Adjacent Vertices Can Be Hard to Find by Quantum Walks

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• 关键词：Quantum walks ; Stationary states ; Multiple marked vertices ; Quantum search ; Exceptional configurations ; General graphs ; Hypercube ; Two ; dimensional grid
• 刊名：Lecture Notes in Computer Science
• 出版年：2017
• 出版时间：2017
• 年：2017
• 卷：10139
• 期：1
• 页码：256-267
• 丛书名：SOFSEM 2017: Theory and Practice of Computer Science
• ISBN：978-3-319-51963-0
• 卷排序：10139

文摘

Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element only. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs \(\varOmega (N)\) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph.