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

• URL: http://arxiv.org/abs/2103.05235v1
• Date: Tue, 9 Mar 2021 05:45:25 GMT
• Title: A new type of spectral mapping theorem for quantum walks with a moving shift on graphs
• Authors: Sho Kubota, Kei Saito, Yusuke Yoshie
• Abstract summary: The spectral mapping theorem for quantum walks can only be applied for walks employing a shift operator whose square is the identity. We acquire a new spectral mapping theorem for the Grover walk with a shift operator whose cube is the identity on finite graphs.
• Score: 0.2578242050187029
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $U$.

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