Quantum centrality testing on directed graphs via PT-symmetric quantum walks

TL;DR

This paper introduces a PT-symmetric quantum walk approach to analyze centrality in directed graphs, enabling probability conservation and reducing computational complexity, with advantages over classical methods in certain cases.

Contribution

It develops a novel PT-symmetric quantum walk framework for directed graphs, extending quantum centrality measures with lower Hilbert space requirements and improved performance.

Findings

The method conserves probability in directed graph quantum walks.

It reduces the Hilbert space needed compared to previous models.

It can outperform classical algorithms like PageRank in specific scenarios.

Abstract

Various quantum-walk based algorithms have been proposed to analyse and rank the centrality of graph vertices. However, issues arise when working with directed graphs --- the resulting non-Hermitian Hamiltonian leads to non-unitary dynamics, and the total probability of the quantum walker is no longer conserved. In this paper, we discuss a method for simulating directed graphs using PT-symmetric quantum walks, allowing probability conserving non-unitary evolution. This method is equivalent to mapping the directed graph to an undirected, yet weighted, complete graph over the same vertex set, and can be extended to cover interdependent networks of directed graphs. Previous work has shown centrality measures based on the CTQW provide an eigenvector-like quantum centrality; using the PT-symmetric framework, we extend these centrality algorithms to directed graphs with a significantly reduced Hilbert space compared to previous proposals. In certain cases, this centrality measure provides an advantage over classical algorithms used in network analysis, for example by breaking vertex rank degeneracy. Finally, we perform a statistical analysis over ensembles of random graphs, and show strong agreement with the classical PageRank measure on directed acyclic graphs.