choice, rogue events are scarce, except under weak disorder. A monotonic threshold is observed between rare and high-probability rogue wave regimes, depending on the quantum coin. We provide a comprehensive analysis of the coin-disorder interplay in regard to rogue wave events. Our findings shed light on the possible control of extreme quantum amplitudes through quantum coins in disordered DTQWs.

DOI: 10.1103/physreva.108.062206

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Z

choice, regardless of the disorder degree. Conversely, for coins near the Pauli-

X

\overline{X}(t)

X ¯ ( t ) , under various initial conditions of the walker and quantum coin operators. The results indicate a significant dependence of the observed phenomena on the choice of initial state, enabling the selective induction of the quantum boomerang effect in both or only one portion of the wavepacket associated with specific internal states. By varying the quantum coin parameter

\theta

θ , we find that the maximum mean position follows a power-law decay near the Pauli-Z coin, characterized by

X_{\text {Max}} \sim \theta ^{-2}

X Max ∼ θ - 2 . Additionally, we identify a scaling behavior

X_{\text {Max}} \sim W^{-2}

X Max ∼ W - 2 , which is consistent with the localization length observed in disordered quantum systems.