2003/11/19 by Norio Inui, Yoshinao Konishi, Norio Konno · 7 citations
Computer Science · Mathematics · Physics and Astronomy · #Eigenvalues and eigenvectors #Lattice (music) #Mathematics #Operator (biology) #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum algorithm #Quantum and electron transport phenomena #Quantum mechanics #Quantum walk #Quantum-Dot Cellular Automata #Random walk #Statistical physics #Statistics #quant-ph
paper · pdf · doi:10.1103/physreva.69.052323
21 pages RevTeX, 4 figures eps
arxiv created 2003/11/19 · openalex publication_date 2004/05/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The Grover walk, which is related to Grover's search algorithm on a quantum computer, is one of the typical discrete time quantum walks. However, a localization of the two-dimensional Grover walk starting from a fixed point is strikingly different from other types of quantum walks. The present paper explains the reason why the walker who moves according to the degree-four Grover operator can remain at the starting point with a high probability. It is shown that the key factor for the localization is due to the degeneration of eigenvalues of the time evolution operator. In fact, the global time evolution of the quantum walk on a large lattice is mainly determined by the degree of degeneration. The dependence of the localization on the initial state is also considered by calculating the wave function analytically.