Average position of quantum walks with an arbitrary initial state

Abstract

We investigated discrete time quantum walks with an arbitrary initial state $\mid {\Psi }_0(\theta ,\varphi ,\phi )⟩=\cos \theta e^{i\varphi }\mid 0L⟩+\sin \theta e^{i\phi }\mid 0R⟩$ with a U(2) coin $U(\alpha ,\beta ,\gamma )$. We discover that the average position $\bar{x}=\max (\bar{x})\cos (\alpha +\gamma +\varphi -\phi )$, with coin operator $U(\alpha ,\pi /4,\gamma )$ and initial state $\mid {\Phi }_0(\pi /4,\varphi ,\phi )⟩=(e^{i\varphi }\mid 0L⟩+e^{i\phi }\mid 0R⟩)\sqrt{2}/2$. If we set initial state and coin operator to $\mid {\Phi }_0⟩(\theta ,\pi /2,0)=i\cos \theta \mid 0L⟩+\sin \theta \mid 0R⟩)$ and coin operator $U(0,\pi /4,0)$, for $\alpha +\gamma +\varphi -\phi =\pi /2$, we discover that $\bar{x}=-\max (\bar{x})\cos (2\theta ).$ Last we verify the result above, and obtain the summarize properties of quantum walks with an arbitrary state. We get that $\bar{x}(\theta ,\varphi ,\phi ,\alpha ,\beta ,\gamma ,t)=\cos 2\theta \ast {\bar{x}}_{|0L⟩}(\beta ,t)+\sin 2\theta \ast \cos (\alpha +\gamma +\varphi -\phi )\ast {\bar{x}}_{(\mid 0L⟩+\mid 0R⟩)\sqrt{2}/2}(\alpha =\gamma =0,\beta ,t)$. If the average positions $\bar{x}$ with initial state $|0L⟩$ and $\mid {\Psi }_0⟩=(\mid 0L⟩+\mid 0R⟩)\sqrt{2}/2$ and coin operator $U(0,\beta ,0)$ are known, we can get the average position result of quantum walks with an arbitrary initial state and a U(2) coin operator.