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.

Google PageRank 是一种流行的算法，用于对网络中节点或网站的重要性进行排名，并且最近提出了 PageRank 算法的量子对应物，表明与 Google PageRank 相比，排名具有更高的准确性。量子 PageRank 算法本质上是基于量子随机游走，可以用 Lindblad 主方程表示，但需要求解一个 O(${否}^{\mathrm{4个}}$) 维，当网络中的节点数N增加到 150 以上时，需要非常大的内存和时间。在这里，我们通过使用 Runge-Kutta 方法将矩阵维数减少到 O(${否}^{\mathrm{2个}}$) 并采用 TensorFlow 进行 GPU 并行计算。我们展示了它在使用 RTX 2060 GPU 解决 Erdös-Rényi 图上的量子随机游走时的性能。在6000个节点的图上测试需要5.5GB的内存和223s，在1000个节点的图上测试需要226MB和3.6s。与当前流行的 Mathematica 求解器QSWalk相比，我们对同一 1000 个节点图的求解器将所需的内存和时间减少到仅 0.2% 和 0.05%。我们将求解器应用于美国主要航空公司网络的量子 PageRank（最多 922 个节点），以及在 2186 个节点的胶合树上的量子随机游走。这种用于大规模量子 PageRank 和量子随机游走的高效求解器将极大地促进现实应用中的量子信息研究。