Generalized Quantum PageRank Algorithm with Arbitrary Phase Rotations

TL;DR

This paper introduces a modified quantum PageRank algorithm with arbitrary phase rotations, analyzing its behavior on small and complex networks, and demonstrating improvements in node differentiation and stability over classical and previous quantum methods.

Contribution

The paper presents a novel quantum PageRank modification using arbitrary phase rotations, enhancing node discrimination and stability in complex networks.

Findings

Decreasing phase reduces standard deviation of PageRank, improving node distinction.

Optimal phase choice balances convergence time and discrimination ability.

Some APR schemes restore residual node degeneracy and better identify true secondary hubs.

Abstract

The quantization of the PageRank algorithm is a promising tool for a future quantum internet. Here we present a modification of the quantum PageRank introducing arbitrary phase rotations (APR) in the underlying Szegedy's quantum walk. We define three different APR schemes with only one phase as a degree of freedom. We have analyzed the behavior of these algorithms in a small generic graph, observing that a decrease of the phase reduces the standard deviation of the instantaneous PageRank, so the nodes of the network can be distinguished better. However, the algorithm takes more time to converge, so the phase can not be decreased arbitrarily. With these results we choose a concrete value for the phase to later apply the algorithm to complex scale-free graphs. In these networks, the original quantum PageRank is able to break the degeneracy of the residual nodes and detect secondary hubs that the classical algorithm suppresses. Nevertheless, not all of the detected secondary hubs are real according to the PageRank's definition. Some APR schemes can overcome this problem, restoring the degeneration of the residual nodes and highlighting the truly secondary hubs of the networks. Finally, we have studied the stability of the new algorithms. The original quantum algorithm was known to be more stable than the classical. We have found that one of our algorithms, whose PageRank distribution resembles the classical one, has a stability similar to the original quantum algorithm.