Quantum walk has emerged as a promising framework for developing innovative quantum algorithms. The efficiency of such algorithms hinges on the interference between multiple paths simultaneously traversed by a quantum walker, as well as the inherent interaction and intrinsic entanglement when multiple quantum walkers are involved. As such, quantum walk has become a subject of intense theoretical and experimental studies. A critical challenge in the field is to establish the advantages of algorithms based on quantum walks in comparison to classical computation. To achieve this goal, an efficient decomposition of the prescribed quantum walk operators is essential. In this presentation, I will discuss the overarching design principles used to create efficient quantum circuits tailored to different types of quantum walks across a wide range of undirected and directed graphs. My aim is to provide some insights into the derivation of such decompositions. If time permits, I will also discuss potential applications with practical significance.
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