2015/06/22 by Yusuke Higuchi, Etsuo Segawa, Higuchi, Yusuke +3 · 11 citations
Computer Science · Mathematics · Physics and Astronomy · #Combinatorics #Discrete mathematics #FOS: Mathematics #FOS: Physical sciences #Hilbert space #Lattice (music) #Mathematical Physics (math-ph) #Mathematical physics #Mathematics #Operator (biology) #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum computer #Quantum mechanics #Quantum walk #Spectral Theory (math.SP) #Unitary operator #Unitary state #math-ph #math.MP #math.SP
paper · pdf · doi:10.48550/arxiv.1506.06457
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2015/06/22 · arxiv created 2016/06/01 · arxiv updated 2016/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Given two Hilbert spaces, H and K, we introduce an abstract unitary operator U on H and its discriminant T on K induced by a coisometry from H to K and a unitary involution on H. In a particular case, these operators U and T become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between U and T via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.