The quantum SearchRank algorithm is a promising tool for a future quantum search engine based on PageRank quantization. However, this algorithm loses its functionality when the $N/M$ ratio between the network size $N$ and the number of marked nodes $M$ is sufficiently large. We propose a modification of the algorithm, replacing the underlying Szegedy quantum walk with a semiclassical walk. To maintain the same time complexity as the quantum SearchRank algorithm we propose a simplification of the algorithm. This new algorithm is called Randomized SearchRank, since it corresponds to a quantum walk over a randomized mixed state. The performance of the SearchRank algorithms is first analyzed on an example network, and then statistically on a set of different networks of increasing size and different number of marked nodes. On the one hand, to test the search ability of the algorithms, it is computed how the probability of measuring the marked nodes decreases with $N/M$ for the quantum SearchRank, but remarkably it remains at a high value around $0.9$ for our semiclassical algorithms, solving the quantum SearchRank problem. The time complexity of the algorithms is also analyzed, obtaining a quadratic speedup with respect to the classical ones. On the other hand, the ranking functionality of the algorithms has been investigated, obtaining a good agreement with the classical PageRank distribution. Finally, the dependence of these algorithms on the intrinsic PageRank damping parameter has been clarified. Our results suggest that this parameter should be below a threshold so that the execution time does not increase drastically.

翻译：量子搜索排名算法是一种基于PageRank量子化的未来量子搜索引擎的有前景工具。然而，当网络规模$N$与标记节点数$M$的比值$N/M$足够大时，该算法会丧失其功能。我们提出对该算法进行改进，将底层的Szegedy量子游走替换为半经典游走。为保持与量子搜索排名算法相同的时间复杂度，我们提出了一种算法简化方案。这种新算法被称为随机化搜索排名算法，因为它对应于在随机混合态上的量子游走。首先在示例网络上分析搜索排名算法的性能，然后在一组规模递增且标记节点数不同的网络上进行统计分析。一方面，为测试算法的搜索能力，我们计算了量子搜索排名算法中测量到标记节点的概率随$N/M$增大而衰减的程度，但值得注意的是，我们的半经典算法中该概率仍保持约$0.9$的高值，从而解决了量子搜索排名问题。同时分析了算法的时间复杂度，获得了相对于经典算法的二次加速。另一方面，我们研究了算法的排序功能，结果表明其与经典PageRank分布具有良好一致性。最后，阐明了这些算法对PageRank固有阻尼参数的依赖性。我们的结果表明，该参数应低于某个阈值，以避免执行时间急剧增加。