基于PT-对称量子游走的有向图的量子中心性检验
Quantum centrality testing on directed graphs via PT-symmetric quantum walks
摘要：
已经提出了各种基于量子行走的算法来分析和排序图顶点的中心性。然而，当使用有向图时会出现问题-由此产生的非厄米哈密顿量导致非幺正动力学，并且量子步行者的总概率不再守恒。在本文中，我们讨论了一种方法来模拟有向图使用PT-对称量子行走，允许概率守恒的非幺正演化。这种方法相当于将有向图映射到同一顶点集上的无向但加权的完整图，并且可以扩展到覆盖有向图的相互依赖网络。以前的工作已经表明，基于CTQW的中心性措施提供了一个本征向量的量子中心性;使用PT对称框架，我们将这些中心性算法扩展到有向图，与以前的建议相比，Hilbert空间显着减少。在某些情况下，这种中心性度量提供了优于网络分析中使用的经典算法的优势，例如通过打破顶点秩退化。最后，我们对随机图的集合进行了统计分析，并与经典的有向无环图的PageRank度量表现出很强的一致性。
英文摘要：
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.
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