Quantum walk on a cylinder

We consider the two-dimensional alternate quantum walk on a cylinder. We concentrate on the study of the motion along the open dimension, in the spirit of looking at the closed coordinate as a small or “hidden” extra dimension. If one starts from localized initial conditions on the lattice, the dynamics of the quantum walk that is obtained after tracing out the small dimension shows the contribution of several components which can be understood from the study of the dispersion relations for this problem. In fact, these components originate from the contribution of the possible values of the quasimomentum in the closed dimension. In the continuous space-time limit, the different components manifest as a set of Dirac equations, with each quasimomentum providing the value of the corresponding mass. We briefly discuss the possible link of these ideas to the simulation of high-energy physical theories that include extra dimensions. Finally, entanglement between the coin and spatial degrees of freedom is studied, showing that the entanglement entropy clearly overcomes the value reached with only one spatial dimension.

Figure 1

Dispersion relation of the AQW in 2D for the conventional case (a) and for the cylinder with $Q=3$ nodes around cyclic dimension (b). Both of them correspond to the Hadamard case ${\theta }_x={\theta }_y=\pi /4$.

Figure 2

Dispersion relations of the AQW on the cylinder for cases $Q=4,5$, and 6, plotted along the quasimomentum of the open dimension $x$. The thick line (orange and black) contributions are always present, but the orange one is present only for even cases $Q=4$ and 6. Thin solid lines correspond to $Q=6$, and the dot-dashed ones to $Q=5$. The dashed horizontal line appears only for $Q=4$.

Figure 3

Marginal probability distribution, as defined by Eq. (7) for an AQW on the cylinder in the Hadamard case after $j=100$ steps, starting from a localized initial condition $\left|\psi \left(0\right)\right.〉=\frac{1}{\sqrt{2}}\left(1,i\right)$ at the lattice origin $(0,0)$. Different values of $Q=4,5$, and 6 are represented, from top to bottom.

Figure 4

Plots of the entanglement entropy for various values of the number $Q$ of nodes on the closed dimension. From left to right and top to bottom: $Q=1,5,6,7$, respectively.