Average position of quantum walks with an arbitrary initial state

We investigated discrete time quantum walks with an arbitrary initial state $\mid\Psi_{0}(\theta,\phi,\varphi)\rangle=\cos\theta e^{i\phi}\mid0L\rangle+\sin\theta e^{i\varphi}\mid0R\rangle$withaU(2)coin$U(\alpha,\beta,\gamma)$\mathrm{.}Wediscoverthattheaverageposition$\bar{x}=\max(\bar{x})\cos(\alpha+\gamma+\phi-\varphi)$,withcoinoperator$U(\alpha,\pi/4,\gamma)$andinitialstate$\mid\Phi_{0}(\pi/4,\phi,\varphi)\rangle=(e^{i\phi}\mid0L\rangle+e^{i\varphi}\mid0R\rangle)\sqrt{2}/2$. If we set initial state and coin operator to $\mid {\mathrm{\Phi }}_0⟩(\theta ,\pi \mathrm{/}2,0)=i\cos \theta \mid 0L⟩+\sin \theta \mid 0R⟩)$ and coin operator $U(0,\pi \mathrm{/}4,0)$, for $\alpha +\gamma +\varphi -\phi =\pi \mathrm{/}2$, we discover that $\bar{x}=-\max (\bar{x})\cos (2\theta )\mathrm{.}$ Last we verify the result above, and obtain the summarize properties of quantum walks with an arbitrary state. We get that $\bar{x}(\theta ,\varphi ,\phi ,\alpha ,\beta ,\gamma ,t)=\cos 2\theta \ast {\bar{x}}_{\mathrm{\mid }0L⟩}(\beta ,t)+\sin 2\theta \ast \cos (\alpha +\gamma +\varphi -\phi )\ast {\bar{x}}_{(\mid 0L⟩+\mid 0R⟩)\sqrt{2}\mathrm{/}2}(\alpha =\gamma =0,\beta ,t)$. If the average positions $\bar{x}$ with initial state $\mathrm{\mid }0L⟩$ and $\mid {\mathrm{\Psi }}_0⟩=(\mid 0L⟩+\mid 0R⟩)\sqrt{2}\mathrm{/}2$ 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.