Quantum walk for SU(1,1)

• URL: http://arxiv.org/abs/2207.04511v1
• Date: Sun, 10 Jul 2022 18:07:39 GMT
• Title: Quantum walk for SU(1,1)
• Abstract summary: We propose a scheme to implement the quantum walk for SU (1,1) in the phase space. The movement of the walker described by the SU (1,1) coherent states can be visualized on the hyperboloid or the Poincar'e disk.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose a scheme to implement the quantum walk for SU(1,1) in the phase space, which generalizes those associated with the Heisenberg-Weyl group. The movement of the walker described by the SU(1,1) coherent states can be visualized on the hyperboloid or the Poincar\'{e} disk. In both one-mode and two-mode realizations, we introduce the corresponding coin-flip and conditional-shift operators for the SU(1,1) group, whose relations with those for Heisenberg-Weyl group are analyzed. The nonorthogonality of the SU(1,1) coherent states precludes the quantum walk for SU(1,1) from the idealized one. However, the overlap between different SU(1,1) coherent states can be reduced by increasing the Bargmann index $k$, which indicates that the two-mode realization provides more possibilities to simulate the idealized quantum walk.

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