and entanglement module: This module mainly adopts the quantum graph convolution layer architecture and proposes an entanglement strategy based on edge features and edge types to further improve the performance of the model. ,distance and relative angles , the circuit consists of two standard parts: a convolutional layer based on node features and an entanglement layer based on edge features . The quantum state of the first layer is expressed as: , in, Used to encode node information and consists of a revolving door; Used to encode edge information, including multiple and This strategy extracts edge information based on angle and distance and uses it as input to superimpose multiple entanglement modules, where entanglement is performed according to different types of edge information in each entanglement module.

[0078] Edges are classified into different types: ,in How many edge types there are based on edge properties.

[0079] For each edge type , select a two-qubit gate. Therefore, the module consists of multiple two-qubit gates, such as 、 , used to represent the information between nodes, and the inter-node connections that represent their distance and angular relationships. The parameters of the two-qubit gate are independent to best adapt to the specific edge characteristics.

[0080] For example, a node contains three Graph data, according to edge and Different types of two-qubit gates are applied and The output of the distance entanglement module Expressed as: , in, It is The edge Trainable parameters in a layer.

[0081] Represents a node. Similarly, the output of the angle entanglement module is expressed as: , in, is a relative angle.

[0082] Strategic Objective: The convolution and entanglement modules of this invention aim to avoid the problem of over-smoothing information between nodes, thereby enhancing the model's ability to capture graph structural features. Crucially, the input to the entanglement layer is all normalized information in the current module structure. This design further makes the quantum circuit essentially invariant under geometric transformations (such as translation and reflection), ensuring that the model can more efficiently meet equivariance constraints on graph data, especially when processing graphs with complex symmetries.

[0083] Geometric Disentanglement Module: The disentanglement module is crucial to ensure the equivariance of the model. This module plays an important role in maintaining the equivariance of quantum graph neural networks, ensuring that the transformation of quantum states conforms to the desired equivariance constraints, which are determined by the relationship between entanglement and disentanglement operations. Introduced entanglement. Deentanglement module is the Hermitian conjugate of the geometric information entanglement module.

[0084] , in , , The revolving gates are their Hermitian conjugate forms.

[0085] The Hadamard gate H is self-adjoint, so its conjugate transpose is itself.

[0086] yes The Hermitian conjugation of the gate corresponds to the reverse control operation.

[0087] Subsequently, the module output is obtained through measurement and input into a classic linear layer to finally generate the complete graph prediction and classification results.

[0088] Derivation and proof of permutation and rotation equivariance: First, if It's a group elements, and yes A representation of The group Mapped to a square matrix, so that for all , we can know that: , If the function About Transformations is equivariant, then: , from which we can deduce: , As the formula shows, a quantum neural network that is equivariant under both transformations g and h will also be equivariant under the combined transformation Therefore, by proving that the model of the present invention is equivariant under rotational and permutation symmetries respectively, its overall equivariance under rotational and permutation symmetries can be effectively proved.

[0089] Permutation equivariance constraint: Given a graph , is a permutation matrix, The corresponding is the reordered matrix, here we need to prove and The transformed graph G is permutation equivariant with respect to the model. The tensor product Mapped to , a permutation equivariant (Quantum circuit) satisfies the following conditions: , Where T represents transpose; represents the nth quantum system, represents the mapping of the nth quantum system.

[0090] In addition, the model of the present invention integrates edge and node information into a 3D graph data structure. ,in and are the set of nodes and the set of edges in the graph data, respectively. It is the 3D coordinate set of the node. The module design is as follows: , This formula shows how to combine node and edge operations in the quantum circuit of the present invention to achieve quantum transformation of the entire graph through tensor products and multiplications.

[0091] in, It is a convolutional layer based on node features. It is an entanglement layer based on edge features; is the number of nodes, j and k are the vertices of the edge (each node uses a quantum gate, and if j and k are connected by an edge, a quantum gate is also used to process it). represents the quantum gate that processes edge jk, Indicates that a total of n u-gates are used on the node layer.

