Quantum walk for SU(1,1)

TL;DR

This paper introduces a scheme to implement quantum walks in phase space using SU(1,1) coherent states, extending the concept beyond the traditional Heisenberg-Weyl group, and analyzes their properties and potential for simulation.

Contribution

The paper develops a novel quantum walk framework based on SU(1,1) group in phase space, including new operators and analysis of state overlaps, expanding quantum walk models.

Findings

Quantum walk for SU(1,1) visualized on hyperboloid and Poincaré disk.

Overlap between SU(1,1) coherent states decreases with increasing Bargmann index k.

Two-mode realization offers better simulation prospects for idealized quantum walks.

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 probability distribution, standard deviation and the von Neumann entropy are employed to describe the walking process. 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.