$N$-dimensional alternate coined quantum walks from a dispersion-relation perspective

We suggest an alternative definition of $N$-dimensional coined quantum walk by generalizing a recent proposal [Di Franco et al., Phys. Rev. Lett. 106, 080502 (2011)]. This $N$-dimensional alternate quantum walk, ${\mathrm{AQW}}_{\left(N\right)}$, in contrast with the standard definition of the $N$-dimensional quantum walk, ${\mathrm{QW}}_{\left(N\right)}$, requires only a coin qubit. We discuss the quantum diffusion properties of ${\mathrm{AQW}}_{\left(2\right)}$ and ${\mathrm{AQW}}_{\left(3\right)}$ by analyzing their dispersion relations that reveal, in particular, the existence of diabolical points. This allows us to highlight interesting similarities with other well-known physical phenomena. We also demonstrate that ${\mathrm{AQW}}_{\left(3\right)}$ generates considerable genuine multipartite entanglement. Finally, we discuss the implementability of ${\mathrm{AQW}}_{\left(N\right)}$.

Figure 1

3D views of the two branches of Eq. (1), for ${\theta }_1={\theta }_2=\pi /4$ (a) and ${\theta }_1=\pi /4\ne {\theta }_2=\pi /3$ (b). Propagation in ${\mathrm{AQW}}_{\left(2\right)}$ after 90 time steps of an initial state with a Gaussian probability distribution of width ${\sigma }_{HWHM}=7$ and coin state $\mathrm{col}(1/\sqrt{2},i/\sqrt{2})$ equal for all populated sites, for ${\theta }_1={\theta }_2=\pi /4$ (c) and ${\theta }_1=\pi /4\ne {\theta }_2=\pi /3$ (d).

Figure 2

Propagation in ${\mathrm{AQW}}_{\left(3\right)}$ after 90 time steps of (a,b) a localized initial state and (c,d) a spatially extended initial state with Gaussian probability distribution of width ${\sigma }_{HWHM}=7$, whose pseudomomentum is centered at one of the DPs [$(u_{\mathrm{DP}},v_{\mathrm{DP}},w_{\mathrm{DP}})=(\pi /4,\pi /4,-3\pi /4)$]. In both cases ${\theta }_1={\theta }_2={\theta }_3=\pi /4$ and the initial coin state is $\mathrm{col}(0,1)$. Panel (e) shows the tripartite negativity $N^{\left(3\right)}$ against the number of time steps $t$ in AQW${}_{\left(3\right)}$, with the walker starting at the origin and initial coin state $\mathrm{col}(1,i)/\sqrt{2}$.