## Abstract

Google PageRank is a prevalent 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 stochastic walks on Erdös-Rényi graphs using an RTX 2060 GPU. The test on the graph of 6000 nodes requires a memory of 5.5 GB and time of 223 s, and that on the graph of 1000 nodes requires 226 MB and 3.6 s. Compared with QSWalk, a currently prevalent Mathematica solver, our solver for the same graph of 1000 nodes reduces the required memory and time to only 0.2% and 0.05%. We apply the solver to quantum PageRank for the USA major airline network with up to 922 nodes, and to quantum stochastic walk on a glued tree of 2186 nodes. This efficient solver for large-scale quantum PageRank and quantum stochastic walks would greatly facilitate studies of quantum information in real-life applications.

## Authors

Hao Tang*

Ruoxi Shi***Tian-Shen He***

Yan-Yan Zhu

Tian-Yu Wang

Marcus Lee

Xian-Min Jin

*Equal contribution

## Paper

The paper is available at https://www.sciencedirect.com/science/article/pii/S2095927320305880.