[0092] Based on the above formula, the model's predicted output is As shown below: , represents the maximization operation, that is, finding j that maximizes the inner product, that is, the solution in the optimization problem; r is the coordinate information (geometric information); x is the feature information (non-geometric information); Represents rotation and permutation equivariant modules; Represents convolution and entanglement modules; Represents the quantum measurement module; conjugate matrices are used for inverse transformation and to adjust the effect of the transformation; the gate operations taken for each quantum bit are exactly the same, thus ensuring that the model is permutation equivariant.

[0093] Rotational equivariant constraint: In the present invention, the model of the present invention It can be expressed as Since rotational equivariance is only related to the geometric information of the data, It is the convolution and entanglement module in the model module. It is the rotation and arrangement equivariant module in the model of the present invention. The module designs a conjugate transpose module to ensure rotational equivariance. At this point, the model is represented as , the output of the model is as follows: , Where r represents coordinate information (geometric information) and x represents feature information (non-geometric information).

[0094] In rotational equivariance, all spherical rotations about the origin form the SO(3) group. The rotational equivariance model should satisfy the condition ,in .

[0095] All spherical rotations about the origin form the SO(3) group; the SO(3) group is the special orthogonal group that describes all rotations in three-dimensional space. The elements of SO(3) represent rotations in three-dimensional space, performed about axes at the origin.

[0096] Applying the rotation transformation g to the model input, the output of the model is: , To ensure that the model is equivariant to rotation transformations, the output after the rotation is consistent with the output before the rotation. Next, we need to design the quantum circuit so that .

[0097] First, the present invention designs a quantum circuit so that , in this step, by the rotation matrix To express the effect of rotation on the model, and to ensure that the output of the model after rotation is consistent with that before rotation, the output of the model is expressed as follows according to the circuit designed by the present invention: , Represents the rotation matrix The Hermitian conjugated form of .

[0098] According to known , , the formula is simplified to: , Then the present invention designs the measurement operator circuit to satisfy the rotation equivariance. Specifically, , so the output of the rotation equivariant model is expressed as: , So far, we can deduce from the above formula , holds true for all g∈SO(3), and it can be deduced that the model of the present invention is rotationally equivariant.

[0099] This example conducts result analysis, specifically including: 1. Molecular property prediction experiment; Experimental Dataset: The QM9 dataset, derived from the GDB17 dataset, contains up to 134,000 small, stable organic molecules, widely used for predicting various molecular properties. These molecules represent a group of atoms connected by four different chemical bond types. Each atom has a 3D position and a 5-dimensional unique hot node embedding (H, C, O, N, F). The quantum properties of each molecule are obtained using density functional theory (DFT) calculations and are also provided.

[0100] Evaluation metrics: The molecular property prediction method is evaluated by mean square error (MAE), which is expressed as follows: , where n is the number of molecules, is the actual property value of the ith molecule, is the attribute value predicted by the model.

[0101] Benchmark Model: This compares the proposed method with state-of-the-art quantum models and classical graph neural network models, and displays detailed information such as the input data type and number of training parameters for each model in graph learning tasks. The benchmark models selected in this example are as follows: LieConv is a new type of convolutional neural network that combines the principles of Lie groups and Lie algebras. It uses the symmetry and invariance represented by Lie groups to perform convolution operations and process them in a more geometrically meaningful way.

[0102] E(n)-GNN is a graph data algorithm with a similar structure to GNN, but ensures that the model operation is equivariant to the symmetry of the input data, thereby maintaining geometric properties in each layer of the network.

[0103] SEGN generalizes EGNNs so that the attributes of nodes and edges are no longer limited to invariant scalars, but can contain covariant information such as vectors or tensors.

[0104] EQGAT is an equivariant GNN that uses Cartesian coordinates to incorporate directionality. In this model, the attention mechanism acts as a content and spatial dependency filter, taking into account the relationship between nodes when propagating information.

[0105] QGCNN is a graph learning method based on quantum computing principles. It uses quantum state preparation to encode graph node features and then applies quantum convolutional layers to achieve efficient and expressive graph feature extraction.

[0106] EQC is a solution for weighted graph learning tasks, suitable for learning tasks with permutation symmetry.

