# Discrete-time quantum walk algorithm for ranking nodes on a network

## TL;DR

This paper introduces a quantum algorithm based on discrete-time quantum walks for ranking nodes in directed networks, potentially applicable to the internet and quantum networks, highlighting quantum interference effects.

## Contribution

It presents a novel quantum PageRank algorithm applicable to all directed networks, demonstrating its potential for practical quantum network analysis.

## Key findings

- Quantum rank hierarchy aligns with classical for trees
- Quantum hierarchy differs for cyclic networks due to interference
- Algorithm is implementable on current quantum systems

## Abstract

We present a quantum algorithm for ranking the nodes on a network in their order of importance. The algorithm is based on a directed discrete-time quantum walk, and works on all directed networks. This algorithm can theoretically be applied to the entire internet, and thus can function as a quantum PageRank algorithm. Our analysis shows that the hierarchy of quantum rank matches well with the hierarchy of classical rank for directed tree network and for non-trivial cyclic networks, the hierarchy of quantum ranks do not exactly match to the hierarchy of the classical rank. This highlights the role of quantum interference and fluctuations in networks and the importance of using quantum algorithms to rank nodes in quantum networks. Another application this algorithm can envision is to model the dynamics on networks mimicking the chemical complexes and rank active centers in order of reactivities. Since discrete-time quantum walks are implementable on current quantum processing systems, this algorithm will also be of practical relevance in analysis of quantum architecture.