Average position of quantum walks with an arbitrary initial state

We investigated discrete time quantum walks with an arbitrary initial state $\mid\Psi_{0}(\theta,\phi,\varphi)\rangle=\cos\theta e^{i\phi}\mid0L\rangle+\sin\theta e^{i\varphi}\mid0R\rangle$ with a U(2) coin $U(\alpha,\beta,\gamma)$. We discover that the average position $\bar{x}=\max(\bar{x})\cos(\alpha+\gamma+\phi-\varphi)$, with coin operator $U(\alpha,\pi/4,\gamma)$ and initial state $\mid\Phi_{0}(\pi/4,\phi,\varphi)\rangle=(e^{i\phi}\mid0L\rangle+e^{i\varphi}\mid0R\rangle)\sqrt{2}/2$. If we set initial state and coin operator to $\mid\Phi_{0}\rangle(\theta,\pi/2,0)=i\cos\theta\mid0L\rangle+\sin\theta\mid0R\rangle)$ and coin operator $U(0,\pi/4,0)$, for $\alpha+\gamma+\phi-\varphi=\pi/2$, 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 $\bar{x}(\theta,\phi,\varphi,\alpha,\beta,\gamma,t)=\cos2\theta*\bar{x}_{|0L\rangle}(\beta,t)+\sin2\theta*\cos(\alpha+\gamma+\phi-\varphi)*\bar{x}_{(\mid0L\rangle+\mid0R\rangle)\sqrt{2}/2}(\alpha=\gamma=0,\beta,t)$. If the average positions $\bar{x}$ with initial state $|0L\rangle$ and $\mid\Psi_{0}\rangle=(\mid0L\rangle+\mid0R\rangle)\sqrt{2}/2$ 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.