[0107] Table 2 Performance comparison of the present invention and the classic benchmark model

[0108]

[0109] Table 3 Performance comparison between the present invention and the quantum benchmark model

[0110]

[0111] As shown in Tables 2 and 3, the model's performance is compared against both classical and quantum benchmarks, respectively, based on MAE. The proposed model demonstrates significant improvements in MAE, achieving superior results compared to other models on certain tasks. While compressing the input graph data can lead to information loss and thus performance degradation, it offers advantages in computational efficiency and trainable parameter size. Compared to the quantum benchmark model, the model achieves approximately 10% performance improvement in most cases. Notably, the proposed model surpasses existing quantum equivariant models on certain tasks, validating the effectiveness of the proposed model structure and algorithmic strategy.

[0112] 2. Point cloud classification experiment;

[0113] Dataset: The ModelNet40 dataset is one of the most popular 3D object datasets, containing 12,311 CAD-generated meshes covering 40 categories (such as airplanes, cars, plants, and lamps). The corresponding point cloud data points are uniformly sampled from the mesh surface and further preprocessed by shifting the data to the origin and scaling it to the unit sphere.

[0114] Evaluation metrics: The performance of the model is evaluated by the accuracy of point cloud classification, which can be expressed as follows: .

[0115] Where N represents the total number of samples in the test set, is an indicator function, when the predicted label Equal to the true label , the value is 1, otherwise the value is 0.

[0116] Benchmark Models: Our method is compared with several state-of-the-art quantum model graph neural networks for point cloud classification tasks.

[0117] QGCNN is a quantum graph neural network, as demonstrated in experiments on the QM9 dataset.

[0118] sQCNN-3D is a 3D pixelated version of a scalable quantum convolutional neural network (QCNN) suitable for point cloud data processing.

[0119] PI-QSVM is a quantum support vector machine with a permutation-invariant encoding that contains an equal quantum superposition of all permutations and is therefore invariant under point-order permutations.

[0120] Table 4 Comparison of classification results between the present invention and existing models

[0121]

[0122] As shown in Table 4, the present invention demonstrates higher accuracy than the quantum model in three classification tasks. In summary, the present invention is able to learn both geometric and non-geometric information of 3D graph data with fewer trainable parameters and is equivariant to rotation and permutation. To further improve the model's prediction and classification accuracy, the present invention introduces an edge entanglement strategy. Ultimately, the present invention demonstrates superior performance in 3D graph learning tasks.

[0123] The present invention provides a 3D graph data processing method based on a rotationally permuted and equivariant quantum graph neural network. There are many methods and approaches to implement this technical solution. The above is only a preferred embodiment of the present invention. It should be noted that those skilled in the art may make various improvements and modifications without departing from the principles of the present invention, and such improvements and modifications should also be considered within the scope of protection of the present invention. Any components not specified in this embodiment may be implemented using existing technologies.

Claims

1. A 3D graph data processing method based on a rotation-permutation equivariant quantum graph neural network, characterized in that: The following steps are involved: Step 1: Establish a rotation-permutation equivariant quantum graph neural network, which includes rotation and permutation equivariant modules. , convolution and entanglement modules , and the geometric disentanglement module ; is the universal symbol for Hermitian conjugation; Among them, the rotation and arrangement equivariant module Used to process geometric information and maintain the rotation and permutation equivariance of quantum graph neural networks, wherein the geometric information includes rotation and permutation equivariance; The geometric information entanglement module For distance and angle entanglement; The convolution and entanglement modules Used to process non-geometric information and retain the relationship between edge features and nodes; the non-geometric information includes edge and node information, that is, feature information; The geometric disentanglement module It is a rotation and permutation equivariant module The conjugate transpose of is used to ensure that the quantum graph neural network remains equivariant under rotation and permutation operations; Step 2: Use the quantum graph neural network for 3D graph data processing.

