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arXiv:1912.08411 · Physical Review Letters

Experimental Parity-Time Symmetric Quantum Walks for Centrality Ranking on Directed Graphs

Tong Wu, Josh Izaac, Zixi Li, Kai Wang, Zhaozhong Chen, Shining N. Zhu, Jingbo B. Wang, Xiao‐Song Ma

Published Dec 8, 2020• 24 citations
arXivHTMLPDF

Abstract

Using quantum walks (QWs) to rank the centrality of nodes in networks, represented by graphs, is advantageous compared to certain widely used classical algorithms. However, it is challenging to implement a directed graph via QW, since it corresponds to a non-Hermitian Hamiltonian and thus cannot be accomplished by conventional QW. Here we report the realizations of centrality rankings of a three-, a four-, and a nine-vertex directed graph with parity-time (PT) symmetric quantum walks by using high-dimensional photonic quantum states, multiple concatenated interferometers, and dimension dependent loss to achieve these. We demonstrate the advantage of the QW approach experimentally by breaking the vertex rank degeneracy in a four-vertex graph. Furthermore, we extend our experiment from single-photon to two-photon Fock states as inputs and realize the centrality ranking of a nine-vertex graph. Our work shows that a PT symmetric multiphoton quantum walk paves the way for realizing advanced algorithms.

DOI
10.1103/physrevlett.125.240501

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PageRank is an algorithm used by Google Search to rank web pages in their search engine results. An important step for quantum networks is to quantize the classical protocol as quantum mechanics provides computational resources that can be used to outperform classical algorithms. In this paper, we experimentally realize continuous-time quantum walks for directed graphs with non-Hermitian adjacency matrices by using linear optical circuits and single photons. We find that the node classical centrality in a directed graph is correlated with the maximum node probability resulting from a continuous-time quantum walk and then demonstrate PageRank. Our work opens up an avenue of applications of quantum information in real-life tasks.
Network centrality has important implications well beyond its role in physical and information transport analysis; as such, various quantum-walk-based algorithms have been proposed for measuring network vertex centrality. In this work, we propose a continuous-time quantum walk algorithm for determining vertex centrality, and show that it generalizes to arbitrary graphs via a statistical analysis of randomly generated scale-free and Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi networks. As a proof of concept, the algorithm is detailed on a four-vertex star graph and physically implemented via linear optics, using spatial and polarization degrees of freedoms of single photons. This paper reports a successful physical demonstration of a quantum centrality algorithm.
Quantum walks are a fundamental tool in the study of quantum information, bridging the gap between quantum computation and modelling complex systems. In this thesis, we present a software package for simulation of multiparticle continuous-time quantum walks (CTQWs) using high performance computation, and detail a method of systematic dimensionalty reduction for fermionic CTQWs. We then consider CTQW applications; proving that the single particle CTQW is unsuited for determining graph isomorphism, and proposing a CTQW-based network centrality algorithm. This algorithm is utilized in the first successful physical quantum centrality implementation. Finally, we extend our centrality algorithm to PT-symmetric directed graphs.
Abstract We experimentally investigate the impact of static disorder and dynamic disorder on the non-unitary dynamics of parity–time (PT)-symmetric quantum walks. Via temporally alternating photon losses in an interferometric network, we realize the passive PT-symmetric quantum dynamics for single photons. Controllable coin operations allow us to simulate different environmental influences, which result in three different behaviors of quantum walkers: a standard ballistic spread, a diffusive behavior, and a localization, respectively, in a PT-symmetric quantum walk architecture.
Abstract In this article, we introduce, test and discuss three quantum methods for computing hub and authority centrality scores in directed networks. The methods are based on unitary, continuous-time quantum walks; the construction of a suitable Hermitian Hamiltonian is achieved by performing a quantum walk on the associated bipartite graph. Two methods, called CQAu and CQAw, use the same evolution operator, inspired by the classical Hyperlink-Induced Topic Search (HITS) algorithm, but with different initial states; the computation of hub and authority scores is performed simultaneously. The third method, called CQG and inspired by classical PageRank, requires instead two separate runs with different evolution operators, one for hub and one for authority scores. The methods are tested on several directed graphs with different sizes and properties; a comparison with other well-established ranking algorithms is provided. CQAw emerges as the most reliable of the three methods and yields rankings that are largely compatible with results from HITS, although CQAu and CQG also present interesting features and potential for applications.