Localizationlike effect in two-dimensional alternate quantum walks with periodic coin operations

Exploiting multidimensional quantum walks as feasible platforms for quantum computation and quantum simulation attracts constantly growing attention from a broad experimental physics community. Here, we propose a two-dimensional quantum walk scheme with a single-qubit coin that presents, in the considered regimes, a strong localizationlike effect on the walker. The result could provide new possible directions for the implementation of quantum algorithms or from the point of view of quantum simulation. We characterize the localizationlike effect in terms of the parameters of a step-dependent qubit operation that acts on the coin space after any standard coin operation, showing that a proper choice can guarantee a nonnegligible probability of finding the walker in the origin even for large times. We finally discuss the robustness to imperfections, a qualitative relation with coherences behavior, and possible experimental realizations of this model with the current state-of-the-art settings.

Figure 1

(a) Probability to find the walker in the origin, against the total number of time steps $t$, for values of $\varphi$ equal to $\pi /6$ (black line), $\pi /4$ (red dashed line), $\pi /3$ (blue dotted line). Only the even time steps are shown, as the probability is zero for all odd steps. (b) Average probability to find the walker in the origin, against the total number of time steps $t$, for values of $\varphi$ equal to $\pi /6$ (black line), $\pi /4$ (red dashed line), $\pi /3$ (blue dotted line).

Figure 2

(a) Average probability to find the walker in the origin, for a total number of time steps $t=40$, against the values of ${\varphi }_x$ and ${\varphi }_y$. (b) Average probability to find the walker in the origin, for a total number of time steps $t=100$ and ${\varphi }_y=0$, against the values of ${\varphi }_x$.

Figure 3

(a) Average probability to find the walker in the origin, against the total number of time steps $t$, for ${\varphi }_x=19\pi /25$ and ${\varphi }_y=0$. (b) Average probability to find the walker in the origin, against the total number of time steps $t$, for a value of ${\varphi }_x=(1+\delta ){\varphi }_0,{\varphi }_0=19\pi /25,\delta$ sampled at each time step from a uniform distribution in the range $[-{10}^{-2},{10}^{-2}]$, and ${\varphi }_y=0$. The black line corresponds to time-dependent disorder, the red dashed line corresponds to position-dependent disorder, the blue dotted line corresponds to disorder depending both on position and time.

Figure 4

Coherence norm against the total number of time steps $t$, for ${\varphi }_x=19\pi /25,{\varphi }_y=0$ (black line), ${\varphi }_x=\pi /4,{\varphi }_y=0$ (red dashed line), and ${\varphi }_x=0,{\varphi }_y=0$ (blue dotted line).