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In this paper, we present a highly accurate and effective theoretical model to study electron transport and interference in quantum cavities with arbitrarily complex boundaries. Based on this model, a variety of quantum effects can be studied and quantified. In particular, this model provides information on the transient state of the system under study, which is important for analyzing nanometer-scale electronic devices such as high-speed quantum transistors and quantum switches. ͓S0163-1829͑99͒02739-3͔

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 highdimensional 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.

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