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Google PageRank is a prevalent and useful algorithm for ranking the significance of nodes or websites in a network, and a recent quantum counterpart for PageRank algorithm has been raised to suggest a higher accuracy of ranking comparing to Google PageRank. The quantum PageRank algorithm is essentially based on quantum stochastic walks and can be expressed using Lindblad master equation, which, however, needs to solve the Kronecker products of an O(N 4 ) dimension and requires severely large memory and time when the number of nodes N in a network increases above 150. Here, we present an efficient solver for quantum PageRank by using the Runge-Kutta method to reduce the matrix dimension to O(N 2 ) and employing TensorFlow to conduct GPU parallel computing. We demonstrate its performance in solving quantum PageRank for the USA major airline network with up to 922 nodes. Compared with the previous quantum PageRank solver, our solver dramatically reduces the required memory and time to only 1% and 0.2%, respectively, making it practical to work in a normal computer with a memory of 4-8 GB in no more than 100 seconds. This efficient solver for large-scale quantum PageRank and quantum stochastic walks would greatly facilitate studies of quantum information in real-life applications.

In recent years, point clouds have earned quite some research interest by the development of depth sensors. Due to different layouts of objects, orientation of point clouds is often unknown in real applications. In this paper, we propose a new point-set learning framework named Pointwise Rotation-Invariant Network (PRIN), focusing on achieving rotation-invariance in point clouds. We construct spherical signals by Density-Aware Adaptive Sampling (DAAS) from sparse points and employ Spherical Voxel Convolution (SVC) to extract rotation-invariant features for each point. Our network can be applied to applications ranging from object classification, part segmentation, to 3D feature matching and label alignment. PRIN shows performance better than state-of-the-art methods on part segmentation without data augmentation. We provide theoretical analysis for what our network has learned and why it is robust to input orientation. Our code is available online 1 .

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