Recent developments in the field of quantum machine learning have promoted the idea of incorporating physical symmetries in the structure of quantum circuits. A crucial milestone in this area is the realization of S n -permutation equivariant quantum neural networks (QNNs) that are equivariant under permutations of input objects. In this paper, we focus on encoding the rotational symmetry of point cloud datasets into the QNN. The key insight of the approach is that all rotationally invariant functions with vector inputs are equivalent to a function with inputs of vector inner products. We provide a structure of the QNN that is exactly invariant to both rotations and permutations, with its efficacy demonstrated numerically in the problems of two-dimensional image classifications and identifying high-energy particle decays, produced by proton-proton collisions, with the SO ( 1 , 3 ) Lorentz symmetry. Published by the American Physical Society 2024
Recent developments in the field of quantum machine learning have promoted the idea of incorporating physical symmetries in the structure of quantum circuits. A crucial milestone in this area is the realization of S n -permutation equivariant quantum neural networks (QNNs) that are equivariant under permutations of input objects. In this paper, we focus on encoding the rotational symmetry of point cloud datasets into the QNN. The key insight of the approach is that all rotationally invariant functions with vector inputs are equivalent to a function with inputs of vector inner products. We provide a structure of the QNN that is exactly invariant to both rotations and permutations, with its efficacy demonstrated numerically in the problems of two-dimensional image classifications and identifying high-energy particle decays, produced by proton-proton collisions, with the SO ( 1 , 3 ) Lorentz symmetry. Published by the American Physical Society 2024
Abstract Despite the great promise of quantum machine learning models, there are several challenges one must overcome before unlocking their full potential. For instance, models based on quantum neural networks (QNNs) can suffer from excessive local minima and barren plateaus in their training landscapes. Recently, the nascent field of geometric quantum machine learning (GQML) has emerged as a potential solution to some of those issues. The key insight of GQML is that one should design architectures, such as equivariant QNNs, encoding the symmetries of the problem at hand. Here, we focus on problems with permutation symmetry (i.e., symmetry group Sn), and show how to build Sn-equivariant QNNs We provide an analytical study of their performance, proving that they do not suffer from barren plateaus, quickly reach overparametrization, and generalize well from small amounts of data. To verify our results, we perform numerical simulations for a graph state classification task. Our work provides theoretical guarantees for equivariant QNNs, thus indicating the power and potential of GQML.
Abstract The Symmetric group S n manifests itself in large classes of quantum systems as the invariance of certain characteristics of a quantum state with respect to permuting the qubits. Subgroups of S n arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, such as reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this study we propose the architectures of equivariant quantum convolutional neural networks (EQCNNs) adherent to S n and its subgroups. We demonstrate that a careful choice of pixel-to-qubit embedding order can facilitate easy construction of EQCNNs for small subgroups of S n . Our novel EQCNN architecture corresponding to the full permutation group S n is built by applying all possible QCNNs with equal probability, which can also be conceptualized as a dropout strategy in quantum neural networks. For subgroups of S n , our numerical results using MNIST datasets show better classification accuracy than non-equivariant QCNNs. The S n -equivariant QCNN architecture shows significantly improved training and test performance than non-equivariant QCNN for classification of connected and non-connected graphs. When trained with sufficiently large number of data, the S n -equivariant QCNN shows better average performance compared to S n -equivariant QNN . These results contribute towards building powerful quantum machine learning architectures in permutation-symmetric systems.
Group-equivariant quantum machine learning has emerged as a promising paradigm by incorporating symmetry into quantum models. However, constructing models simultaneously equivariant to both rotational and permutational symmetries in a principled manner remains a bottleneck. In this work, we develop a general framework for dual-equivariant gates under rotations and permutations and analyze the dimension of the resulting gate space using group representation theory. Based on this, we introduce HyQuRP, a hybrid quantum-classical neural network with dual equivariance. On 3D point cloud classification benchmarks in the sparse-point regime, HyQuRP outperforms strong classical and quantum baselines. For example, when six subsampled points are used, HyQuRP ($\sim$1.5K parameters) achieves 76.13% accuracy on the 5-class ModelNet benchmark, compared with 72.54%, 71.09%, and 71.03% for Tensor Field Network, PointNet, and PointMamba with similar parameter counts. These results highlight HyQuRP's strong data efficiency and suggest the potential of equivariant quantum machine learning approaches in symmetry-sensitive tasks.
Quantum neural network architectures that have little to no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks the action of which commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNNs) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNNs) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a S U ( 2 ) -equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity. Published by the American Physical Society 2024
Exploiting the power of quantum computation to realize superior machine learning algorithms has been a major research focus of recent years, but the prospects of quantum machine learning (QML) remain dampened by considerable technical challenges. A particularly significant issue is that generic QML models suffer from so-called barren plateaus in their training landscapes—large regions where cost function gradients vanish exponentially in the number of qubits employed, rendering large models effectively untrainable. A leading strategy for combating this effect is to build problem-specific models that take into account the symmetries of their data in order to focus on a smaller, relevant subset of Hilbert space. In this work, we introduce a family of rotationally equivariant QML models built upon the quantum Fourier transform, and leverage recent insights from the Lie-algebraic study of QML models to prove that (a subset of) our models do not exhibit barren plateaus. In addition to our analytical results we numerically test our rotationally equivariant models on a dataset of simulated scanning tunneling microscope images of phosphorus impurities in silicon, where rotational symmetry naturally arises, and find that they dramatically outperform their generic counterparts in practice. Published by the American Physical Society 2024
Equivariant quantum graph neural networks (EQGNNs) offer a potentially powerful method to process graph data. However, existing EQGNN models only consider the permutation symmetry of graphs, and failing to fully exploit the geometric and non-geometric information in graphs, resulting in suboptimal performance when processing 3D graph data. To address these limitations, we derive constraints of rotation and permutation equivariance, and then propose a novel rotation- and permutation-equivariant quantum graph neural network (RP-EQGNN). An equivariant module is designed to extract the geometric information. Then, a convolution and entanglement module is constructed to extract non-geometric information. To improve performance of our model, an edge entanglement strategy is designed to perform distinguishable entanglement operations based on edge heterogeneity. The experiment results demonstrate that RP-EQGNN is significantly better for graph regression on the QM9 dataset and the OC20 dataset than Q3DGL and EQC in MAE and achieves results comparable to those of EquiformerV2, Geoformer, SO3KRATES and HEGNN. It also has advantage for point cloud classification on the ModelNet40 dataset over quantum models, including sQCNN-3D and PI-QSVM. RP-EQGNN introduces an innovative approach for processing 3D graph data, establishing a basis for future investigations into symmetries within graph neural networks.