Elephant quantum walk

We introduce an analytically treatable discrete time quantum walk in a one-dimensional lattice which combines non-Markovianity and hyperballistic diffusion associated with a Gaussian whose variance ${\sigma }_t^2$ grows cubicly with time $\sigma \propto t^3$. These properties have have been numerically found in several systems, namely, tight-binding lattice models. For its rules, our model can be understood as the quantum version of the classical non-Markovian “elephant random walk” process for which the quantum coin operator only changes the value of the diffusion constant although, contrarily, to the classical coin.

Figure 1

The panels on the left are for the elephant quantum walk, and the panel on the right is for the quantum walk for $\theta =\frac{\pi }{4}$. (Top) Time evolution of the probability distribution, whose cones are superlinear and linear, respectively. The initial state is localized at the origin with a coin state $|s\rangle =\frac{1}{\sqrt{2}}{(1,1)}^T$. (Bottom) Probability distribution at different time steps.

Figure 2

Left: The squares (blue) represent the trace distance $D\left(t\right)$ vs the number of time steps between two initial states with ${\gamma }_A=0$ and ${\gamma }_B=\pi ,\varphi =0$ for an initial Gaussian packet of width $\delta =0.001$. In case of noise, the trace distance decreases rapidly after the second time step (circles, red). The inset shows the velocity of the trace distance. Right: Diffusion coefficient—and in the inset the exponent $\eta =\alpha /2$—vs $\theta$ for the same initial condition.

Figure 3

Top: Depiction of the elephant random walk model. Bottom: Depiction of the elephant quantum walk model.