A new type of spectral mapping theorem for quantum walks with a moving shift on graphs

• Main Authors: Kubota, S. (Author), Saito, K. (Author), Yoshie, Y. (Author)
• Format: Article
• Language: English
• Published: Springer 2022
• Subjects: Adjoint matrix; Eigen-value; Eigenvalues and eigenfunctions; Graph theory; Mapping theorem; Photomapping; Quantum walk; Shift operators; Spectral graph theory; Spectral mapping theorem; Spectral mappings; Time evolutions; Unit circles
• Online Access: View Fulltext in Publisher

 The conventional spectral mapping theorem for quantum walks can only be applied for walks employing a shift operator whose square is the identity. This theorem gives most of the eigenvalues of the time evolution U by lifting the eigenvalues of an induced self-adjoint matrix T onto the unit circle on the complex plane. We acquire a new spectral mapping theorem for the Grover walk with a shift operator whose cube is the identity on finite graphs. Moreover, graphs we can consider for a quantum walk with such a shift operator is characterized by a triangulation. We call these graphs triangulable graphs in this paper. One of the differences between our spectral mapping theorem and the conventional one is that lifting the eigenvalues of T- 1 / 2 onto the unit circle gives most of the eigenvalues of U. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.