Using quantum walks (QW) to rank the centrality of nodes in networks, represented by graphs, is advantageous comparing 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 graphs 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 PT symmetric multiphoton quantum walk paves the way for realizing advanced algorithms.

中文翻译：

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在有向图上进行中心性排名的实验奇偶时间对称量子游走

与某些广泛使用的经典算法相比，使用量子行走（QW）对网络中节点的中心性（由图形表示）进行排序是有利的。但是，通过QW实现有向图非常困难，因为它对应于非Hermitian哈密顿量，因此无法通过常规QW来实现。在这里，我们报告具有奇偶时间（PT）对称量子游走的三，四和九顶点有向图的中心排名的实现。通过使用高维光子量子态，多个级联干涉仪和与尺寸有关的损耗来实现这些。我们通过打破四顶点图中的顶点秩简并性，通过实验证明了QW方法的优势。{此外，我们将实验从单光子Fock状态扩展为两光子Fock状态，