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.

As mobile, handheld computing devices become more common and are used for an ever-increasing variety of tasks, new mechanisms for data entry must be investigated. Personal digital assistants often provide a small stylus-activated soft keyboard, as do some mobile phones that include touch screens. However, there is little data regarding the importance of keyboard size or the users' tasks, the effectiveness of these keyboards, or user reactions to these keyboards. In this article, an experiment designed to investigate these issues in the context of a palm-style QWERTY keyboard is described. In this study, 30 novices completed 6 realistic tasks using either a small, medium, or large soft keyboard. The results not only confirm that keyboard size does not affect data entry rates but that making the keyboard smaller does not increase error rates or negatively impact preference ratings. However, tasks that required users to switch between the alphabetic keyboard and the numeric keyboard do result in significantly slower data entry rates. A model that accurately predicts the time required to enter predefined text is presented, and directions for future research are discussed.