Abstract
Quantum walks on graphs have shown prioritized benefits and applications in wide areas. In some scenarios, however, it may be more natural and accurate to mandate high-order relationships for hypergraphs, due to the density of information stored inherently. Therefore, we can explore the potential of quantum walks on hypergraphs. In this paper, by presenting the one-to-one correspondence between regular uniform hypergraphs and bipartite graphs, we construct a model for quantum walks on bipartite graphs of regular uniform hypergraphs with Szegedy’s quantum walks, which gives rise to a quadratic speed-up. Furthermore, we deliver spectral properties of the transition matrix, given that the cardinalities of the two disjoint sets are different in the bipartite graph. Our model provides the foundation for building quantum algorithms on the strength of quantum walks on hypergraphs, such as quantum walks search, quantized Google’s PageRank, and quantum machine learning.
Highlights
As a quantum-mechanical analogs of classical random walks, quantum walks have become increasingly popular in recent years, and have played a fundamental and important role in quantum computing
By analyzing the mathematical formalism of hypergraphs and three existing discrete-time quantum walks[40], we find that discrete-time quantum walks on regular uniform hypergraphs can be transformed into Szegedy’s quantum walks on bipartite graphs that are used
Let V = {v1, v2, ..., vn} and E = {e1, e2, ..., em}. where n = |V| is used to denote the number of vertices in the hypergraph and m = |E| the number of hyperedges
Summary
Quantum walks on graphs have shown prioritized benefits and applications in wide areas. Szegedy[7] proposed a quantum walks model that quantizes the random walks, and its evolution operator is driven by two reflection operators on a bipartite graph. Paying attention to quantum walks on hypergraphs is a natural choice Inspired by these latter developments, we focus on discrete-time quantum walks on regular uniform hypergraphs. We can study Szegedy’s quantum walks on bipartite graphs instead of the corresponding quantum walks on regular uniform hypergraphs. We construct a model for quantum walks on bipartite graphs of regular uniform hypergraphs with Szegedy’s quantum walks. Our work generalizes quantum walks on regular uniform hypergraphs by extending the classical Markov chain, due to Szegedy’s quantum walks. In Sec. Results: Quantum walks on hypergraphs, we construct a method for quantizing Markov chain to create discrete-time quantum walks on regular uniform hypergraphs.
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