2. The method according to claim 1, characterized in that In step 1, the rotation and arrangement equivariant module It contains the geometry information encoding module and geometric information entanglement module ; The geometric information encoding module Used to encode node coordinates, including: Given a 3D graph data containing n nodes ,in and are the set of nodes and the set of edges in the graph data, respectively. is the 3D coordinate set of the node, each node All have corresponding attributes, including node characteristics and 3D coordinates , the node feature is non-geometric information, the 3D coordinate is geometric information, and i ranges from 1 to n; The geometric information encoding module Taking the node geometry information as input, set For the current node, when learning the current node When the geometric information is expressed in the quantum representation Convert to Euler angle configuration , current node With other nodes The distance between It is expressed as: ， in represents the 3D coordinates of the jth node; Establish the unit representation of the symmetry group, the rotation matrix Determined by the selected data encoding form; at the same time, the revolving gate is used to rotate the graph data, and the characteristics of each graph node are encoded into a quantum bit. The formula is: （1）， in Indicates right arrow state; middle, Represents the rotation operation around the Z axis, represents the coordinate of the i-th node on the Z axis, middle, Represents the rotation operation around the Y axis, represents the coordinate of the i-th node on the Y axis, middle, Represents the rotation operation around the X axis, Represents the coordinate of the i-th node on the X-axis; The node coordinate information Input into formula (1), the quantum bit of each node will undergo rotation operations around the X axis, around the Y axis, and around the Z axis; The revolving door is used to rotate the graph data so that the graph data can be mapped to the Hilbert space. It can be expressed as any rotation on Hilbert space, represented by ; express Is a The matrix of The group is a unitary group; The geometric information encoding module In the process of encoding the node coordinates, each node represents a point in space and corresponds to a separate quantum bit.

3. The method according to claim 2, characterized in that In step 1, the geometric information entanglement module It is a quantum entanglement module, through a double quantum control gate To construct, the information of adjacent nodes is applied as a group to the operation of the quantum gate, which is mathematically expressed as: ， in yes The identity matrix represents the control qubit as When , the target quantum bit does not change; is the operation applied to the target qubit, when the control qubit is When the target quantum bit is applied ; For a 3D graph data containing n nodes , geometric information entanglement module It is expressed as: ， Among them, n represents the number of nodes, that is, the number of quantum bits, Indicates that the Hadamard gate acts on the i-th node, i and j are both node indices; It means that the double quantum control gate acts on the j-th node and the n-th node.

4. The method according to claim 3, characterized in that In step 1, the convolution and entanglement modules It uses a quantum graph convolutional layer architecture and an entanglement strategy based on edge features and edge types, including: known node features , the distance between nodes and relative angle The quantum graph convolutional layer architecture includes a convolutional layer based on node features. and an entanglement layer based on edge features , the quantum state of the lth layer in the quantum graph convolutional layer architecture It is expressed as: ， Among them, the convolution layer based on node features Used to process node features. , and is the trainable parameter of layer l; and It indicates that entanglement is increasing; It indicates the entanglement enhancement for distance information; It indicates the entanglement enhancement for the angle information; In the convolution and entanglement modules In , edges are classified into different types: ,in , m is the total number of edge types based on edge characteristics; represents the kth edge type; e represents the set of types; For each edge type , choose a two-qubit gate whose parameters are independent; The entanglement layer Including distance entanglement module and angle entanglement module ; For a network consisting of three nodes Graph data, according to the edge and Different types of two-qubit gates are used and , distance entanglement module Output It is expressed as: ， in is the output quantum state; Represents the distance between the i-th node and the j-th node; Represents the trainable parameters of the edge connecting the i-th node and the j-th node; is the input quantum state; Angle entanglement module Output It is expressed as: ， in, is the relative angle between the i-th node and the j-th node; i and j are 1, 2, and 3.

5. The method according to claim 4, characterized in that In step 1, the geometric disentanglement module The calculation formula is: ， Revolving door , , They are , ,and The Hermitian conjugate form of ; H is the Hadamard gate; It is a dual quantum control gate The Hermitian conjugate of , corresponding to the reverse control operation.

6. An electronic device, characterized in that: The method comprises a processor and a memory, wherein the memory stores program codes, and when the program codes are executed by the processor, the processor executes the steps of the method according to any one of claims 1 to 5.

7. A storage medium, characterized in that: A computer program or instruction is stored, and when the computer program or instruction is run on a computer, the steps of the method according to any one of claims 1 to 5 are executed.

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