N -dimensional alternate coined quantum walks from a dispersion-relation perspective
Eugenio Roldán, Carlo D. Franco, Fernando Silva, Germán J. de Valcárcel · Physical Review A · 2013
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, ${\mathrm{AQW}}_{(N)}$, in contrast with the standard definition of the $N$-dimensional quantum walk, ${\mathrm{QW}}_{(N)}$, requires only a coin qubit. We discuss the quantum diffusion properties of ${\mathrm{AQW}}_{(2)}$ and ${\mathrm{AQW}}_{(3)}$ 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 ${\mathrm{AQW}}_{(3)}$ generates considerable genuine multipartite entanglement. Finally, we discuss the implementability of ${\mathrm{AQW}}_{(N)}$.