Alternate two-dimensional quantum walk with a single-qubit coin

We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the nonlocalized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of $x$-$y$ spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.

Figure 1

Sketch of the alternate quantum walk: the walker is allowed to move alternately in two orthogonal directions of a two-dimensional lattice; a coin operation is performed before each movement.

Figure 2

Spatial probability distribution after $t=50$ time steps of the two-dimensional Grover walk with the initial state of the coin as in Eq. (5). Only the sites with even $x$ and $y$ are shown, as the probability is zero for all odd sites.

Figure 3

Spatial probability distribution after $t=50$ time steps of the alternate quantum walk with the initial state of the coin as in Eq. (25).

Figure 4

Spatial probability distribution after $t=50$ time steps of the alternate quantum walk with the initial state of the coin as in Eq. (26).

Figure 5

Entanglement between $x$ and $y$ position of the walker in the alternate quantum walk, with the initial conditions of the coin as in Eq. (24), $\phi =0$, and after a number of steps $t=10$.

Figure 6

Entanglement between $x$ and $y$ position of the walker in the alternate quantum walk, with the initial conditions of the coin as in Eq. (24), $\phi =\pi /8$, and after a number of steps $t=10$.

Figure 7

Entanglement between $x$ and $y$ position of the walker in the alternate quantum walk, with the initial conditions of the coin as in Eq. (24), $\phi =\pi /2$, and after a number of steps $t=10$.

Figure 8

Entanglement between $x$ and $y$ position of the walker in the alternate quantum walk against both $\theta$ and $\phi$, with the initial conditions of the coin as in Eq. (24) and after a number of steps $t=10$.

Figure 9

Limit density function $f(x,y)$ for the alternate quantum walk with $\gamma =\pi /4$ and $|{\psi }_0(0,0)\rangle ={}^T[1/\sqrt{2},i/\sqrt{2}]$.